
==== Front
Arch Ration Mech Anal
Arch Ration Mech Anal
Archive for Rational Mechanics and Analysis
0003-9527
1432-0673
Springer Berlin Heidelberg Berlin/Heidelberg

1984
10.1007/s00205-024-01984-y
Article
Regularity of the Optimal Sets for a Class of Integral Shape Functionals
Buttazzo Giuseppe 1
Maiale Francesco Paolo 2
Mazzoleni Dario 3
Tortone Giorgio 1
http://orcid.org/0000-0003-4968-3087
Velichkov Bozhidar bozhidar.velichkov@unipi.it

1
1 https://ror.org/03ad39j10 grid.5395.a 0000 0004 1757 3729 Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5, 56127 Pisa, Italy
2 https://ror.org/043qcb444 grid.466750.6 0000 0004 6005 2566 Gran Sasso Science Institute, Viale F. Crispi 7, 67100 L’Aquila, Italy
3 https://ror.org/00s6t1f81 grid.8982.b 0000 0004 1762 5736 Dipartimento di Matematica, University of Pavia, Via Ferrata 5, 27100 Pavia, Italy
Communicated by A. Braides.

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© The Author(s) 2024
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We prove the first regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain Ω is obtained as the integral of a cost function j(u, x) depending on the solution u of a certain PDE problem on Ω. The main feature of these functionals is that the minimality of a domain Ω cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions j(u,x)=-g(x)u+Q(x), where u is the solution of the PDE -Δu=f with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal u from the inwards/outwards optimality of Ω and then we use the stability of Ω with respect to variations with smooth vector fields in order to study the blow-up limits of the state function u. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose ∂Ω into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of ∂Ω we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove C∞ regularity of the regular part of the free boundary when the data are smooth.

http://dx.doi.org/10.13039/100019180 HORIZON EUROPE European Research Council 853404 Buttazzo Giuseppe issue-copyright-statement© Springer-Verlag GmbH Germany, part of Springer Nature 2024
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pmcIntroduction

This paper is dedicated to the regularity of the optimal shapes, solutions to shape optimization problems of the formmin{J(A):A∈A},

where A is an admissible class of open, Lebesgue measurable or quasi-open subsets of Rd, and where J:A→R is a given shape functional, with J(A) usually depending on the solution of a PDE on the domain A. This kind of minimization problems arise in different models in Biology, Engineering and Physics (see for example [6, 21] for an overview) and have been extensively studied from both numerical and theoretical points of view. In particular, there are two classes of shape optimization problems of the form above with long history, both leading to overdetermined elliptic PDE problems with Dirichlet boundary conditions.

The first class involves the so-called spectral functionals, that is, functionals depending on the eigenvalues of the Dirichlet Laplacian asJ(A)=φ(λ1(A),⋯,λk(A))+|A|,

where φ:Rk→R is a real-valued function and |A| denotes the Lebesgue measure of A. The associated shape optimization problemsmin{φ(λ1(A),⋯,λk(A))+|A|:A⊂Rd},

have a long history and are related to the classical question “Can one hear the shape of the drum?” and, more generally, to the interplay between the geometry of the domains and the spectrum of the Dirichlet Laplacian. The first results on the characterization of the optimal shapes, for the first and the second Dirichlet eigenvalues, go back to the works of Faber–Krahn (1922) and Krahn–Szegö (1923), and consist in finding explicit minimizers (balls and unions of disjoint balls), which is only possible in some special cases as J=λ1 and J=λ2. Today, thanks to theory developed by Buttazzo and Dal Maso [10], and to the more recent results [5, 30], it is well-known that, for monotone functionals φ, minimizers exist in a class of measurable (quasi-open) sets. The regularity of the optimal shapes has also been extensively studied; we refer to [4] and [34] for the case of optimal sets of λ1 in a box, to [32] for the optimal sets of λ2 in a box, and to [8, 25, 26, 31] (see also [15, 32]) for functionals involving higher eigenvalues λk.

The second class of functionals involves integral shape functionals, namely, for every bounded open set A⊂Rd we defineJ(A):=∫Aj(uA,x)dx,

where the cost functionj:R×Rd→R

is fixed and the state function uA is the (weak) solution of the PDE1.1 -Δu=finA,u∈H01(A),

the force term f:Rd→R being also a prescribed function.

Optimization problems for integral shape functionals arise in Optimal Control and in models from Mechanics, in which the optimization criteria j(uA,x) takes into account external factors and forces that might appear not immediately, but only after the design is complete and the state function uA is already fixed. This type of problems pops up also in population dynamics, when one aims to optimize the population size. As in the case of spectral functionals, also for integral functionals with monotone cost, the existence of optimal shapes in bounded domains follows from the general theory of Buttazzo and Dal Maso, and again the solutions belong to the large class of measurable (or quasi-open) sets. This existence problem was studied, in a more general framework, in [13]. A general existence result in the class of open sets, was proved recently in [11] and in [12]. Precisely, it was shown that if D is a bounded open set in Rd and if the function j satisfies some suitable growth assumptions, then the shape optimization problemmin{∫Aj(uA,x)dx:Aopen,A⊂D},

has a solution Ω⊂D, Ω-open.

On the other hand, even if the existence theory is quite well understood, there is no regularity theory for the minimizers of integral shape functionals, even in the simplest casej(u,x)=-g(x)u+Q(x),withg≠f,

the regularity of the optimal sets was out of reach.

In this paper we prove the first general regularity result for the optimal shapes of integral functionals. In order to make our main result (Theorem 1.2) easier to read, we introduce the following definition.

Definition 1.1

Let D be an open set in Rd. For k∈N\{0}, α∈[0,1] and N∈N, we call a set Ω⊂D (k,α,N)-regular in D if the free boundary ∂Ω∩D is the disjoint union of a regular part Reg(∂Ω) and a (possibly empty) singular part Sing(∂Ω) such that:Reg(∂Ω) is a relatively open subset of ∂Ω∩D and locally a graph of a Ck,α-regular function;

Sing(∂Ω) is a closed subset of ∂Ω∩D and has the following properties:If d<N, then Sing(∂Ω) is empty.

If d≥N, then the Hausdorff dimension of Sing(∂Ω) is at most d-N, namely Hd-N+ε(Sing(∂Ω))=0for everyε>0.

Moreover, if the regular part Reg(∂Ω) is C∞, then we say that Ω is (∞,N)-regular in D.

The main result of the present paper is the following.

Theorem 1.2

Let D be a bounded open set in Rd, where d≥2. Letf:D→R,g:D→R,Q:D→R,

be given non-negative functions. Suppose that the following conditions hold: f,g∈L∞(D)∩C2(D);

there are constants C1,C2>0 such that 1.2 0≤C1g≤f≤C2ginD.

Q∈C2(D) and there are a positive constants cQ,CQ such that 0<cQ≤Q(x)≦CQfor everyx∈D.

Then, there is α∈(0,1) such that every solution Ω⊂D to the shape optimization problem1.3 min{∫A(-g(x)uA+Q(x))dx:A⊂D,Aopen},

is (1,α,5)-regular in D. Moreover, if f,g,Q∈C∞(D), then Ω is (∞,5)-regular in D.

Proof

The definitions of the regular and the singular parts, Reg(∂Ω) and Sing(∂Ω), of the free boundary ∂Ω∩D are given in Sect. 5. The C1,α regularity of Reg(∂Ω) is proved in Theorem 6.4, while the C∞ regularity follows from Proposition 6.5. The bounds on the dimension of Sing(∂Ω) are given in Theorem 8.1. □

Remark 1.3

(On the bound on the dimension of the singular set Sing(∂Ω)) In Sect. 7 we develop a theory about the regularity of the stable global solutions of the one-phase Bernoulli problem (the definition of global stable solution is given in Definition 7.3); we show that there is a critical dimension d∗ (see Definition 7.7), in which a singular global solution appears for the first time (see Theorem 7.8), and we prove that the d∗ can be only 5, 6, or 7 (see Theorem 7.9). In Sect. 8 (Theorem 8.1) we use this results to show the following bounds on the singular part Sing(∂Ω) of an optimal set Ω, solution to (1.3):if d<d∗, then Sing(∂Ω) is empty;

if d≥d∗, then the Hausdorff dimension of Sing(∂Ω) is at most d-d∗.

In particular, since (by Theorem 7.9) d∗≥5, we get that:if d<5, then Sing(∂Ω) is empty;

if d≥5, then the Hausdorff dimension of Sing(∂Ω) is at most d-5.

In other words, the precise statement of Theorem 1.2 is that under the conditions (a)–(b)–(c), the optimal sets are (1,α,d∗)-regular, where d∗ is the critical dimension from Definition 7.7.

Remark 1.4

(On the assumptions (a)–(b)–(c) in Theorem 1.2) The C2 regularity assumption in (a) is technical and is related to the use we make of the second order variations of the functional J along vector fields (see Sect. 2). The assumption (b) is used in the proofs of the Lipschitz continuity and the non-degeneracy of the state function uΩ; we notice that (b) is automatically satisfied when f and g are both bounded from above and from below by positive constants. The bounds from above and below on the weight Q in (c) are usual in the class of Bernoulli-type free boundary problems; these bounds are necessary for the Lipschitz continuity and the non-degeneracy of uΩ (see Sect. 3), which are essential ingredients for the blow-up analysis in Sect. 4. The C2 regularity of Q, on the other hand, is used again in the computation of the second variation in Sect. 2 and the passage to the blow-up limit in the proof of Theorem 8.1.

Remark 1.5

(On the existence of optimal sets) In [11, Theorem 1.1] it was proved that if D is a bounded open subset of Rd and if f, g, Q satisfy the following conditions:f,g∈L∞(D);

there are positive constants C1≤C2 such that 0≤C1g≤f≤C2g in D;

Q∈L∞(D), Q≥0 in D,

then, there is an open set Ω⊂D solution to the shape optimization problem (1.3).

The presence of the inclusion constraint Ω⊂D is essential for the existence theory for general shape optimization problems (see for instance [10] and [11]). In the case of integral functionals with affine cost, as the one in (1.3), the inclusion constraint can be removed. In the next theorem, which we prove in Sect. 9, we show that optimal sets exist in Rd. Moreover, we prove that the optimal sets in Rd are bounded, which implies that they are solutions to (1.3) in some sufficiently large ball D:=BR, so the regularity of the free boundary in Rd is a consequence of Theorem 1.2.

Theorem 1.6

In Rd, d≥2, let f,g,Q:Rd→R be non-negative functions.

Suppose that the following conditions hold: f,g∈L∞(Rd)∩L1(Rd)∩C2(Rd) and that f(x)>0andg(x)>0foreveryx∈Rd.

Q∈C2(Rd) and there are positive constants cQ,CQ such that 0<cQ≤Q(x)≦CQfor everyx∈Rd.

Then, there is an open set Ω⊂Rd, which is a solution to the shape optimization problem1.4 min{∫A(-g(x)uA+Q(x))dx:A⊂Rd,Aopen,|A|<+∞},

and every solution Ω to (1.4) is bounded and (1,α,5)-regular in Rd. Moreover, if f,g,Q∈C∞(Rd), then Ω is (∞,5)-regular in Rd.

Integral Shape Functionals in the Case f=cg

In this section we briefly discuss the case in which f and g are proportional, which is the only instance of integral functional studied in the literature. Precisely, we claim that if f and g are such thatg=12λ2ffor some constantλ>0,

the shape optimization problem (1.3) is equivalent to the Bernoulli free boundary problem1.5 min{∫D(12|∇u|2-f(x)u+λ2Q(x)1{u≠0})dx:u∈H01(D)}.

Fix a solution u∈H01(D) to (1.5), for which the set {u≠0} is open, and fix an optimal set Ω for (1.3) with a state function uΩ. Since u satisfies-Δu=fin{u≠0}u∈H01({u≠0}),

by integrating by parts, we have that∫D(12|∇u|2-f(x)u+λ2Q(x)1{u≠0})dx=∫D(-12f(x)u+λ2Q(x)1{u≠0})dx=λ2∫Ωu(-g(x)u+Q(x))dx=λ2J({u≠0}).

Analogously, since {uΩ≠0}⊂Ω and Q is positive, we have∫D(12|∇uΩ|2-f(x)uΩ+λ2Q(x)1{uΩ≠0})dx≤λ2J(Ω).

Thus, if u is a solution to (1.5), then J({u≠0})≤J(Ω) and so, {u≠0} is a solution to (1.3). Conversely, if Ω minimizes (1.3), then uΩ is a minimizer of (1.5). Thus, for proportional f and g, the problem (1.3) is equivalent to (1.5); moreover, we notice that the argument above does not require the positivity of f and g, so the equivalence of the problems (1.5) and (1.3) holds also when f and g change sign.

The regularity of the solutions to the free boundary problem (1.5) is nowadays well-understood (at least when Q satisfies the condition (c) of Theorem 1.2). When f≥0, the regularity of ∂Ω follows from the regularity theory for the one-phase Bernoulli problem (see [2, 14, 18–20, 22, 39]). If the right-hand side f changes sign, then (1.5) becomes a two-phase Bernoulli problem, for which the regularity of the free boundary was obtained recently in [35] and [16].

Finally, we notice that when f and g are not proportional, the state function uΩ of an optimal set Ω (that is, a solution to the shape optimization problem (1.3)) is not a minimizer of a free boundary functional as the one in (1.5). In particular, this implies that one can test the optimality of uΩ only with functions u~ which are themselves state functions of some Ω~. This means that a function u~, that differs from u only in a small ball Br, cannot be used to test the optimality of uΩ (truncations, harmonic replacements and radial extensions in small balls are not admitted), which makes most of the classical free boundary regularity results impossible to apply.

Adjoint State and Optimality Condition on the Free Boundary

Let us go back to the general case when f and g are not proportional. We will show that the optimality condition on the boundary ∂Ω∩D of an optimal open set Ω for (1.3) leads to a free boundary problem involving the state function uΩ. In order to see this, we introduce the adjoint state function vΩ as follows: for every open set A⊂D we will denote by vA the weak solution to the problem1.6 -ΔvA=ginA,vA∈H01(A).

By an integration by parts one can see that∫DguAdx=∫D∇uA·∇vAdx=∫DfvAdx,

which means that the two state variables uA and vA are interchangeable. Precisely, an open set Ω⊂D is a solution to (1.3) if and only if it minimizesmin{∫A(-f(x)vA+Q(x))dx:A⊂D,Aopen}.

Sometimes, it is more convenient to consider simultaneously the two state functions, by using the following equivalent formulation, which is symmetric in vΩ and uΩmin{∫A(∇uA·∇vA-f(x)vA-g(x)uA+Q(x))dx:A⊂D,Aopen}.

Throughout the paper we will denote the functional from (1.3) by F. Precisely, we set1.7 F(Ω;D):=∫D(-g(x)uΩ+Q(x)1Ω)dx,

which after an integration by parts has the symmetric formF(Ω;D)=∫D(∇uΩ·∇vΩ-g(x)uΩ-f(x)vΩ+Q(x)1Ω)dx,

so if ξ∈Cc∞(D;Rd) is a smooth compactly supported vector field in D and Ωt:=(Id+tξ)(Ω), then the first variation of F along ξ is given by (see Lemma 2.6)δF(Ω;D)[ξ]:=∂∂t|t=0F(Ωt,D)=∫Ω((∇uΩ·∇vΩ+Q(x))divξ-∇uΩ·((∇ξ)+(Dξ))∇vΩ)dx-∫Ω(uΩdiv(gξ)+vΩdiv(fξ))dx.

Moreover, if Ω is a minimizer and ∂Ω is smooth, then an integration by parts gives thatδF(Ω;D)[ξ]=∫∂Ω(ν·ξ)(Q-|∇uΩ||∇vΩ|)dHd-1=0for everyξ∈Cc∞(D;Rd),

so the state functions uΩ and vΩ are (at least formally) solutions to the free boundary system1.8 -ΔuΩ=finΩ,-ΔvΩ=ginΩ,uΩ=vΩ=0and|∇uΩ||∇vΩ|=QonD∩∂Ω.

Remark 1.7

An epsilon-regularity theorem for viscosity solutions of the above system was proved recently in [28]. Precisely, it was shown that if:uΩ,vΩ:D→R are continuous and non-negative functions such that Ω={uΩ>0}={vΩ>0};

(Ω,uΩ,vΩ) is a viscosity solutions to (1.8) ;

uΩ and vΩ are ε-flat (in a suitable sense) in a ball Br(x0)⊂D centered on ∂Ω;

then the free boundary ∂Ω is C1,α-regular in Br/2(x0).

We notice that an epsilon-regularity theorem for a similar free boundary system, with f≡g≡0, was studied in [1] in the context of a different shape optimization problem. Precisely, in [1], by using the minimality of Ω, it was shown that it is an NTA domain, so by the Boundary Harnack Principle for harmonic functions, the system reduces to a one-phase problem for the function uΩ, for which the epsilon-regularity theorem is known by [2] and [18]. In Sect. A we will detail how our results can be applied to the problem of [1].

Remark 1.8

The free boundary system (1.8) emphasizes that, despite the use of the adjoint variable vΩ, it is not possible to reformulate the shape optimization problem (1.3) in terms of an integral functional depending on uΩ or the couple (uΩ,vΩ). Precisely, no functional of the form∫F(∇u,∇v,u,v,x)+Q(x)1{u2+v2>0}dx

has Euler-Lagrange equations given by (1.8). The same holds for the equations satisfied by the blow-up limits of (uΩ,vΩ). Indeed, since the couple (uΩ,vΩ) satisfies (1.8), any 1-homogeneous blow-up limit (u0,v0) would be a solution of the same system with f=g≡0 and Q≡const>0, but this system is not the Euler-Lagrange system of an integral functional of the above type. This peculiarity represents one of the major challenges for studying the regularity of the free boundary and requires rewriting the optimality and stability conditions in terms of inner variations.

Regularity of the Free Boundary and Dimension of the Singular Set

The key steps in the proof of Theorem 1.2 are the following:Prove that if Ω is an optimal set for (1.3), then it is a viscosity solution to (1.8). This, in combination with the epsilon-regularity result cited in Remark 1.7, will imply that the flat part of the free boundary is smooth.

Show that there exists a critical dimension d∗∈{5,6,7} such that in dimension d<d∗ all the points on ∂Ω∩D are regular.

Prove that a Federer-type dimension reduction principle holds for solutions to (1.3).

There are two main difficulties in following the program outlined in the three points above.

The first difficulty is in the fact that the first order optimality conditionδF(Ω;D)[ξ]=0for everyξ∈Cc∞(D;Rd),

is not leading to a monotonicity formula for uΩ and vΩ; in particular, we do not know if the blow-up limits of uΩ and vΩ are in general homogeneous.

The second obstruction comes from the impossibility to make external perturbations of uΩ and vΩ (that is, in light of Remark 1.8 no perturbations of the form u~=uΩ+ϕ and v~=vΩ+ψ are allowed), so the only information conserved along blow-up sequences is the one contained in the internal variations of Ω along smooth vector fields.

We overcome these difficulties by using the first and the second variation of F. Indeed, suppose that Ω is optimal in D and consider the flow Φt associated to a compactly supported vector field ξ∈Cc∞(D;Rd). Then, setting Ωt:=Φt(Ω), we get that the functiont↦F(Ωt;D),

has a minumum in t=0, so we have1.9 ∂∂t|t=0F(Ω;D)=0and∂2∂t2|t=0F(Ω;D)≥0,

that is, Ω is a stable critical point of the shape functionalΩ↦F(Ω;D).

We prove that this notion is stable under blow-up limits at free boundary points x0∈∂Ω:u0(x)=limn→∞uΩ(x0+rnx)rnandv0(x)=limn→∞vΩ(x0+rnx)rn.

Thus Ω0={u0>0}={v0>0} is a stable critical point for the same functional, but this time with f≡g≡0 and Q≡Q(x0). Now, since u0,v0:Rd→R are harmonic on Ω0, we can apply the Boundary Harnack Principle from [29] to obtain that the ratiou0v0:Ω0→R

is Hölder continuous in Ω0, up to ∂Ω0. After a second blow-up (this time in zero), we get that the positivity set Ω00 of the functionsu00(x)=limm→∞u0(rmx)rmandv00(x)=limm→∞v0(rmx)rm,

is a stable critical point (in every ball BR⊂Rd) for the functional F(·,BR), still with f≡g≡0 and Q≡Q(x0). Moreover, by the Boundary Harnack Principle, we also have that u00 and v00 are proportional. Thus, u00 is (up to a constant) a stable critical point (in the sense of (1.9)) for the Alt–Caffarelli’s one-phase functionalG(u;BR):=∫BR|∇u|2dx+|{u>0}∩BR|in every ballBR⊂Rd,

so the third blow-up in zerou000(x)=limk→∞u00(rkx)rk

is a 1-homogeneous stable critical point for the Alt–Caffarelli’s functional G.

The existence of a homogeneous blow-up u000 is a key element in the proof of Theorem 1.2. Indeed, it allows to prove (see Proposition 6.3) that the state functions uΩ and vΩ are viscosity solutions to the system (1.8) by showing that if we take a free boundary point admitting a one-sided tangent ball, then the homogeneous blow-up limits u000 and v000 constructed above are half-plane solutions. This, in combination with the epsilon-regularity theorem cited in Remark 1.7, implies that, in a neighborhood of any boundary point admitting a half-plane solution as blow-up limit, the free boundary is C1,α-regular; we will call these points regular points (see Sect. 5), while the remaining part of the free boundary (if any) will be called singular. Finally, we notice that the smoothness of the regular part requires only the criticality of Ω (the first part of (1.9)).

It is natural to expect that the stability of Ω (the second part of (1.9)) leads to an estimate on the dimension of the singular set; in fact, the bounds on the critical dimension (the dimension in which a singularity appears for the first time) for minimizers of the one-phase Alt–Caffarelli functional rely (see [14] and [22]) on the well-known stability inequality of Caffarelli-Jerison-Kenig, which was originally obtained in [14] through a particular second order variation of the functional G. On the other hand, this stability inequality is not easy to handle when it comes to passing to blow-up limits and developing a dimension reduction principle. In Sect. 7, we give a different formulation of the stability, which uses the second variation along vector fields, as defined in (1.9). Again in Sect. 7, we show that our notion of stability allows to develop a dimension reduction principle and that there is a critical dimension d∗, which is the smallest dimension admitting one-homogeneous stable solutions with singularity (see Theorem 7.8). Then, in Proposition 7.12, we prove that on smooth cones (that is, cones with isolated singularity) our notion of stability is equivalent to the stability inequality of Caffarelli-Jerison-Kenig. Thus, we obtain the bound 5≤d∗≤7 on the critical dimension d∗ as a consequence of the results of Jerison-Savin [22] and De Silva-Jerison [19]. Finally, in Theorem 8.1, we prove the bounds on the singular set from Theorem 1.2 by (again) a triple blow-up argument, which allows to transfer the dimension bounds from Theorem 7.8 to the singular set of the solutions to (1.3).

Remark 1.9

The methodology we developed for the analysis of (1.3) can be applied, with some natural adjustments, to various free boundary and shape optimization problems. In fact, once the local behavior of the state variables close to the free boundary is known, the triple blow-up analysis combined with the notion of stability along domain variations can be applied in an almost straightforward manner. We mention some recent contributions where the authors leveraged our methodology, adapting it to different problems, see [7, 23, 33]. In particular, in [7] the ε-regularity theory was applied to a spectral shape optimization problem, while in [33] and [23] our approach to the stable critical points was applied to free boundary problems.

Plan of the Paper

In Sect. 2 we compute the first and second variations of the functional F. In Sect. 3 we prove the non-degeneracy and Lipschitz continuity of the state variables. In Sect. 4 we perform a blow-up analysis by proving the existence of triple blow-up sequences converging to a homogeneous limit.

In Sect. 5, we use the blow-up limits from Sect. 4 to define the decomposition of the free boundary into a regular part and a singular part.

In Sect. 6 we prove that the state functions uΩ,vΩ of an optimal set Ω for (1.3) are viscosity solutions of the free boundary system (1.8). Using this information, in Theorem 6.4, we prove that the regular part of the free boundary is C1,α-smooth.

In Sect. 7 we define the notion of a global stable solution to the one-phase Bernoulli (Alt–Caffarelli) problem and we study the dimension of the singular sets for these global stable solutions. We notice that this section can also be read indipendently and that, together with Sect. 2, it contains the key results for the analysis of the singular set.

In Sect. 8 we use the first and the second variations from Sect. 2, the triple blow-up procedure from Sect. 4 and the theory from Sect. 7 in order to prove the dimension bounds on the singular set. This concludes the proof of Theorem 1.2, which follows from Theorem 8.1, Theorem 6.4 and Proposition 6.5.

In Sect. 9 we address the existence of optimal sets in Rd and we prove Theorem 1.6. Ultimately, in Appendix A we apply the analysis of Sects. 7 and 8 to optimal sets arising sets arising from the heat conduction problem studied in [1].

Notations

In the whole paper we use the notationBr(x0)={x∈Rd:|x-x0|<r},

for the ball of radius r centered at point x0 and, when x0=0, we write Br=Br(0); we denote by ωd the Lebesgue measure of a ball of radius one in Rd. For any set A⊂Rd, we set1A(x):=1,ifx∈A,0,ifx∈Rd\A.

Given a non-negative function u we often denote its positivity set as Ωu:={u>0}.

We denote by H1(Rd) the set of Sobolev functions in Rd, that is, the closure of the smooth functions with bounded support Cc∞(Rd) with respect to the usual Sobolev norm‖φ‖H12=∫Rd(|∇φ|2+φ2)dx.

Given an open set Ω⊂Rd, we define the Sobolev space H01(Ω) as the closure, with respect to ‖·‖H1, of the space Cc∞(Ω) of smooth functions compactly supported in Ω. Thus, every u∈H01(Ω) is identically zero outside Ω and we have the inclusion H01(Ω)⊂H1(Rd).

We will sometimes use the following notation for minimum and maximum of two real numbers:min{a,b}=a∧b,max{a,b}=a∨b,a,b∈R.

Given a function u:Rd→R we will denote by ∇u and Du the vectors column and row with components the partial derivatives of u, while D2u will be the Hessian matrix of u. Given a vector field F:Rd→Rd with components Fk, k=1,⋯,d we will denote by ∇F the d×d matrix with columns ∇Fk, k=1,⋯,d and rows (∂jF1,∂jF2,⋯,∂jFd). By convention DF:=(∇F)T, where for any matrix M∈Rd×d, we will denote by MT its transpose. Given a vector field V:Rd→Rd and a matrix with variable coefficients M=(mij)ij:Rd→Rd×d we will denote by (V·∇)(M) the d×d matrix with variable coefficients V·∇mij.

First and Second Variations Under Inner Perturbations

In this section we compute the first and the second variations (with respect to perturbations with compactly supported vector fields) of the functional F from (1.7). Both variations will be fundamental tools in the study of the blow-up limits of the state variables uΩ and vΩ on a domain Ω, which is optimal for (1.3).

First and Second Variation of the State Function

In Lemma 2.5 we compute the expansion of the state variable uΩ (solution to (1.1)) with respect to smooth perturbations of a set Ω. We first prove Lemma 2.2 and Lemma 2.3, where we compute the expansion of a one-parameter family of solutions to PDEs on the same domain Ω.

Remark 2.1

In what follows we will denote by Rd×d the space of d×d square matrices with real coefficients. Given a real matrix A=(aij)ij∈Rd×d, we define its norm in the space Rd×d as‖A‖Rd×d:=(∑i=1d∑j=1daij2)1/2,

and we notice that for every vector V∈Rd, we have |AV|≤‖A‖Rd×d|V|, where |V| is the usual Euclidean norm of V. Next, let Ω be a measurable set in Rd. Given a matrix A:Ω→Rd×d with variable coefficients aij:Ω→R, we say thatA∈L∞(Ω;Rd×d),

if aij∈L∞(Ω) for every 1≤i,j≤d. We define the norm ‖·‖L∞(Ω;Rd×d) as‖A‖L∞(Ω;Rd×d):=‖‖A‖Rd×d‖L∞(Ω)=‖∑i=1d∑j=1daij2‖L∞(Ω)1/2.

Lemma 2.2

(First order expansion of solutions to PDEs) Let Ω be a bounded open set in Rd. Let the functionsf:R→L2(Ω),t↦ft,A:R→L∞(Ω;Rd×d),t↦At,

be such that: At(x) is a symmetric matrix for every (t,x)∈R×Ω and there is a symmetric matrix δA∈L∞(Ω;Rd×d) such that At=Id+t(δA)+o(t)inL∞(Ω;Rd×d).

there is δf∈L2(Ω) such that ft=f0+t(δf)+o(t)inL2(Ω).

Then, for every t small enough there is a unique solution ut to the problem2.1 -div(At∇ut)=ftinΩ,ut∈H01(Ω),

andut=u0+t(δu)+o(t)inH01(Ω),

where δu is the unique weak solution in H01(Ω) to the PDE2.2 -Δ(δu)-div((δA)∇u0)=δfinΩ,δu∈H01(Ω).

Proof

Clearly u0∈H01(Ω) is the solution to -Δu0=f0 in Ω. We set wt:=1t(ut-u0). We will prove that wt converges to δu strongly in H01(Ω).

We notice that (2.1) can be written as-div((Id+(At-Id))∇(u0+twt))=f0+(ft-f0)inΩ.

So, using the equation for u0 and dividing by t, we get2.3 -Δwt-div(1t(At-Id)∇u0)-div((At-Id)∇wt)=1t(ft-f0)inΩ.

If we fix ε>0, we can choose t small enough such that‖At-Id‖L∞(Ω;Rd×d)≤ε1t(At-Id)-δAL∞(Ω;Rd×d)≤ε1t(ft-f0)-δfL2(Ω)≤ε.

Thus, by testing (2.3) with wt, we obtain∫Ω|∇wt|2dx=-∫Ω∇wt·1t(At-Id)∇u0dx-∫Ω∇wt·(At-Id)∇wtdx+∫Ω1t(ft-f0)wtdx≤(ε+‖δA‖L∞(Ω;Rd×d))‖∇wt‖L2‖∇u0‖L2+ε‖∇wt‖L22+(ε+‖δf‖L2(Ω))‖wt‖L2.

Now, by the Poincaré inequality,‖φ‖L2(Ω)2≤Cd|Ω|2/d‖∇φ‖L2(Ω)2for everyφ∈H01(Ω),

and the equation for u0, we have that‖∇u0‖L2(Ω)2=∫Ωf0u0dx≤‖f0‖L2(Ω)‖u0‖L2(Ω)≤‖f0‖L2(Ω)Cd|Ω|1/d‖∇u0‖L2(Ω),

which gives the bound‖∇u0‖L2(Ω)≤Cd|Ω|1/d‖f0‖L2(Ω).

Thus, we deduce that∫Ω|∇wt|2dx≤Cd|Ω|1/d‖f0‖L2(Ω)(ε+‖δA‖L∞(Ω;Rd×d))‖∇wt‖L2+ε‖∇wt‖L22+(ε+‖δf‖L2(Ω))Cd|Ω|1/d‖∇wt‖L2,

and so, for t (and ε<1) small enough∫Ω|∇wt|2dx1/2≤Cd|Ω|1/d(1-ε)(1+‖f0‖L2(Ω)+‖f0‖L2(Ω)‖δA‖L∞(Ω;Rd×d)+‖δf‖L2(Ω)).

Thus, for every sequence tn→0, there is a subsequence for which wtn converges as n→∞, strongly in L2(Ω) and weakly in H01(Ω), to some function w∞. Passing to the limit the Eq. (2.3) we get that w∞ is also a solution to (2.2). Thus w∞=δu. In particular, this implies that wt converges as t→0, strongly in L2(Ω) and weakly in H01(Ω), to δu. Finally, in order to prove that the convergence is strong, we test again (2.3) with wt:lim supt→∞∫Ω|∇wt|2dx=limt→0∫Ω∇wt·1t(At-Id)∇u0dx+limt→0∫Ω1t(ft-f0)wtdx=∫Ω∇(δu)·δA∇u0dx+∫Ωδfδudx=∫Ω|∇(δu)|2dx.

Combining this estimate with the lower semi-continuity of the H1 norm, we getlimt→∞∫Ω|∇wt|2dx=∫Ω|∇(δu)|2dx

which implies that wt converges to δu strongly in H01(Ω). □

Lemma 2.3

(Second order expansion of solutions to PDEs) Let Ω be a bounded open set in Rd. Let the functionsf:R→L2(Ω)andA:R→L∞(Ω;Rd×d)

be such that: At(x) is a symmetric matrix for every (t,x)∈R×Ω and there are symmetric matrices δA∈L∞(Ω;Rd×d) and δ2A∈L∞(Ω;Rd×d) such that At=Id+tδA+t2δ2A+o(t2)inL∞(Ω;Rd×d);

there are δf∈L2(Ω) and δ2f∈L2(Ω) such that ft=f0+tδf+t2δ2f+o(t2)inL2(Ω).

Then, for every t small enough there is a unique solution ut to the problem-div(At∇ut)=ftinΩ,ut∈H01(Ω),

andut=u0+tδu+t2δ2u+o(t2)inH01(Ω),

where δu∈H01(Ω) is the solution to (2.2) and where δ2u∈H01(Ω) solves the PDE2.4 -Δ(δ2u)=div((δA)∇(δu))+div((δ2A)∇u0)+δ2finΩ,δ2u∈H01(Ω).

Proof

Let wt:=1t(ut-u0) be as in the proof of Lemma 2.2. We setvt:=1t(wt-δu)∈H01(Ω).

We will prove that vt converges strongly in H01(Ω) to δ2u. From the equation for wt, we have-Δ(δu+tvt)-div(1t(At-Id)∇u0)-div((At-Id)∇(δu+tvt))=1t(ft-f0).

Thus, using the Eq. (2.2) for δu (-Δ(δu)-div(δA∇u0)=δf), we get-Δvt-div(1t2(At-Id-tδA)∇u0)-div(1t(At-Id)∇(δu+tvt))=1t2(ft-f0-tδf).

Now, reasoning as in Lemma 2.2, we get that the family vt is uniformly bounded in H01(Ω) and converges as t→0 strongly in L2(Ω) and weakly in H01(Ω) to the solution δ2u of (2.4). In order to obtain the strong H1 convergence, we computelim supt→∞∫Ω|∇vt|2dx=-limt→0∫Ω∇vt·1t2(At-Id-tδA)∇u0dx-limt→0∫Ω∇vt·1t(At-Id)∇(δu)dx+limt→0∫Ω1t2(ft-f0-tδf)vtdx=-∫Ω∇(δ2u)·(δ2A)∇u0dx-∫Ω∇(δ2u)·(δA)∇(δu)dx+∫Ω(δ2f)(δ2u)dx=∫Ω|∇(δ2u)|2dx,

which concludes the proof. □

Remark 2.4

We recall that if M=(mij)1≤i,j≤d∈Rd×d is a matrix with constant coefficients, then the following Taylor expansions in Rd×d (with respect to the norm ‖·‖Rd×d)(Id+tM)-1=Id-tM+t2M2+o(t2),det(Id+tM)=1+ttr(M)+t22((tr(M))2-tr(M2))+o(t2),

where Id is the identity matrix in Rd×d and tr(M) is the trace tr(M):=∑i=1dmii. As a consequence, if M∈L∞(Ω;Rd×d) is a matrix with variable coefficients, then we have the expansions(Id+tM)-1=Id-tM+t2M2+o(t2)inL∞(Ω;Rd×d),det(Id+tM)=1+ttr(M)+t22((tr(M))2-tr(M2))+o(t2)inL∞(Ω;Rd×d).

In particular, this implies that given two matrices M,N∈L∞(Ω;Rd×d), we have:(Id+tM+t2N+o(t2))-1=Id-tM+t2(M2-N)+o(t2);det(Id+tM+t2N+o(t2))=det(Id+tM)det(Id+t2N)+o(t2)=1+ttr(M)+t22((tr(M))2-tr(M2)+2tr(N))+o(t2),

and so,(Id+tM+t2N+o(t2))-1(Id+tM+t2N+o(t2))-Tdet(Id+tM+t2N+o(t2))=Id+t(-M-MT+tr(M)Id)+t2(MMT+M2+(MT)2-(N+NT)-(M+MT)tr(M))+t22((tr(M))2-tr(M2)+2tr(N))Id+o(t2).

Proposition 2.5

Let D be a bounded open set in Rd and let f∈C2(D) with f≥0 in D. Given an open set Ω⊂D and a compactly supported vector field ξ∈Cc∞(D;Rd), we consider the associated flow Φ:R×Rd→Rd, determined by the family of ODEs (for every x∈D)2.5 ∂tΦt(x)=ξ(Φt(x))for everyt∈RΦ0(x)=x.

We consider the family of open sets Ωt:=Φt(Ω) and the corresponding state variables uΩt given by (1.1). We define δuΩ and δ2uΩ to be the weak solutions in H01(Ω) to the PDEs-Δ(δuΩ)=div((δA)∇uΩ)+δfinΩ,δuΩ∈H01(Ω),

and-Δ(δ2uΩ)=div((δA)∇(δuΩ))+div((δ2A)∇uΩ)+δ2finΩ,δ2uΩ∈H01(Ω),

where the matrices δA∈L∞(D;Rd×d) and δ2A∈L∞(D;Rd×d) are given by2.6 δA:=-Dξ-∇ξ+(divξ)Id,δ2A:=(Dξ)(∇ξ)+12(∇ξ)2+12(Dξ)2-12(ξ·∇)[∇ξ+Dξ]-(∇ξ+Dξ)divξ+Id(divξ)2+ξ·∇(divξ)2,

while the variations δf∈L2(D) and δ2f∈L2(D) of the right-hand side f are:2.7 δf:=div(fξ),δ2f:=12ξ·(D2f)ξ+12∇f·Dξ[ξ]+f(divξ)2+ξ·∇[divξ]2+(∇f·ξ)divξ.

Then,uΩt∘Φt=uΩ+t(δuΩ)+t2(δ2uΩ)+o(t2)inH01(D).

Proof

We set ut:=uΩt∘Φt. Then, ut∈H01(Ω) and uΩt=ut∘Φt-1.

Moreover, by a change of variables, we have that ut is satisfies the PDE-div(At∇ut)=ftinΩ,ut∈H01(Ω),

where the matrix At and the function ft are defined asft:=f(Φt)|det(DΦt)|andAt:=(DΦt)-1(DΦt)-T|det(DΦt)|,

where for a vector field F:Rd→Rd with components Fk, k=1,⋯,d, we denote by

DF the matrix with rows DFk=(∇Fk)T. We next compute the second order Taylor expansion of DΦt in t=0. By differentiating the equation for the flow Φt, we get∂t(DΦt)=Dξ(Φt)DΦtfor everyt∈R,DΦ0=Id.

Then, taking another derivative in t, we get∂tt(DΦt)=∂t[Dξ(Φt)DΦt]=∂t[Dξ(Φt)]DΦt+Dξ(Φt)∂t[DΦt]=(∂tΦt·∇)[Dξ](Φt)DΦt+Dξ(Φt)Dξ(Φt)DΦt=(ξ(Φt)·∇)[Dξ](Φt)DΦt+Dξ(Φt)Dξ(Φt)DΦt,

where for a vector field F and a matrix M=(mij)ij, we use the notation (F·∇)[M] for the matrix with coefficients F·∇mij. Finally, taking t=0, we get∂2∂t2|t=0(DΦt)=(ξ·∇)[Dξ]+(Dξ)2,

and the Taylor expansionDΦt=Id+tDξ+t22((ξ·∇)[Dξ]+(Dξ)2)+o(t2).

By the expansions from Remark 2.4, we getAt=Id+t(δA)+t2(δ2A)+o(t2)inL∞(D;Rd×d),ft=f+t(δf)+t2(δ2f)+o(t2)inL2(D),

where δA, δ2A, δf, δ2f are given by (2.6) and (2.7). Thus, the claim follows from Lemmas 2.2 and 2.3. □

First and Second Variation of F

In the next lemma, we compute the first derivative of the functional F along inner variations with compact support in D.

Lemma 2.6

(First variation of F along inner perturbations) Let D be a bounded open set in Rd and let f,g,Q∈C1(D). Let Ω⊂D be open and ξ∈Cc∞(D;Rd) be a vector field with compact support in D. Let Φt be the flow of the vector field ξ defined by (2.5) and set Ωt:=Φt(Ω). Then2.8 ∂∂t|t=0F(Ωt,D)=∫Ω∇uΩ·∇vΩ+Qdivξ+ξ·∇Q-∇uΩ·((∇ξ)+(Dξ))∇vΩdx-∫ΩuΩdiv(gξ)+vΩdiv(fξ)dx.

Moreover, if ∂Ω is C2-regular in a neighborhood of the support of ξ, then2.9 ∂∂t|t=0F(Ωt,D)=∫∂Ω(ν·ξ)(Q-|∇uΩ||∇vΩ|)dHd-1,

where ν is the outer unit normal to ∂Ω.

Proof

By applying Lemma 2.5 to ut=uΩt∘Φt and vt=vΩt∘Φt, we get thatut=uΩ+t(δuΩ)+o(t)andvt=vΩ+t(δvΩ)+o(t)inH01(D),

where δuΩ and δvΩ are the solutions to2.10 -Δ(δuΩ)=div((δA)∇uΩ)+δfinΩ,δuΩ∈H01(Ω),-Δ(δvΩ)=div((δA)∇vΩ)+δginΩ,δvΩ∈H01(Ω),

with δf:=div(fξ), δg:=div(gξ), and δA as in Lemma 2.5. Therefore, settingft:=f(Φt)|det(DΦt)|gt:=g(Φt)|det(DΦt)|Qt:=Q(Φt)|det(DΦt)|andAt:=(DΦt)-1(DΦt)-T|det(DΦt)|,

we getF(Ωt,D)=∫Ωt(∇uΩt·∇vΩt-guΩt-fvΩt+Q)dy=∫Ω(∇ut·At∇vt-gtut-ftvt+Qt)dx=F(Ω,D)+t∫Ω(∇(δuΩ)·∇vΩ+∇uΩ·∇(δvΩ)-g(δuΩ)-f(δvΩ))dx+t∫Ω(∇uΩ·(δA)∇vΩ-uΩ(δg)-vΩ(δf)+(Qdivξ+∇Q·ξ))dx+o(t)=F(Ω,D)+t∫Ω(∇uΩ·(δA)∇vΩ-uΩ(δg)-vΩ(δf)+div(Qξ))dx+o(t)

where in the first equality we applied the change of variables y=Φt(x) and in the last one we use the equations -ΔuΩ=f and -ΔvΩ=g in Ω. Substituting with the expression for δA from (2.6), we obtain (2.8).

Suppose now that ∂Ω is C2-smooth. Since in Ω we have the identity∇uΩ·∇vΩdivξ-∇uΩ·((∇ξ)+(Dξ))∇vΩ=div(ξ(∇uΩ·∇vΩ)-(∇uΩ·ξ)∇vΩ-(∇vΩ·ξ)∇uΩ)+(∇vΩ·ξ)ΔuΩ+(∇u·ξ)ΔvΩ,

by integrating by parts we get2.11 δF(Ω,D)[ξ]==∫Ωdiv(ξ((∇uΩ·∇vΩ)+Q)-(∇uΩ·ξ)∇vΩ-(∇vΩ·ξ)∇uΩ)dx-∫Ω((∇uΩ·ξ)g+uΩdiv(gξ)+(∇vΩ·ξ)f+vΩdiv(fξ))dx=∫∂Ω((ν·ξ)((∇u·∇v)+Q)-(∇u·ξ)(ν·∇v)-(∇v·ξ)(ν·∇u))dHd-1.

Since uΩ and vΩ are positive in Ω and vanish on ∂Ω, we have that∇uΩ=-ν|∇uΩ|and∇vΩ=-ν|∇vΩ|on∂Ω,

and soδF(Ω,D)[ξ]=∫∂Ω(ν·ξ)(|∇uΩ||∇vΩ|+Q)-|∇uΩ|(ν·ξ)|∇vΩ|-|∇vΩ|(ν·ξ)|∇uΩ|dHd-1=∫∂Ω(ν·ξ)(Q-|∇uΩ||∇vΩ|)dHd-1,

which concludes the proof. □

Remark 2.7

(First variation and stationary domains) Given a bounded open set D, functions f, g and Q on D, and the functional F defined in (1.7), we will use the notation δF(Ω,D)[ξ] for the first variation of F at Ω along a smooth compactly suppported vector field ξ∈Cc∞(D;Rd). Precisely, we setδF(Ω,D)[ξ]:=∫Ω∇uΩ·∇vΩ+Qdivξ+ξ·∇Q-∇uΩ·((∇ξ)+(Dξ))∇vΩdx-∫ΩuΩdiv(gξ)+vΩdiv(fξ)dx.

We will say that an open set Ω⊂D is stationary (or a critical point) for F in D ifδF(Ω,D)[ξ]=0for everyξ∈Cc∞(D;Rd).

By (2.9), if Ω is stationary in D and the boundary ∂Ω∩D is smooth, then|∇uΩ||∇vΩ|=Qon∂Ω∩D.

Finally, we notice that, by Lemma 2.5, any minimizer of (1.3) is stationary for F.

Proposition 2.8

(Second variation of F along inner perturbations) Let D⊂Rd be a bounded open set in Rd and let f,g,Q∈C2(D). Let Ω⊂D be an open set and ξ∈Cc∞(D;Rd) be a smooth vector field with compact support. Let Φt be the flow of the vector field ξ defined by (2.5) and set Ωt:=Φt(Ω). Then2.12 12∂2∂t2|t=0F(Ωt,D)=∫Ω(∇uΩ·(δ2A)∇vΩ-∇(δuΩ)·∇(δvΩ)-(δ2f)vΩ-(δ2g)uΩ+δ2Q)dx,

where δ2A,δ2f, δ2g, δuΩ and δvΩ are the ones defined in Lemma 2.5 and whereδ2Q:=(ξ·∇Q)divξ+12ξ·D2Qξ+12Q(divξ)2+12Qξ·∇(divξ).

Proof

By applying Lemma 2.5 to ut:=uΩt∘Φt and vt:=vΩt∘Φt, we get thatut=uΩ+t(δuΩ)+t2(δ2uΩ)+o(t2),vt=vΩ+t(δvΩ)+t2(δ2vΩ)+o(t2)inH01(Ω).

with δuΩ,δvΩ satisfying (2.10) and δ2uΩ,δ2vΩ∈H01(Ω) such that-Δ(δ2uΩ)=div((δA)∇(δuΩ))+div((δ2A)∇uΩ)+δ2finΩ,δ2uΩ∈H01(Ω),-Δ(δ2vΩ)=div((δA)∇(δvΩ))+div((δ2A)∇vΩ)+δ2ginΩ,δ2vΩ∈H01(Ω).

Thus, by computing the second order (in t) Taylor expansion ofF(Ωt,D)=∫Ωt(∇ut·At∇vt-gtut-ftvt+Qt)dx,

we get that12∂2∂t2|t=0F(Ωt,D)==∫Ω∇(δ2uΩ)·∇vΩ+∇uΩ·(δ2A)∇vΩ+∇uΩ·∇(δ2vΩ)dx+∫Ω∇(δuΩ)·∇(δvΩ)+∇(δuΩ)·(δA)∇vΩ+∇uΩ·(δA)∇(δvΩ)dx-∫Ωf(δ2v)+(δf)(δvΩ)+(δ2f)vΩ+g(δ2uΩ)+(δg)(δuΩ)+(δ2g)uΩdx+∫Ωδ2Qdx.

Thus, we only have to show that we can write the above expression as in (2.12). By using δvΩ as a test function in the equation for δuΩ and vice versa, we obtain∫Ω∇(δvΩ)·(δA)∇uΩdx-∫Ω(δvΩ)(δf)dx=-∫Ω∇(δuΩ)·∇(δvΩ)dx∫Ω∇(δuΩ)·(δA)∇vΩdx-∫Ω(δuΩ)(δg)dx=-∫Ω∇(δuΩ)·∇(δvΩ)dx.

Then, by testing the equations for uΩ and vΩ respectively with δ2vΩ and δ2uΩ, we get∫Ω∇(δ2uΩ)·∇vΩdx=∫Ωfδ2vdxand∫Ω∇(δ2vΩ)·∇uΩdx=∫Ωgδ2uΩdx.

Using this identities in the expression of the second derivative, we get precisely (2.12). □

Remark 2.9

(Second variation and stable critical domains) Given a bounded open set D, functions f, g and Q on D, and the functional F from (1.7), we will indicate by δ2F(Ω,D)[ξ] the second variation of F at Ω along a smooth compactly supported vector field ξ∈Cc∞(D;Rd). Precisely, we setδ2F(Ω,D)[ξ]:=∫Ω∇uΩ·(δ2A)∇vΩ-∇(δuΩ)·∇(δvΩ)-(δ2f)vΩ-(δ2g)uΩ+δ2Qdx.

In particular, we notice that for every open set Ω, we haveF(Ωt,D)=F(Ω,D)+tδF(Ω,D)[ξ]+t2δ2F(Ω,D)[ξ]+o(t2),

where Ωt=Φt(Ω) (with Φt the flow associated to the vector field ξ) and δF(Ω,D)[ξ] is the first variation (2.11).

We will say that an open set Ω⊂D is a stable critical point for F in D ifδF(Ω,D)[ξ]=0andδ2F(Ω,D)[ξ]≥0for everyξ∈Cc∞(D;Rd).

By Lemmas 2.5 and 2.8, any minimizer of (1.3) is a stable critical point for F.

Lipschitz Regularity and Non-degeneracy of the State Functions

In this section we study the regularity of the state functions uΩ and vΩ on an optimal domain Ω, as well as their behavior close to the free boundary ∂Ω. Consequently, we prove that the set Ω satisfies some density estimates.

Assumptions on D, f, g, and Q

Throughout this subsection we will assume that:D is a open subset of Rd, with d≥2;

f,g∈L∞(D) are two functions such that ‖f‖L∞+‖g‖L∞≤Mand0≤C1g≤f≤C2gonD,

for some positive constants M,C1,C2>0.

Q∈L∞(D) is such that 0<cQ≤Q≤CQonD.

Inwards and Outwards Minimality Conditions

We will use the following notation. Given a set A⊂Rd, and functions f∈L2(A) and φ∈H1(A), we setEf(φ,A):=12∫A|∇φ|2dx-∫Af(x)φdx.

Proposition 3.1

Let D⊂Rd and f,g,Q∈L∞(D) be as in Sect. 3.1 and let Ω⊂D be an open set that minimizes (1.3) in D. Then, the solution uΩ to (1.1) has the following properties. (i) Outwards minimality. For every open set Ω~⊂D such that Ω⊂Ω~ we have Ef(u,D)+C2CQ2|Ω|≤Ef(ϕ,D)+C2CQ2|Ω~|foreveryϕ∈H01(Ω~).

In particular, for every Br(x0)⊂D, we have 3.1 Ef(u,Br(x0))≤Ef(ϕ,Br(x0))+C2CQ2ωdrd,

for every ϕ∈H1(Br(x0)) such that ϕ-uΩ∈H01(Br(x0)).

(ii) Inwards minimality. For every open set ω⊂Ω we have 3.2 Ef(u,D)+C1cQ2|Ω|≤Ef(ϕ,D)+C1cQ2|ω|foreveryϕ∈H01(ω).

Proof

Suppose that the open set Ω~⊂D contains Ω. Then, the optimality of Ω implies that∫Ω(-g(x)uΩ+Q(x))dx≤∫Ω~(-g(x)uΩ~+Q(x))dx,

which can be written as∫Dg(x)(uΩ~-uΩ)dx≤∫Ω~\ΩQ(x)dx.

Now, the positivity of f implies that uΩ~≥uΩ on D, so we get1C2∫Df(x)(uΩ~-uΩ)dx≤∫Dg(x)(uΩ~-uΩ)dx≤∫Ω~\ΩQ(x)dx≤CQ|Ω~\Ω|,

which after rearranging the terms gives-12∫Ωf(x)uΩ+C2CQ2|Ω|≤-12∫Ω~f(x)uΩ~+C2CQ2|Ω~|,

which after an integration by parts on Ω~ and Ω, reads asEf(uΩ,D)+C2CQ2|Ω|≤Ef(uΩ~,D)+C2CQ2|Ω~|.

Finally, since uΩ~ minimizes the energy Ef(·,D) among all functions in H01(Ω~), we obtain (i).

The proof of (ii) is similar. Let ω⊂Ω and let uω be the associated state function. Then, using the optimality of Ω and the bounds on f, g and Q, we getcQ(|Ω|-|ω|)≤∫Ω\ωQ(x)dx≤∫Dg(uΩ-uω)dx≤1C1∫Df(uΩ-uω)dx=2C1(Ef(uω,D)-Ef(uΩ,D)),

which implies (ii) since uω minimizes Ef(·,D) in H01(ω). □

Remark 3.2

The state variable vΩ satisfies analogous inwards/outwards minimality conditions for the functional Eg(·,D), where the constants C1 and C2 are replaced by 1/C2 and 1/C1.

Lipschitz Continuity and Non-degeneracy

As a consequence of Proposition 3.1 we obtain the Lipschitz continuity and the non-degeneracy of uΩ (and of vΩ).

Corollary 3.3

Let D⊂Rd and f,g,Q∈L∞(D) be as in Sect. 3.1. If Ω⊂D is optimal for (1.3), then the state functions uΩ and vΩ are locally Lipschitz in D with Lipschitz constants depending on d, C1, C2, M and CQ.

Proof

By Proposition 3.1 (i), uΩ satisfies (3.1) for any Br(x0)⊂D, so uΩ is an almost-minimizer in the sense of [8, Definition 3.1]. Thus, by [8, Theorem 3.3], uΩ is locally Lipschitz continuous in D with Lipschitz constant depending on d and the bounds from above on fL∞ and C2CQ. □

Lemma 3.4

Let D⊂Rd and f,g,Q∈L∞(D) be as in Sect. 3.1 and let Ω⊂D be an optimal set for (1.3) and uΩ be the associated state function. Then, there are constants C0,r0>0, depending on d, C1, cQ and M, such that the following implication holds(uΩL∞(Br(x0))≦C0r)⇒(uΩ≡0inBr/2(x0)),

for every x0∈Ω¯∩D and every r∈(0,r0]. In other words, if x0∈Ω¯ and Br(x0)⊂D, thensupBr(x0)uΩ≥C0r.

Proof

By Proposition 3.1 (ii), we have that for every ω⊂ΩEf(uΩ,D)+cQC12|Ω|≤Ef(uω,D)+cQC12|ω|.

Thus, the claim follows by [2, Lemma 4.4], [9, Lemma 3.3], or [20, Lemma 2.8]. □

Density Estimates on the Boundary of Ω

An a consequence of the Lipschitz continuity and the non-degeneracy of uΩ and vΩ, we obtain density estimates for the optimal set Ω.

Proposition 3.5

Let D⊂Rd and f,g,Q∈L∞(D) be as in Sect. 3.1. Then, there are ε0,r0>0 (depending on C1, C2, M, d, cQ, CQ) such that for every set Ω⊂D optimal for (1.3)3.3 ε0|Br|≦|Br(x0)∩Ω|≦(1-ε0)|Br|.

for every ball Br(x0)⊂D of radius r≤r0 centered on ∂Ω.

Proof

Assume that x0=0∈∂Ω. The lower estimate is an immediate consequence of the Lipschitz continuity (Lemma 3.3) and the non-degeneracy (Lemma 3.3) of uΩ. The upper bound can be obtained as in [2]. Precisely, consider the solution h to-Δh=MinBrh=uΩonD\Br.

Since ΔuΩ+f≥0 in Rd, we get that -Δ(h-uΩ)≥M-f≧0 in Br. In particular, we have that uΩ≦h and {uΩ>0}⊂{h>0} in Br. Thus, testing the optimality (3.1) of uΩ with h,C2CQ2|Br∩{uΩ=0}|≥Ef(uΩ,Br)-Ef(h,Br)=12∫Br|∇(uΩ-h)|2dx+∫Br(∇h·∇(uΩ-h)-f(uΩ-h))dx≥12∫Br|∇(uΩ-h)|2dx.

By the Poincaré and Cauchy-Schwarz inequalities, we have∫Br|∇(uΩ-h)|2dx≥Cd|Br|1r∫Br(h-uΩ)dx2,

so in order to prove the upper bound in (3.3), we only need to show that 1rd+1∫Br(h-uΩ)dx is bounded from below by a positive constant. Notice that, by the non-degeneracy of uΩ, we haveC~r≦supBr/2uΩ≦supBr/2h.

On the other hand, since h(x)+M2d|x|2 is harmonic in BR, the Harnack inequality in Br impliesC~r≤supBr/2h≤Cd(h(x)+Mr2)for everyx∈Br/2.

Thus, by taking r0 such that 2Cdr0M≤C~, we get that h≧CdC~r=C¯r in Br/2. On the other hand, if L is the Lipschitz constant of uΩ, then for any ε∈(0,1), uΩ≦Lεr in Bεr. Then∫Br(h-uΩ)dx≥∫Bεr(h-uΩ)dx≥(C¯r-Lεr)|Bεr|,

which concludes the proof after choosing ε≤1/2 small enough. □

An Estimate on the Level Sets of uΩ

We conclude the section with this auxiliary result that will play a crucial role in Sect. 4.3 in the proof of the existence of homogeneous blow-up limits.

Lemma 3.6

Let D⊂Rd and f,g,Q∈L∞(D) be as in Sect. 3.1; let Ω be a solution to (1.3) and let uΩ be the associated state function. Then, there are constants C>0 and r0>0, depending only on d,M,C1,C2,cQ,CQ, such that3.4 |{0<u<rt}∩Br(x0)|≦Ct|Br|,

for every Br(x0)⊂D centered on ∂Ω and every t∈(0,1).

Proof

The estimate is contained in the proof of [11, Theorem 1.10]; we sketch the idea for the sake of completeness. Let x0=0∈∂Ω and t>0. We fix a function η∈Cc∞(B2r) such that 0≤η≤1 in B2r and η≡1 in Br, and we use the competitorϕ=η(uΩ-rt)++(1-η)uΩ.

to test the optimality condition (3.2) in B2r. Sinceϕ=uΩ-rtηin{uΩ>rt}andϕ=(1-η)uΩin{0≦uΩ≦rt},

we getEf(ϕ,B2r)=∫{uΩ>rt}∩B2r(12|∇uΩ|2+(rt)22|∇η|2-rt∇uΩ·∇η-fuΩ+frtη)dx+∫{0≦uΩ≦rt}∩B2r((1-η)22|∇uΩ|2+uΩ22|∇η|2-(1-η)uΩ∇uΩ·∇η-fuΩ+fuΩη)dx≦Ef(uΩ,B2r)-rt∫{uΩ>rt}∩B2r∇uΩ·∇ηdx+∫B2r((rt)22|∇η|2+frtη)dx+∫{0≦uΩ≦rt}∩B2r((1-η)2-12|∇uΩ|2-(1-η)uΩ∇uΩ·∇η)dx≦Ef(uΩ,B2r)+Cd(t‖∇uΩ‖L∞+rtfL∞+t2)|Br|.

Thus (3.2) impliesC1cQ2|{0<uΩ≤rt}∩Br|≤C1cQ2(|{uΩ>0}∩B2r|-|{φ>0}∩B2r|)≤Ef(ϕ,D)-Ef(uΩ,D)≤Cd(t‖∇u‖L∞+rtfL∞+t2)|Br|.

Therefore, we getC1cQ2|Br∩{0≦u≦rt}|≦Ct|Br|

with C>0 depending on d, ‖uΩ‖L∞(B2r), M, C1 and cQ. □

Compactness and Convergence of Blow-Up Sequences

Take an optimal set Ω for (1.3) in some D⊂Rd, and consider the corresponding state functions u=uΩ and v=vΩ. For any x0∈∂Ω∩D and any sequence rk→0+, we setux0,rk(x):=1rku(x0+rkx),vx0,rk(x):=1rkv(x0+rkx),Ωx0,rk:=Ω-x0rk.

Since u and v vanish in x0 and are Lipschitz (in a neighborhood of x0), we have that ux0,rk and vx0,rk vanish in 0 and (for large k) are uniformly Lipschitz in any ball BR. Thus, there are functions u0,v0:Rd→R and subsequences of ux0,rk and vx0,rk that converge locally uniformly in Rd respectively to u0 and v0. As usual we say that u0 and v0 are blow-up limits of u and v in x0; and we recall that they might depend on the sequence rk. We notice that the blow-up limits of u and v will always be taken along the same sequence rk→0.

The main results are Propositions 4.3 and 4.7. In Proposition 4.3 we list the properties of any couple of functions u0,v0 obtained as blow-up limits of u, v, while in Proposition 4.7 we show that there is at least one sequence rk→0 that provides blow-up limits which are 1-homogeneous and stationary for the one-phase Alt–Caffarelli functional.

A General Lemma About the Convergence of Blow-Up Sequences

The construction of the blow-up limit from Proposition 4.7 will require taking three consecutive blow-ups. We give here a general lemma, which we will use several times in this section.

Lemma 4.1

Let B2R be a ball in Rd. Let un:B¯2R→R be a sequence of non-negative Lipschitz functions converging uniformly to a Lipschitz function u∞:B¯2R→R, and suppose that there is a constant L>0 such that‖∇un‖L∞(B2R)≤Lfor everyn≥1.

Then, the following holds. (i) Suppose that there is a constant C~>0 such that, for every n≥1, 4.1 supBr(x0)un≥C~rfor everyx0∈BR∩{un>0}¯and everyr∈(0,R).

Then 4.2 supBr(x0)u∞≥C~rfor everyx0∈BR∩{u∞>0}¯and everyr∈(0,R),

and 4.3 1{un>0}→1{u∞>0}pointwise a.e. inBR.

(ii) Suppose that there is a constant M~>0 such that, for every n≥1, we have the bound 4.4 ∫B2R∇un·∇φdx≤M~‖φ‖L∞(B2R)for everyφ∈Cc0,1(B2R),

where Cc0,1(B2R) is the space of Lipschitz functions with compact support in B2R. Then un converges to u∞ strongly in H1(BR).

Proof

We first prove (i). Suppose that x0∈BR∩{u∞>0}¯. Then, there is a sequence xn→x0 of points xn∈BR∩{u∞>0}¯. Let rn:=r-|xn-x0|. By (4.1), there is a point yn∈Brn(xn)⊂Br(x0) such that un(yn)≥C~rn. Thus, by the uniform convergence of un, we get (4.2). In order to prove (4.3), we first notice that by the pointwise convergence of un to u∞, we have that1{u∞>0}(x0)=1⇒1{un>0}(x0)=1forlargen,

so it is sufficient to prove that the setS:={x0∈BR:1{u∞>0}(x0)=0andlim supn→∞1{un>0}(x0)=1},

is of measure zero. Now, by (4.1), we get that for every r∈(0,R) there is a sequence yn∈B¯r(x0) such that un(yn)≥C~r. Then, by the uniform convergence of un, we have that there is y∞∈B¯r(x0) such that u∞(y∞)≥C~r. Then, by the Lipschitz continuity of u∞, the ball BC~r/L(y∞) is contained in {u∞>0}. Since r is arbitrary, we get that the Lebesgue density of {u∞>0} in x0 cannot be zero, so the set S has zero measure.

We next prove (ii). Since un is uniformly bounded in H1(B2R), we have that un converges to u∞ weakly in H1(B2R). Thus, the estimate (4.4) holds also for u∞, that is∫B2R∇u∞·∇φdx≤M~‖φ‖L∞(B2R)for everyφ∈Cc0,1(B2R).

Choose a function φ∈Cc∞(B2R) such that φ≡1 in BR. Then,∫BR|∇(un-u∞)|2dx≤∫B2R|∇(φ(un-u∞))|2dx=∫B2R|∇φ|2(un-u∞)2dx+∫B2R∇(un-u∞)·∇(φ2(un-u∞))dx≤∫B2R|∇φ|2(un-u∞)2dx+2M~‖φ2(un-u∞)‖L∞(B2R).

Since the right-hand side converges to zero, we get the claim. □

Finally, we notice that the above lemma can be applied to the state functions uΩ and vΩ of an optimal domain. This is a consequence of the following lemma.

Lemma 4.2

Let B2R⊂Rd and u∈H1(B2R) be a non-negative function with the following properties. There is a function f∈L∞(B2R) such that -Δu=finΩu:={u>0},u=0on∂Ωu∩B2R,

in the sense that ∫B2R∇u·∇φdx=∫B2Rφfdxfor everyφ∈H01(B2R)withφ=0inB2R\Ωu.

There is Λ>0 such that, for every non-negative φ∈H01(B2R), Ef(u,B2R)+Λ2|B2R∩{u>0}|≤Ef(u+φ,B2R)+Λ2|B2R∩{u+φ>0}|.

Then, for every φ∈Cc0,1(BR), we have∫BR∇u·∇φdx≤Cd(1+Λ+R‖f‖L∞(B2R))Rd-1‖φ‖L∞(BR).

Proof

We only sketch the proof and we refer to [38, Chapter 3] for the details. By (a) we have thatμ:=Δu+|f|

is a positive Radon measure on B2R, where∫B2Rφdμ:=∫B2R(-∇u·∇φ+φ|f|)dxfor everyφ∈Cc0,1(B2R).

By testing the optimality condition in (b) with a function φ∈Cc0,1(B2R), we get∫B2Rφdμ=∫B2R(-∇u·∇φ+φf)dx≤∫B2R|∇φ|2dx+Λ|B2R|,

and choosing φ=Rϕ with ϕ≡1 in BR, we obtainμ(BR)≤∫B2Rϕdμ≤Cd(1+Λ)Rd-1.

As a consequence, for every φ∈Cc0,1(BR), we have∫BR∇u·∇φdx≤∫BR|φ|dμ+∫BR|φ||f|dx≤Cd(1+Λ+R‖f‖L∞)Rd-1‖φ‖L∞(BR).

which concludes the proof. □

First Blow-Up

In this subsection we list the properties of any couple of blow-upsu0,v0:Rd→R

of the state functions uΩ and vΩ at a boundary point x0∈∂Ω∩D. The qualitative properties (Lipschitz continuity, non-degeneracy, density estimates) of uΩ and vΩ are conserved under blow-up limits; the stationarity condition also passes to the limit; the main difference is that u0 and v0 are harmonic where they are positive.

Proposition 4.3

Let D⊂Rd be a bounded open set, let Ω be a solution to (1.3) with f, g, Q as in Theorem 1.2, and let u:=uΩ and v:=vΩ be the state functions on Ω defined in (1.1) and (1.6). We consider a point x0∈∂Ω∩D and blow-up sequences ux0,rk,vx0,rk of u, v converging locally uniformly in Rd to blow-up limits u0,v0∈C0,1(Rd). Then taking C1 and C2 to be the constants from (1.2), we have C1v0≤u0≤C2v0onRd;

the functions u0 are v0 are harmonic in the open set Ω0:={u0>0}={v0>0}.

there are constants ε0>0 and C>0 such that 4.5 ε0|Br|≤|Br(x0)∩Ω0|≤(1-ε0)|Br|,

for every x0∈∂Ω0, r>0, and 4.6 |{0<u0<rt}∩Br(x0)|≦Ct|Br|,

for every x0∈∂Ω0,r>0,t>0;

0∈∂Ω0 and there is a constant C0>0 such that supBr(x0)u0≥C0randsupBr(x0)v≥C0rfor everyx0∈Ω¯0and everyr>0;

there is a constant Λ>0 depending on C1,C2,CQ and d such that, for every R>0, ∫BR∇u0·∇φdx+∫BR∇v0·∇φdx≤ΛRd-1‖φ‖L∞(BR);

for every compactly supported smooth vector field ξ∈Cc∞(Rd;Rd), we have ∫Rd(-∇u0·((∇ξ)+(Dξ))∇v0+(∇u0·∇v0+Q(x0)1Ω0)divξ)dx=0.

Proof

For simplicity, we setuk:=ux0,rk,vk:=vx0,rk,fk:=fx0,rk,gk:=gx0,rkandΩk:=(Ω-x0)/rk,

and we notice that-Δuk=rk2fkand-Δvk=rk2gkinΩk.

The first two claims are just a consequence of the locally uniform convergence of uk and vk to u0 and v0. By Proposition 3.1, Corollary 3.3 and Lemma 3.4, we already know that u and v fulfill the assumptions of Lemma 4.2. Therefore, (4) and (5) follow from Lemma 4.1, while (6) follows from Lemma 2.6 and Remark 2.7, and the strong H1 convergence of uk and vk.

We next prove (3). By Proposition 3.5 and rescaling, we know that, for every k>0,ε0|Br|≤|Br(x0)∩{uk>0}|≤(1-ε0)|Br|,forr<r0/rk,x0∈∂Ωk,

so, by the strong convergence of 1Ωk to 1Ω0 in Lloc1(Rd) (see Lemma 4.1), we get the density estimate (4.5) for Ω0. Similarly, by rescaling (3.4) we obtain|{0<uk<rt}∩Br(x0)|≦Ct|Br|forr<r0rk,x0∈∂Ωk,t>0,

which, passing to the limit as k→∞, gives (4.6). □

Second Blow-Up

Consider blow-up limits u0,v0 as in the previous subsection and letu00,v00:Rd→R.

be blow-up limits of u0 and v0 in zero. Then u00 and v00 are still blow-up limits of the state functions uΩ and vΩ at x0∈∂Ω∩D and Proposition 4.3 still applies. On the other hand, [29, Theorem 1.2] applies to the domain Ω0 and the function u0, so the Boundary Harnack Principle (see [29, Definition 1.1]) holds on Ω0; since u0 and v0 are harmonic in Ω0, we get that the ratio u0/v0 is Hölder continuous up to the boundary ∂Ω0. This, in particular means that the second blow-ups u00 and v00 at any boundary point (and thus in zero) are proportional.

Lemma 4.4

Let u0,v0:Rd→R be non-negative Lipschitz functions on Rd with the same positivity set and let Ω0:={u0>0}={v0>0}. Suppose that u0 and v0 satisfy the conditions (1)-(6) from Proposition 4.3 and let u00,v00:Rd→R be blow-ups of u0,v0 at zero. Then there is a constant λ∈(C1,C2) such that u00=λv00 on Rd;

the function u00 is harmonic in the open set Ω00:={u00>0};

there are constants ε0>0 and C>0 such that ε0|Br|≦|Br(x0)∩Ω00|≦(1-ε0)|Br|,

for every x0∈∂Ω00, r>0, and |{0<u00<rt}∩Br(x0)|≦Ct|Br|,

for every x0∈∂Ω00,r>0,t>0;

0∈∂Ω00 and there is a constant C0>0 such that supBr(x0)u00≥C0rfor everyx0∈Ω¯00and everyr>0;

there is a constant Λ>0 such that, for every R>0, ∫BR∇u00·∇φdx≤ΛRd-1‖φ‖L∞(BR);

for every compactly supported smooth vector field ξ∈Cc∞(Rd;Rd), we have 4.7 ∫Rd(-∇u00·((∇ξ)+(Dξ))∇u00+(|∇u00|2+λQ(x0)1Ω00)divξ)dx=0.

Proof

Let R>0 be fixed and let ϕ:=u0. In order to show that the Boundary Harnack Principle holds on Ω0, we check that Ω0 and ϕ satisfy the list of assumptions (a)–(g) from [29, Theorem 1.2] in the ball BR: by definition of Ω0, we have ϕ>0 in Ω0 and ϕ≡0 on BR\Ω0;

by hypothesis ϕ is Lipschitz continuous in Rd and so, in BR;

since ϕ is harmonic in Ω0 and satisfies the condition (4) from Proposition Proposition 4.3, we can apply [38, Lemma 6.8]; thus, there is a constant κ>0 such that ϕ≥κdistBR\Ω0inBR/2;

since ϕ≧0 and Δϕ=0 in Ω0, we have that Δϕ≥0 in Rd;

for every x0∈∂Ω∩BR, we have |Br(x0)\Ω0|≥ε0|Br(x0)|for everyr∈(0,R-|x0|);

for every x0∈∂Ω0∩BR and every r∈(0,R-|x0|), we have |{0<ϕ<rt}∩Br(x0)|≤Ct|Br|for everyt>0;.

by (4) of Proposition 4.3, for every x0∈∂Ω0∩BR and every r∈(0,R-|x0|), we have supBr(x0)ϕ≥C0r.

Therefore, all the assumptions of [29, Theorem 1.2] are fulfilled and so, the ratiou0v0:Ω0→R,

can be extended to a Hölder continuous function on Ω¯0∩BR/2. Thus, the blow-ups u00 and v00 are proportional and all the other claims follow as in Proposition 4.3. □

Third Blow-Up

Take the state functions uΩ and vΩ on an optimal domain Ω. Letu00,v00:Rd→R,

be the second blow-up limits of uΩ and vΩ at a free boundary point x0∈∂Ω∩D. Then, u00 and v00 are proportional and satisfy the conditions listed in Lemma 4.4. We will show that if we perform a further blow-up in zero, then we obtain functions u000,v000:Rd→R that still satisfy the conditions from Lemma 4.4 but are also 1-homogeneous. Before we give the precise statement, we notice that the stationarity condition (4.7) implies the monotonicity of the associated Weiss’ boundary adjusted energy from [39].

Lemma 4.5

(Monotonicity formula) Let BR⊂Rd, u∈H1(BR) be a continuous non-negative function, and let Ω:={u>0}. Suppose that for every smooth compactly supported vector field ξ∈Cc∞(Rd;Rd), we have:∫Ω(divξ|∇u|2+Λ-∇u·((∇ξ)+(Dξ))∇u)dx=0.

Then, for every x0∈∂Ω,r>0 the mapr↦WΛ(ux0,r):=∫B1|∇ux0,r|2dx+Λ|{ux0,r>0}∩B1|-∫∂B1ux0,r2dHd-1,

is non-decreasing in (0,+∞) and∂∂rWΛ(ux0,r)≧2r∫∂B1|x·∇ux0,r-ux0,r|2dHd-1,

In particular, if r↦WΛ(ux0,r) is constant, then u is 1-homogeneous.

Proof

See for instance [38, Proposition 9.9]. □

Lemma 4.6

Let u00,v00:Rd→R be non-negative Lipschitz functions on Rd with the same positivity set and let Ω00:={u00>0}={v00>0}. Suppose that u00 and v00 satisfy the conditions (1)-(6) from Lemma 4.4 and let u000,v000:Rd→R be blow-ups of u00,v00 at zero. Then: u000=λv000 on Rd;

for every compactly supported smooth vector field ξ∈Cc∞(Rd;Rd), we have 4.8 ∫Rd(-∇u000·((∇ξ)+(Dξ))∇u000+(|∇u000|2+λQ(x0)1Ω000)divξ)dx=0;

u000 is 1-homogeneous in Rd.

Proof

Set Λ:=λQ(x0) and for simplicity, let u:=u00. By Lemma 4.5, the functionr↦WΛ(ur),

is non-decreasing in r and so, it admits a limit Θ as r→0; moreover, by the Lipschitz continuity of u, Θ is finite. Let rk→0 be such that urk→u000. Then, by Lemma 4.1, urk converges to u000 strongly in Hloc1 and the level sets {urk>0} converge in Lloc1 to Ω000. Thus, for any s>0Θ:=limk→+∞WΛ(usrk)=WΛ((u000)s).

Moreover, using again the strong convergence of urk and their level sets, we get that u000 satisfies (4.8). Thus, using again Lemma 4.5 and the fact that s↦WΛ((u000)s) is constantly equal to Θ, we get that u000 is homogeneous. □

As an immediate consequence, we obtain the following proposition.

Proposition 4.7

(Existence of stationary 1-homogeneous blow-ups) Let D⊂Rd be a bounded open set and Ω be a solution to (1.3) with f, g, Q as in Theorem 1.2. Let u:=uΩ and v:=vΩ be the state functions on Ω defined in (1.1) and (1.6) and let x0∈∂Ω∩D. Then, there is a sequence rk→0 such that the corresponding blow-up limitsu0:=limk→∞ux0,rkandv0:=limk→∞vx0,rk

satisfy the following conditions: (i) u0 and v0 are 1-homogeneous in Rd and u0=λv0 for some constant λ∈(C1,C2);

(ii) for every compactly supported smooth vector field ξ∈Cc∞(Rd;Rd), we have ∫Rd(-∇u0·((∇ξ)+(Dξ))∇u0+(|∇u0|2+λQ(x0)1Ω0)divξ)dx=0.

Remark 4.8

The blow-up sequence arising from Proposition 4.7 is obtained by a diagonal argument between the three different blow-ups defined respectively in Sects. 4.2, 4.3 and 4.4. Therefore, by combining all the previous result, we can see that u0 and v0 fulfill the conditions (a), (b), (c), d and (e) in Definition 7.3.

Regular and Singular Parts of the Free Boundary

Let D⊂Rd be a bounded open set and let Ω be a solution to the problem (1.3). As in Sect. 4, we denote by u, v the associated state variables.

Definition 5.1

(Regular and singular points) We will say that a boundary point x0∈∂Ω∩D is regular if there is a blow-up limit (u0,v0) of (u, v) at x0 such thatu0(x)=α(x·ν)+andv0(x)=β(x·ν)+

for some unit vector ν∈Rd and some α>0 and β>0 such that αβ=Q(x0). If such a blow-up limit does not exist, then we will say that x0 is singular.

We will denote by Reg(∂Ω) the set of all regular points on ∂Ω∩D and by Sing(∂Ω) the set of all singular points on ∂Ω∩D. Clearly, we have thatReg(∂Ω)∩Sing(∂Ω)=∅andReg(∂Ω)∪Sing(∂Ω)=∂Ω∪D.

In Sect. 6 we will show that the regular part Reg(∂Ω) is in fact, locally, a smooth manifold (and in particular, a relatively open subset of ∂Ω∩D), while in Sect. 8 we will give an estimate on the dimension of the singular set Sing(∂Ω).

Regular and Singular Parts in Dimension Two

One can easily show that in dimension d=2 the free boundary ∂Ω∩D is composed only of regular points; in particular, in dimension two the proof of Theorem 1.2 is concluded already in Sect. 6, while the results from Sect. 8 are needed only when d≥3.

Lemma 5.2

Let D be a bounded open set in R2 and let Ω be a solution to (1.3). Then, every point x0∈∂Ω∩D is a regular point in the sense of Definition 5.1.

Proof

Let rk→0 be a sequence such that the blow-up limitsu0:=limk→∞urk,x0andv0:=limk→∞vrk,x0,

are, as in Proposition 4.7, proportional (u0=λv0), non-negative and 1-homogeneous functions on Rd, which are harmonic on the positivity set Ω0:={u0>0}={v0>0}. Now, reasoning as in [38, Proposition 9.13], we write u0 and v0 in polar coordinates asu0(r,θ)=rϕ(θ)andv0(r,θ)=rψ(θ),

where (since u0 and v0 are harmonic) ϕ and ψ are solutions to-ϕ′′(θ)=ϕ(θ)and-ψ′′(θ)=ψ(θ)in∂Ω0∩∂B1.

Since the only solutions of this equations (up to a rotation) are multiples of sinθ, and since H1(Ω0∩∂B1)<2π (this follows from the density estimate in Proposition 4.3), we get thatu0(x)=α(x·ν)+andv0(x)=β(x·ν)+

for some unit vector ν∈Rd and some α>0 and β>0. Moreover, by Proposition 4.7, the function u0 is critical point for the one-phase problem (that is, (4.8) holds). Thus,|∇u0|2=λQ(x0)on∂Ω0={x∈R2:x·ν=0},

where λ=α/β. Since |∇u0|2=α2, we get that αβ=Q(x0), which concludes the proof. □

A Geometric Condition for the Regularity in Every Dimension

By an argument similar to the one in Lemma 5.2, we have that the free boundary points admitting one-sided tangent ball are regular points. This result holds in every dimension and will be useful in Sect. 6.

Lemma 5.3

Let D be a bounded open set in Rd, d≥2, and let Ω be a solution to (1.3). Suppose that there is a one-sided tangent ball at the boundary point x0∈∂Ω∩D in the sense that:5.1 there isBr(y0)⊂Ωwithx0∈∂Br(y0)or there isBr(z0)⊂Rd\Ω¯withx0∈∂Br(z0).

Then, x0 is a regular point in the sense of Definition 5.1.

Proof

As in Lemma 5.2, we consider a sequence rk→0 for which the blow-up limitsu0:=limk→∞urk,x0andv0:=limk→∞vrk,x0,

are 1-homogeneous, non-negative, harmonic on their positivity set Ω0:={u0>0} and are such that u0=λv0. The one-sided ball condition implies that there is a unit vector ν∈Rd such thatΩ0⊂{x∈Rd:x·ν>0}orΩ0⊇{x∈Rd:x·ν>0}.

Since Ω0 satisfies an exterior density estimate (by Proposition 4.3, claim Item (3)), the only possibility is that Ω0={x∈Rd:x·ν>0} andu0(x)=α(x·ν)+andv0(x)=β(x·ν)+

for some α>0 and β>0 with α/β=λ. As in Lemma 5.2, since u0 satisfies (4.8), we get that|∇u0|2=αβQ(x0)on∂Ω0={x∈R2:x·ν=0},

which gives that αβ=Q(x0). □

Regularity of Reg(∂Ω)

In this section we prove that the regular part Reg(∂Ω) of the boundary of an optimal set Ω is locally the graph of a smooth function.

Viscosity Formulation

In this subsection we prove that on the free boundary ∂Ω∩D of an optimal set Ω, solution to (1.3), we have the following optimality condition|∇uΩ||∇vΩ|=Qon∂Ω,

in viscosity sense, as in [28, Section 2], in terms of the blow-up limit of the state functions uΩ and vΩ at free boundary points (see Remark 6.2).

Definition 6.1

(Viscosity solutions) Let D be a bounded open set in Rd and let f,g∈L∞(Rd). Let u,v:D→R be two non-negative continuous functions with the same supportΩ:={u>0}={v>0},

on which they satisfy the PDE6.1 -Δu=fand-Δv=ginΩ∩D.

We say that the boundary condition|∇u||∇v|=Qon∂Ω∩D,

holds in viscosity sense if at any point x0∈∂Ω∩D, at which Ω admits a one-sided tangent ball in the sense of (5.1), there exist: a decreasing sequence rk→0;

two positive constants α,β>0 such that αβ=Q(x0);

a unit vector ν∈Rd;

such that the rescalingsux0,rk(x):=u(x0+rkx)rkandvx0,rk(x):=v(x0+rkx)rk,

converge uniformly in every ball BR⊂Rd respectively to the blow-up limits6.2 u0(x):=αx·ν+andv0(x):=βx·ν+.

Remark 6.2

In [28] the authors addressed the ε-regularity theory for viscosity solutions of (1.8). In particular, in [28, Lemma 2.9] they proved that if the free boundary condition is satisfied in the sense of Definition 6.1, then for any smooth function φ∈C∞(D) the following holds: (i) If φ+ touches uv from below at a point x0∈D∩∂Ω, then |∇φ(x0)|≤Q(x0).

(ii) If φ+ touches uv from above at a point x0∈D∩∂Ω, then |∇φ(x0)|≥Q(x0).

(iii) If a and b are constants such that a>0,b>0andab=Q(x0),

and if φ+ touches wab:=12(au+bv) from above at x0∈D∩∂Ω, then |∇φ(x0)|≥Q(x0).

Proposition 6.3

Let D⊂Rd be a bounded open set and let f,g,Q:D→R be as in Theorem 1.2. Let Ω be a solution to (1.3). Then the state variables u:=uΩ and v:=vΩ satisfy6.3 |∇u||∇v|=Qon∂Ω∩D,

in the sense of Definition 6.1.

Proof

It follows as in the proof of Lemma 5.3. □

We can now prove that Reg(∂Ω) is C1,α-regular for some α∈(0,1) by exploiting the ε-regularity theory developed in [28].

Theorem 6.4

(Regularity of Reg(∂Ω)) Let D be a bounded open set in Rd, where d≥2. Letf:D→R,g:D→R,Q:D→R,

be given non-negative functions. Suppose that the following conditions hold: f,g∈L∞(D);

there are constants C1,C2>0 such that 0≤C1g≤f≤C2ginD.

Q∈C0,αQ(D), for some αQ>0, and there are a positive constants cQ,CQ such that 0<cQ≤Q(x)≦CQforeveryx∈D.

Let Ω be a solution to (1.3). Then, the regular part Reg(∂Ω), defined in Sect. 5, is locally the graph of a C1,α function, for some α>0.

Proof

The proof follows by the epsilon-regularity theory developed in [28]. Indeed, if x0∈Reg(∂Ω) and ux0,ρk,vx0,ρk are the blow-up sequences from the definition of Reg(∂Ω). Then, there are α,β>0,ν∈Rd such that αβ=Q(x0),|ν|=1 andlimk→∞ux0,ρk-α(x·ν)+L∞(B1)=0andlimk→∞vx0,ρk-β(x·ν)+L∞(B1)=0.

Moreover, the Lipschitz continuity and the non-degeneracy imply that for every ε>0 there exists k0>0 such that for every k≧k0 we haveα(x·ν-ε)+≤ux0,ρk≤α(x·ν+ε)+for everyx∈B1,β(x·ν-ε)+≤vx0,ρk≤β(x·ν+ε)+for everyx∈B1,

that is, ux0,ρk,vx0,ρk are ε-flat in the direction ν (see [28, Definition 1.2]). By rescaling the associated state equations, we have-Δux0,ρk=ρk2fx0,ρkand-Δvx0,ρk=ρk2gx0,ρkinB1∩{ux0,ρk>0},

whereΔux0,ρkL∞(B1)+Δvx0,ρkL∞(B1)≦ρkfL∞(B1)+gL∞(B1).

On the other hand, since both ux0,ρk and vx0,ρk still satisfy (6.3) in viscosity sense, by applying [28, Theorem 3.1], ∂{ux0,ρk>0} is C1,α in B1/2. Finally, the result follows by rescaling back to the original problem. □

Higher Regularity

We can pass from C1,α to C∞-regularity of Reg(∂Ω) by exploiting the higher order Boundary Harnack Principle for solutions to (1.8).

Proposition 6.5

(Higher regularity of Reg(∂Ω)) Let D be a bounded open set in Rd and let f, g, Q be as in Theorem 6.4. If f,g,Q∈Ck,α(D) for some k≥1 and α>0, then the regular part Reg(∂Ω) of the free boundary ∂Ω∩D of any solution Ω to (1.3) is locally the graph of a C1+k,α function for some α>0. In particular, if f, g, Q are C∞, then Reg(∂Ω) is locally the graph of a C∞ function.

Proof

We use a bootstrap argument as in [31, Section 5.4]. Suppose that Reg(∂Ω) is locally the graph of a Ck,α-regular function, for some k≧1 (when k=1 the claim follows from Theorem 6.4). Let x0∈∂Ω and r>0 be such that ∂Ω∩Br(x0)=Reg(∂Ω). Since u and v satisfy (6.1) in Ω and since v is non-degenerate, we can apply the higher order Boundary Harnack Principle (for PDEs with right-hand side) from [27, Theorem 1.3] and [36, Theorem 1.3], obtaining a non-negative Ck,α-regular function w:Br(x0)∩Ω¯→R satisfying u=wv in Br(x0)∩Ω¯. Then, we have:|∇u||∇v|=Qand|∇u|=w|∇v|on∂Ω∩Br(x0).

Thus, u is a solution to the problem-Δu=finΩ∩Br(x0)|∇u||∇u|=wQon∂Ω∩Br(x0),

which by [24, Theorem 2] implies that Reg(∂Ω) is locally the graph of a Ck+1,α function. □

Stable Homogeneous Solutions of the One-Phase Bernoulli Problem

In this section we study the singular set of the stable global solutions of the one-phase Bernoulli problem (see Definition 7.3 for the definition of global stable solution). The main results are Theorems 7.8 and 7.9, which we will use in Theorem 8.1 in order to estimate the dimension of the singular set of the free boundary of the optimal sets for (1.3). The present section can be read separately from the rest of the paper; we will use only the Taylor expansions from Sect. 2 and the general results from Sect. 4.

Solutions of PDEs in Unbounded Domains

In Rd, d≥3, we defineH˙1(Rd):={u∈L∗(Rd):∇u∈L2(Rd;Rd)}where2∗:=2dd-2.

It is well known that H˙1(Rd) is a Hilbert space equipped with the norm ‖u‖H˙1(Rd):=‖∇u‖L2(Rd;Rd) and that Cc∞(Rd) is dense in H˙1(Rd). Moreover, given an open (bounded or unbounded) set Ω⊂Rd, we define the space H˙01(Ω) as the closure of Cc∞(Ω) with respect to ‖·‖H˙1(Rd).

Let Ω be an open (bounded or unbounded) subset of Rd and let F∈L2(Rd;Rd) be a given vector field. We say that w is a weak solution of the PDE7.1 -Δw=divFinΩ,w∈H˙01(Ω),

if w∈H˙01(Ω) and∫Rd∇w·∇φdx=-∫Rd∇φ·Fdxfor everyφ∈H˙01(Ω).

It is standard to check that w is a solution to (7.1) if and only if w minimizes the functional7.2 J(φ):=12∫Rd|∇φ|2dx+∫Rd∇φ·Fdx,

among all functions φ∈H˙01(Ω). Since it is immediate to check that a minimizer of (7.2) in H˙01(Ω) exists and is unique, we get that also the solution to (7.1) exists and is unique. Finally, we notice that if w∈H˙01(Ω) is the solution to (7.1), then∫Rd|∇w|2dx=-∫Rd∇w·F≤‖F‖L2‖∇w‖L2,

which gives7.3 ∫Rd|∇w|2dx≤∫Rd|F|2dx.

Remark 7.1

(Exterior density estimate and the space H˙01) We say that an open set Ω⊂Rd satisfies a uniform exterior density estimate with a constant c>0 if7.4 |Br(x)\Ω|≥c|Br|for everyr∈(0,1)and everyx∈Rd\Ω.

It is known (see for example [17]) that if an open set Ω⊂Rd satisfies (7.4) then the space H˙01(Ω) can be characterized as:7.5 H˙01(Ω)={u∈H˙1(Rd):u=0a.e.onRd\Ω}.

Lemma 7.2

(Convergence of solutions) Let Ωn be a sequence of open sets in Rd, d≥3 such that: there is a constant c>0 such that, for every n≥1, Ωn satisfies the exterior density estimate (7.4);

there is an open set Ω∞⊂Rd satisfying the exterior density estimate (7.4) with the constant c>0 and such that the sequence of characteristic functions 1Ωn converges pointwise almost-everywhere in Rd to 1Ω∞.

Let Fn∈L2(Rd;Rd) be a sequence of vector fields converging strongly in L2(Rd;Rd) to the vector field F∞∈L2(Rd;Rd). For every n≥1, let wn be the solution to the PDE-Δwn=divFninΩnwn∈H˙01(Ωn).

Then, wn converges strongly in H˙1(Rd) to the solution w∞ of-Δw∞=divF∞inΩ∞w∞∈H˙01(Ω∞).

Proof

First of all, we notice that the sequence wn is bounded in H˙1(Rd) (by (7.3)). Thus, we can extract a subsequence, that we still denote by wn, which converges weakly in H˙1(Rd) and pointwise almost-everywhere on Rd to a function w∈H˙1(Rd). We will show that w=w∞.

First, notice that for almost every x∈Rd\Ω∞ we have thatwn(x)→w(x)and1Ωn(x)→1Ω∞(x)=0.

But then wn(x)=0 (by (7.5)) and so w(x)=0. Thus, using again (7.5), we get w∈H˙01(Ω∞).

Now, let φ∈Cc∞(Ω∞). We will show that for large enough n, φ∈Cc∞(Ωn). Indeed, let δ>0 be a constant such that Bδ(x)⊂Ω∞ for every x in the support of φ. Suppose by contradiction that there is a sequence xn∈{φ≠0}¯ such that xn∉Ωn. Then, by the density estimate for Ωn, |Bδ(xn)∩Ωn|≤(1-c)|Bδ|. Now, up to a subsequence, xn→x∞∈{φ≠0}¯. But then,(1-c)|Bδ|≥limn→∞|Bδ(xn)∩Ωn|=|Bδ(x∞)∩Ω∞|=|Bδ|,

which is a contradiction. Thus, for large n, {φ≠0}¯⊂Ωn. Now, using the equation for wn,∫Rd∇wn·∇φdx=-∫Rd∇φ·Fndx

and passing to the limit, we get that∫Rd∇w∞·∇φdx=-∫Rd∇φ·F∞dx,

that is w=w∞.

Finally, in order to prove that the convergence is strong, we use the equations for wn and w∞ and the strong convergence of Fn to F∞:∫Rd|∇wn|2dx=-∫Rd∇wn·Fndx→-∫Rd∇w∞·F∞dx=∫Rd|∇w∞|2dx,

which concludes the proof. □

Global Stable Solutions of the One-Phase Problem

We define the functionalsδG:C0,1(R)×Cc∞(Rd;Rd)→Randδ2G:C0,1(R)×Cc∞(Rd;Rd)→R,

as follows. Given:a Lipschitz function u:Rd→R,

a smooth compactly supported vector field ξ∈Cc∞(Rd;Rd),

we set:δG(u)[ξ]:=∫Rd(∇u·δA∇u+1Ωudivξ)dx,

7.6 δ2G(u)[ξ]:=∫Rd2∇u·(δ2A)∇u-2|∇(δu)|2dx+∫Rd1Ωu((divξ)2+ξ·∇(divξ))dx,

where Ωu:={u>0}, and where δA,δ2A are defined as in (2.6) and where δu∈H˙01(Ωu) is the weak solution to the PDE7.7 -Δ(δu)=div((δA)∇u)inΩu,δu∈H˙01(Ωu).

Definition 7.3

(Global stable solutions of the one-phase problem) We say that a function u:Rd→R is a global stable solution of the one-phase problem if, for every compactly supported smooth vector field ξ∈Cc∞(Rd;Rd), we have:7.8 δG(u)[ξ]=0andδ2G(u)[ξ]≥0,

and if the following conditions hold: u is globally Lipschitz continuous and non-negative on Rd;

u is harmonic in the open set Ωu:={u>0};

there is a constant c>0 such that |Br(x0)∩Ωu|≦(1-c)|Br|,

for every x0∈Rd\Ωu and every r>0;

0∈∂Ωu and there is a constant η>0 such that supBr(x0)u≥ηrfor everyx0∈Ω¯uand everyr>0;

there is a constant C>0 such that, for every R>0, ∫BR∇u·∇φdx≤CRd-1‖φ‖L∞(BR)for everyφ∈Cc∞(BR).

Remark 7.4

The functionals δG and δ2G correspond to the first and the second variation along vector fields of the one-phase Alt–Caffarelli functional1G(u)=∫Rd(|∇u|2+1{u>0})dx,

so one may expect that the natural definition of a global stable solution is a function u:Rd→R that satisfies (7.8). Unfortunately, the condition (7.8) alone seems to be quite weak. For instance,u(x,y)=1+|x|eu(x,y)=1+|xy|

satisfy (7.8), but they are not even harmonic in {u>0}, while the functionu(x,y)=|xy|,

is harmonic in {u>0}, but the free boundary ∂{u>0} has a cross-like singularity in zero; more generally, the solutions of optimal partition problems satisfy (7.8) and the structure of their nodal sets can be very different from the one of the one-phase free boundaries.

We add the conditions (a), (b), (c), d, (e) in order to have a regularity theory for one-phase stable solutions, which is similar to the one available for minimizers of the Alt–Caffarelli functional. The Lipschitz continuity (a) and the non-degeneracy d guarantee the existence of non-trivial blow-up limits obtained by 1-homogeneous rescalings of u. The condition (e) is needed for the strong convergence of the blow-up sequences (see Lemma 4.1), which together with the exterior density estimate (c) allows to transfer the stability condition (7.8) to the blow-up limits of u (thanks to Lemma 7.2). We also notice that the conditions (e) and (b) imply the Lipschitz continuity of u, so we could actually avoid adding (a) to the list.

Finally, we highlight that the conditions (a)–(b)–(c)-d–(e) are satisfied by the blow-ups of numerous one-phase problems, for instance by the state functions on the domains minimizing (1.3) (see Sect. 4), and of course, by the global minimizers of the classical one-phase Bernoulli problem.

Remark 7.5

(Blow-ups of global stable solutions) We notice that if u:Rd→R satisfies (a), (b), (c), d and (e), then any blow-up u0:Rd→R of u at x0∈∂Ωu,u0=limn→∞ux0,rnwithux0,rn(x):=u(x0+rnx)rnandlimn→∞rn=0,

still satisfies (a), (b), (c), d and (e). In particular, by Lemma 4.1, this means that the convergence ux0,rn→u0 is strong in Hloc1 and thus, by Lemmas 7.2 and 4.5, if u is a global stable solution, then u0 is a 1-homogeneous global stable solution.

Remark 7.6

(Decomposition of the free boundary) Let u:Rd→R be a global stable solution in the sense of Definition 7.3 and let Ωu:={u>0}. We decompose the free boundary ∂Ωu as∂Ωu=Reg(∂Ωu)∪Sing(∂Ωu),

where the regular part Reg(∂Ωu) consists of all points x0∈∂Ωu at which there is a blow-up limit u0 which is a half-space solution, that is,u0(x)=(x·ν)+for some unit vectorν∈Rd,

while the singular part is given by Sing(∂Ωu)=∂Ωu\Reg(∂Ωu). As in Sect. 6, it is immediate to check that the global stable solutions satisfy the optimality condition|∇u|=1on∂Ωu

in viscosity sense, so the ε-regularity theorem of [18] holds and we have that the regular part is a relatively open subset of ∂Ωu and a C∞ manifold. We notice that this decomposition is precisely the one from Sect. 5 with α=β=Q=1, u=v, and f=g=0.

Definition 7.7

(Critical dimension for stable solutions) We define d∗ to be the smallest dimension admitting a 1-homogeneous global stable solution (in the sense of Definition 7.3) u:Rd→R with 0∈Sing(∂Ωu).

Theorem 7.8

(Dimension reduction for stable solutions) Suppose that u:Rd→R is a 1-homogeneous global stable solution of the one-phase problem (in the sense of Definition 7.3). (i) If d<d∗, then u is a half-plane solution.

(ii) If d=d∗, then Sing(∂Ωu)={0} or u is a half-plane solution.

(iii) If d>d∗, then the Hausdorff dimension of Sing(∂Ωu) is at most d-d∗, that is, Hd-d∗+ε(Sing(∂Ωu))=0for everyε>0,

where d∗ is the critical dimension from Definition 7.7.

Proof

The proof follows from a classical argument that can be found for instance in [38] (in particular, [38, Proposition 10.13]). □

Theorem 7.9

(Bounds on the critical dimension for stable solutions) 5≤d∗≤7, where d∗ is the critical dimension from Definition 7.7.

Proof

The claim follows from Propositions 7.11 and 7.12 below. □

Global Minimizers and Global Stable Solutions

Given an open set D⊂Rd and a function u∈H1(D), we define:G(u,D):=∫D(|∇u|2+1{u>0})dx.

Definition 7.10

(Global minimizers) We say that a function u:Rd→R is a global minimizer of the Alt–Caffarelli functional, if:u is non-negative and u∈Hloc1(Rd);

G(u,BR)≤G(v,BR), for every BR⊂Rd and every v:Rd→R such that v-u∈H01(BR).

Proposition 7.11

Suppose that u:Rd→R is a global minimizer in the sense of Definition 7.10. Then, u is a global stable solution in the sense of Definition 7.3. In particular, d∗≤7, where d∗ is the critical dimension from Definition 7.7.

Proof

It is well-known that the global minimizers satisfy the conditions (a)-(b)-(c)-d-(e) of Definition 7.3 (see for instance [38]). Moreover, by [38, Lemma 9.5 and Lemma 9.6], the global minimizers are critical points, that is,δG(u)[ξ]=0for everyξ∈Cc∞(Rd;Rd).

Thus, it only remains to prove the positivity of the second variation:δ2G(u)[ξ]≥0for everyξ∈Cc∞(Rd;Rd).

Let ξ∈Cc∞(Rd;Rd),Φt be the associated defined by (2.5) and let Ωt:=Φt(Ω), for t∈R. In BR, we consider the solution ut to the problem-Δut=0inΩt∩BR,ut=0on∂Ωt∩BR,ut=uon∂BR.

By the optimality of ut, we have thatG(ut,BR)≥G(u,BR)for everyt∈R,

so,ddt|t=0G(ut,BR)=0andd2dt2|t=0G(ut,BR)≥0.

By Lemma 2.8, we have thatd2dt2|t=0G(ut,BR):=∫BR2∇u·(δ2A)∇u-2|∇wR|2+1Ωu((divξ)2+ξ·∇(divξ))dx,

where Ωu={u>0} and wR is the solution to the PDE-ΔwR=div((δA)∇u)inΩu∩BRwR∈H01(Ωu∩BR).

Thus, for every R>0,∫Rd|∇wR|2dx≥∫BR∇u·(δ2A)∇u+121Ωu((divξ)2+ξ·∇(divξ))dx.

Since by Lemma 7.2 the sequence wR→δu strongly in H˙1(Rd) as R→∞, we get that∫Rd|∇(δu)|2dx≥∫BR∇u·(δ2A)∇u+121Ωu((divξ)2+ξ·∇(divξ)))dx,

which is precisely the inequality δ2G(u)[ξ]≥0. Finally, the bound d∗≤7 follows by the example of a singular 1-homogeneous global minimizer constructed by De Silva and Jerison in [19]. □

Global Stable Solutions and the Stability Inequality of Caffarelli–Jerison–Kenig

Let u:Rd→R be a 1-homogeneous global stable solution of the one-phase problem with an isolated singularity in zero, that is,Sing(∂Ωu)={0}.

In particular, we have that the regular partReg(∂Ωu):=∂Ωu\{0},

is a smooth C∞ manifold and the function u is C∞ in Ω¯u\{0}, up to the boundary ∂Ωu\{0}. Thus, u is a classical solution to the PDE7.9 Δu=0inΩu,|∇u|=1on∂Ωu\{0}.

Together with the homogeneity of u this implies (see for instance [22]) thatH>0on∂Ωu\{0},

where H is the mean curvature of ∂Ωu oriented towards the complement of Ωu.

We will say that Ωu supports the stability inequality of Caffarelli-Jerison-Kenig if7.10 ∫Ωu|∇φ|2dx≥∫∂ΩuHφ2dHd-1for everyφ∈Cc∞(Rd\{0}).

In [14] and [22] it was shown that if:d=3 (see [14]) or d=4 (see [22]);

u:Rd→R is a 1-homogeneous non-negative Lipschitz function;

∂Ωu\{0} is C∞ smooth;

u is a solution of the one-phase Bernoulli problem (7.9);

Ωu supports the stability inequality (7.10);

then u is a half space solution, that is,u(x)=(x·ν)+for some unit vectorν∈Rd.

Thus, in order to show that in dimension 3 and 4 there are no global stable solutions (in the sense of Definition 7.3) with singularities, it is sufficient to prove the following proposition.

Proposition 7.12

(The global stable cones satisfy the stability inequality) Let u:Rd→R be a 1-homogeneous global stable solution (in the sense of Definition 7.3) with Sing(∂Ωu)={0}. Then, Ωu supports the stability inequality (7.10). In particular, d∗≥5, where d∗ is the critical dimension from Definition 7.7.

Remark 7.13

Following the proof of Proposition 7.12, it is immediate to check that also the converse is true. Precisely, if u:Rd→R is a 1-homogeneous function, if ∂Ωu\{0} is smooth and if u is a solution to (7.9) such that Ωu supports the stability inequality (7.10), then u is a global stable solution in the sense of Definition 7.3.

Proof

For any bounded open set D⊂Rd and any function u∈H1(D), we define:G(u,D):=∫D|∇u|2dx+|D∩{u>0}|.

Moreover, for any R>1, we call the annulusAR:=BR\B¯1/R.

We fix a smooth vector field ξ∈Cc∞(AR,Rd) and we define the open setΩt:=Φt(Ωu)for everyt∈R,

where Φt is the flow of ξ defined by (2.5). Let ut:AR→R be the solution of the PDEΔut=0inΩt∩AR,ut=0on∂Ωt∩AR,ut=uinΩt∩∂AR.

Step 1. We will show that7.11 d2dt2|t=0G(ut,AR)≥δ2G(u)[ξ],

where δ2G is defined in (7.6). Following Lemma 2.8 we have thatd2dt2|t=0G(ut,AR)=∫AR∩Ωu2∇u·(δ2A)∇u+((divξ)2+ξ·∇(divξ))-2|∇wR|2dx.

where δA and δ2A are defined in (2.6) and wR is the solution of the PDE7.12 -ΔwR=div((δA)∇u)inΩu∩ARwR∈H01(Ωu∩AR).

Now, let δu be the solution to (7.7). By the variational characterization of (7.7) in H˙01(Ωu), the fact that wR∈H˙01(Ωu), and an integration by parts, we have that-12∫BR|∇(δu)|2dx=12∫BR|∇(δu)|2dx+∫BR∇(δu)·(δA)∇udx≤12∫BR|∇wR|2dx+∫BR∇wR·(δA)∇udx=-12∫BR|∇wR|2dx,

which gives (7.11).

Step 2. We set u′ to be the solution of the PDEΔu′=0inΩu∩AR,u′=ξ·νon∂Ωu∩AR,u′=0onΩu∩∂AR,

where ν is the outer normal to ∂Ωu. We will first show thatu′=wR-ξ·∇u.

Indeed, since ξ is supported in AR and since ∇u=-ν on ∂Ωu∩AR, we have:u′=wR-ξ·∇uon∂(Ωu∩AR).

In order to show that u′ is harmonic in Ωu, we compute (using the repeated index summation convention)div((δA)∇u)=∂j[-∂iξj∂iu-∂jξi∂iu+∂iξi∂ju]=-∂ijξj∂iu-∂iξj∂iju-∂jjξi∂iu-∂jξi∂iju+∂ijξi∂ju=-2∂iξj∂iju-∂jjξi∂iu=-∂j[∂jξi∂iu+ξi∂iju]=-Δ(ξ·∇u),

and then we use the equation (7.12) for wR.

Step 3. We next compute the second derivative of G(ut,AR) in terms of u′. For the sake of simplicity, through the rest of the proof we use the notations introduced in Sect. 1.5.-∫Ωu∩AR|∇wR|2dx=∫AR-|∇(u′+ξ·∇u)|2dx=∫Ωu∩AR-|∇u′|2-2∇u′·∇(ξ·∇u)-|∇(ξ·∇u)|2dx=-∫Ωu∩AR(|∇u′|2+|∇(ξ·∇u)|2)dx-2∫∂(Ωu∩AR)∂u′∂ν(ξ·∇u)dHd-1=-∫Ωu∩AR(|∇u′|2+|∇(ξ·∇u)|2)dx-2∫∂(Ωu∩AR)∂u′∂νu′dHd-1=∫Ωu∩AR(|∇u′|2-|∇(ξ·∇u)|2)dx.

On the other hand∫Ωu∩AR∇u·(δ2A)∇udx=∫Ωu∩AR|Dξ(∇u)|2+∇u·(Dξ)2(∇u)dx-∫Ωu∩AR∇u·((ξ·∇)[Dξ])∇udx-∫Ωu∩AR2divξ∇u·Dξ∇udx+∫Ωu∩AR12|∇u|2div(ξdivξ)dx=∫Ωu∩AR|Dξ(∇u)|2+∇u·(Dξ)2(∇u)dx-∫Ωu∩AR∇u·((ξ·∇)[∇ξ])∇udx-∫Ωu∩AR2divξ∇u·∇ξ∇udx-∫Ωu∩ARD2u(∇u)·ξ(divξ)dx+∫∂Ωu∩AR12|∇u|2divξ(ξ·ν)dHd-1.

Notice that-∇u·((ξ·∇)[∇ξ])∇u=-∂iuξk∂kiξj∂ju=-∂k[∂iuξk∂iξj∂ju]+∂kiuξk∂iξj∂ju+∂iuξk∂iξj∂kju+(divξ)∂iu∂iξj∂ju

and-D2u(∇u)·ξ(divξ)=-∂iju∂juξi(divξ)=-∂j[∂iu∂juξi(divξ)]+∂iu∂ju∂jξi(divξ)+∂iu∂juξi∂j(divξ),

which leads to-∇u·((ξ·∇)[∇ξ])∇u-2divξ∇u·∇ξ(∇u)-D2u(∇u)·ξ(divξ)=-∂k[∂iuξk∂iξj∂ju]-∂j[∂iu∂juξi(divξ)]+∂kiuξk∂iξj∂ju+∂iuξk∂iξj∂kju+ξi∂iu∂ju∂j(divξ).

Similarly, we can compute-|∇(ξ·∇u)|2=-∂k(ξi∂iu)∂k(ξj∂ju)=-(∂kξi∂iu+ξi∂kiu)(∂kξj∂ju+ξj∂kju)=-|Dξ(∇u)|2-ξi∂kiuξj∂kju-2∂kξi∂iuξj∂kju,

and so|∇ξ(∇u)|2+∇u·(∇ξ)2(∇u)-|∇(ξ·∇u)|2=∂iu∂iξj∂jξk∂ku-2∂kξi∂iuξj∂kju-ξi∂kiuξj∂kju=∂iu∂iξj∂jξk∂ku-2∂kξi∂iuξj∂kju-∂k[ξi∂iuξj∂kju]+∂kξi∂iuξj∂kju+ξi∂iu∂kξj∂kju=∂iu∂iξj∂jξk∂ku-2∂kξi∂iuξj∂kju-∂k[ξi∂iuξj∂kju]+∂kξi∂iuξj∂kju+∂j[ξi∂iu∂kξj∂ku]-∂jξi∂iu∂kξj∂ku-ξi∂iju∂kξj∂ku-ξi∂iu∂k(divξ)∂ku.

Finally, by collecting the previous computations, we obtain the identity

Thus, integrating by parts and using that ∇u=-ν on ∂Ωu∩AR, we obtain

where H is the mean curvature of ∂Ωu∩AR oriented towards the complement of Ωu.

Step 4. Conclusion. Given any φ∈Cc∞(Rd\{0}), we consider an annulus AR containing the support of φ and the vector field ξ:=φ∇u. Thus, by the minimality of u′ in Ωu∩AR, we get∫Ωu|∇φ|2dx-∫∂Ωuφ2HdHd-1≥∫Ωu∩AR|∇u′|2dx-∫∂Ωu∩AR(u′)2HdHd-1=12d2dt2|t=0G(ut,AR)≥12δ2G(u)[ξ]≥0,

where the last inequality follows from (7.11). □

Hausdorff Dimension of Sing(∂Ω)

In this section we will estimate the dimension of the singular set Sing(∂Ω) of a solution Ω to the shape optimization problem (1.3); our main result is the following.

Theorem 8.1

(Dimension of the singular set) Let D be a bounded open set in Rd and let f, g, Q be as in Theorem 1.2, namely: f,g∈C2(D)∩L∞(D);

there are constants C1,C2>0 such that 0≤C1g≤f≤C2g in D;

Q∈C2(D) and there are constants cQ,CQ such that 0<cQ≤Q≦CQ on D.

Let Ω be a solution to (1.3) and let the singular part Sing(∂Ω) of the free boundary ∂Ω∩D be as in Sect. 5. Then, the following holds: (i) If d<d∗, then Sing(∂Ω)=∅.

(ii) If d≥d∗, then the Hausdorff dimension of Sing(∂Ω) is at most d-d∗, that is, Hd-d∗+ε(Sing(∂Ω))=0for everyε>0.

In order to prove Theorem 8.1 above, we will first show that the stability of Ω, expressed in terms of the state functions uΩ and vΩ as in Lemma 2.8, passes to a blow-up limit.

Lemma 8.2

(Stability of the blow-up limits) Let D be a bounded open set in Rd and let f, g, Q be as in Theorems 1.2 and 8.1. Let Ω be an optimal set for (1.3) and let u:=uΩ and v:=vΩ be the state functions defined in (1.1) and (1.6). Suppose that the couple u0,v0:Rd→R is a blow-up limit of u, v at a point x0∈∂Ω∩D. Then,8.1 ∫Ω0(∇u0·(δ2A)∇v0-∇(δu0)·∇(δv0)+Q(x0)(divξ)2+ξ·∇(divξ)2)dx≥0,

where Ω0:={u0>0}={v0>0}, and δu0 and δv0 are the solutions respectively to the PDEs8.2 -Δ(δu0)=div((δA)∇u0)inΩ0,δu0∈H˙01(Ω0),-Δ(δv0)=div((δA)∇u0)inΩ0,δv0∈H˙01(Ω0),

in the sense explained in Sect. 7.1. In particular, if u0 and v0 are proportional, then there is a constant λ>0 such that λu0 is a global stable solution of the one-phase Bernoulli problem in the sense of Definition 7.3.

Proof

Let rk→0, be a sequence such thatuk(x):=1rku(x0+rkx)andvk(x):=1rkv(x0+rkx),

converge respectively to u0 and v0 locally uniformly and (by Proposition 4.3) strongly in Hloc1(Rd). We define:fk(x):=rkf(x0+rkx),gk(x):=rkg(x0+rkx),Qk(x):=Q(x0+rkx).

Then, the functions uk and vk satisfy the PDEs-Δuk=fkinΩkuk∈H01(Ωk),-Δvk=gkinΩkvk∈H01(Ωk),

whereΩk:=1rk(-x0+Ω).

Moreover, Ωk is optimal in the rescaled domainDk:=1rk(-x0+D),

for the functionalFk(A):=∫A(∇uA·∇vA-uAgk-vAfk+Qk)dx

where, this time, by uA and vA we denote the solutions to-ΔuA=fkinAuA∈H01(A),-ΔvA=gkinAvA∈H01(A).

We fix a compactly supported smooth vector field ξ∈Cc∞(Rd;Rd). Since rk→0, for k large enough the support of ξ is contained in Dk and the stability of Ωk (Lemma 2.8) reads as8.3 ∫Rd(∇uk·(δ2A)∇vk-∇(δuk)·∇(δvk)-(δ2fk)vk-(δ2gk)uk+δ2Qk)dx≥0,

where δuk and δvk are the solutions to-Δ(δuk)=div((δA)∇uk)+δfk=div((δA)∇uk+fkξ)inΩkδuk∈H01(Ωk),-Δ(δvk)=div((δA)∇vk)+δgk=div((δA)∇vk+gkξ)inΩkδvk∈H01(Ωk),

and where we used the following notation:δA and δ2A are the matrices defined in (2.6) (we notice that δA and δ2A are defined in terms of ξ only);

the variations δfk and δ2fk (δgk and δ2gk are defined analogously) are given by; δfk(x):=div(fkξ)=rk2∇f(x0+rkx)·ξ+rkf(x0+rkx)divξ,δ2fk(x):=rk32ξ·(D2f(x0+rkx))ξ+rk22∇f(x0+rkx)·Dξ[ξ]+rkf(x0+rkx)(divξ)2+ξ·∇[divξ]2+rk2(∇f(x0+rkx)·ξ)divξ,

δ2Qk is given by δ2Qk(x):=(ξ·∇Qk)divξ+12ξ·D2Qkξ+Qk12((divξ)2+ξ·∇(divξ))=rk(ξ·∇Q(x+rkx))divξ+rk22ξ·D2Q(x0+rkx)ξ+12Q(x0+rk)((divξ)2+ξ·∇(divξ)),

where in all the formulas above, the field ξ and its derivatives are all computed in x.

We notice that δfk and δ2fk vanish outside the support of ξ. Moreover, since f and g are C2, we get that δfk and δ2fk converge to zero uniformly. Similarly, since Q is C2, we get thatδ2Qk→12Q(x0)((divξ)2+ξ·∇(divξ))strongly inL1(Rd).

Next, we notice that by Proposition 4.3, uk and vk converge strongly in Hloc1(Rd) to the blow-up limits u0, v0. Thus, since the support of δ2A is compact, we getlimk→∞∫Rd∇uk·(δ2A)∇vkdx=∫Rd∇u0·(δ2A)∇v0dx.

Similarly, we have that(δA)∇uk+fkξ→(δA)∇u0and(δA)∇vk+gkξ→(δA)∇v0,

strongly in L2(Rd;Rd). Thus, by Lemma 7.2, we get that δuk and δvk converge respectively to the solutions δu0 and δv0 of (8.2). Now, (8.1) follows by passing to the limit (8.3). □

Proof of Theorem 8.1

The strategy is similar to the one for minimizers of the one-phase problem. We split the proof in two different cases.

Case 1: d<d∗. Let x0∈∂Ωu∩D and rk→0+ be the infinitesimal sequence such that the rescalings ux0,rk and vx0,rk converge to the 1-homogeneous blow-up limits u0 and v0 given by Proposition 4.7. Then, given λ∈(C1,C2) as in Proposition 4.7, the blow-up limits8.4 u0:=limk→∞1λQ(x0)ux0,rkandv0:=limk→∞λvx0,rk

coincide up to a multiplicative constant (that is v0=Q(x0)v0) and, by Lemma 8.2 and Proposition 4.7, u0 is a 1-homogeneous stable solutions of the one-phase problem in the sense of Definition 7.3. Therefore, by definition of d∗, Sing(∂Ωu0)=∅ and u0 is a half space solution:u0(x)=(x·ν)+for some unit vectorν∈Rd.

Finally, by rewriting the result in terms of the blow-up sequence of Proposition 4.7, we deduce the existence α>0 and β>0 such that αβ=Q(x0) such that1rku(x0+rkx)→α(x·ν)+,and1rkv(x0+rkx)→β(x·ν)+

as k→∞. Thus, by definition, x0∈Reg(∂Ω). Since this is true at every free boundary point x0∈∂Ω∩D, we get that Sing(∂Ω)=∅.

Case 2: d≧d∗. Given ε>0, let us prove thatHd-d∗+ε(Sing(∂Ω))=0.

We will apply consecutively the three blow-ups from Sect. 4. By contradiction, assume thatHd-d∗+ε(Sing(∂Ω))≥C>0.

Therefore, by [37, Lemma 10.5], there are x0∈Sing(∂Ω) and a sequence rk→0 such thatHd-d∗+ε(Sing(∂Ω)∩Brk(x0))≥Crkd-d∗+ε.

By the first blow-up analysis from Sect. 4.2 (see Proposition 4.3), we deduce that the blow-up sequences ux0,rk,vx0,rk and Ωx0,rk converge to some limits u0,v0 and Ω0 such thatHd-d∗+ε(Sing(∂Ω0)∩B1)≥C,

where u0, v0 and Ω0 are as in Proposition 4.3 and where u0 and v0 satisfy the stability condition (8.1) from Lemma 8.2. By applying again [37, Lemma 10.5], there exists a point x00∈Sing(∂Ω0) and another sequence rk→0+ such thatHd-d∗+ε(Sing(∂Ω0)∩Brk(x00))≥Crkd-d∗+ε.

Hence, by applying the second blow-up analysis of Sect. 4.3 to the sequences1rkλQ(x0)u0(x00+rkx),λrkv0(x00+rkx)and1rk(Ω0-x00),

we get thatHd-d∗+ε(Sing(∂Ω00)∩B1)≥C,

where v00=Q(x0)u00 and u00 satisfies (2)-(3)-(4)-(5)-(6) in Lemma 4.4. Moreover, by the strong convergence of the blow-up sequence, u00 and v00 still satisfy (8.1), so u00 is a global stable solution of the one-phase problem in the sense of Definition 7.3. Finally, by applying for the last time [37, Lemma 10.5], we deduce that8.5 Hd-d∗+ε(Sing(∂Ω000)∩B1)≥C,

in which Ω000={u000>0} and u000 is a blow-up limit of u00 at some free boundary point x000∈∂Ω00. But now u000 is a 1-homogeneous stable solution of the one-phase problem in the sense of Definition 7.3, so (8.5) contradicts Theorem 7.8. □

Existence of Optimal Sets in Rd and Proof of Theorem 1.6

In this section we prove Theorem 1.6; we prove the existence of an optimal set for (1.4) and then we show how to obtain the regularity of the optimal sets as a consequence from Theorem 1.2.

Statement of the Problem in the Class of Measurable Sets

One can not obtain the existence of optimal sets (for (1.4) or (1.3)) directly in the class of open sets. Thus, we extend the definition of the shape functional to the class of measurable sets. Precisely, if Ω is a Lebesgue measurable set of finite measure in Rd, we define the space H~01(Ω) of all H1(Rd) functions vanishing (Lebesgue-)almost-everywhere outside Ω. We will say that u is a (weak) solution to the problem9.1 -Δu=finΩ,u∈H~01(Ω),

if u∈H~01(Ω) and9.2 ∫Ω∇v·∇udx=∫Ωvfdxfor everyv∈H~01(Ω).

It is immediate to check that u satisfies (9.2) if and only if9.3 12∫Rd|∇u|2dx-∫Ωuf(x)dx≤12∫Rd|∇v|2dx-∫Ωvf(x)dxfor everyv∈H~01(Ω).

From now on, we will denote the unique solution to (9.1) by u~Ω.

We will first prove the following lemma, which in particular implies that for every open set Ω⊂Rd of finite measure and non-negative functions f,g∈L2(Ω), we have∫Ω(-g(x)u~Ω+Q(x))dx≤∫Ω(-g(x)uΩ+Q(x))dx,

where uΩ is the weak solution to (1.1).

Lemma 9.1

Let Ω be a bounded open set in Rd and f∈L2(Ω) a non-negative function. Then9.4 0≤uΩ≤u~Ω,

where u~Ω∈H~01(Ω) is the solution to (9.1) and uΩ∈H01(Ω) is the weak solution to (1.1).

Moreover, if Ω satisfies the following exterior density estimate:9.5 there are constantsr0>0andc>0such thatfor everyx0∈∂Ωand everyr≤r0|Br(x0)\Ω|≥c|Br|,

then H01(Ω)=H~01(Ω) and uΩ=u~Ω.

Proof

First we notice that uΩ and u~Ω are the unique minimizers of the functional (9.3) in H01(Ω) and H~01(Ω). Since f≥0, by testing the optimality of uΩ with uΩ∨0∈H01(Ω) and the optimality of u~Ω with u~Ω∨0∈H~01(Ω), we get that uΩ≥0 and u~Ω≥0 in Ω. Now, a standard argument (see for instance [38, Lemma 2.6]) gives that ΔuΩ+f≥0 on Rd in the sense of distributions, precisely:9.6 -∫Rd∇uΩ·∇φdx+∫Rdφfdx≥0for everyφ≥0,φ∈H1(Rd).

Next, we notice that for every open set Ω, we have the inclusion H01(Ω)⊂H~01(Ω). Thus, if we take φ to be the negative part of u~Ω-uΩ,φ:=-((u~Ω-uΩ)∧0),

we have that φ∈H~01(Ω). So, using the (weak) equation for u~Ω, we get∫Rd∇φ·∇u~Ωdx=∫Ωφf(x)dx

On the other hand, by the positivity of φ and (9.6),∫Rd∇φ·∇uΩdx≤∫Ωφf(x)dx.

Combining the two, we obtain∫Rd|∇φ|2dx=-∫Rd∇φ·∇(u~Ω-uΩ)dx≤0,

which gives that φ≡0 and so (9.4) holds. Finally, it is known (see for instance [17, Proposition 4.7]) that in the presence of the density estimate (9.5), we have that H01(Ω)=H~01(Ω). Thus, the equality uΩ=u~Ω follows by the fact that the minimizer of the functional in (9.3) is unique. □

Existence of Optimal Measurable Sets in Rd

For all measurable set Ω⊂Rd, we setJ~(Ω):=∫Ω-g(x)uΩ+Q(x)dx.

We can prove the following existence result for the minimization of J~.

Lemma 9.2

If d≥3, let f,g∈L2(Rd), while if d=2, let f,g∈L1(R2)∩L∞(R2) be nonnegative functions, and let Q:Rd→R be a measurable function bounded from below by a positive constant cQ>0. Then, the following variational problem has a solution9.7 min{∫Ω(-g(x)u~Ω+Q(x))dx:Ω⊂Rd,Ωmeasurable,|Ω|<+∞}.

Proof

Let d≥3, let Ωn be a minimizing sequence of sets of finite Lebesgue measure for (9.7). We set un:=u~Ωn. By the Poincaré inequality, the equation for un and the Hölder inequality, there is a dimensional constant Cd such that‖un‖L2(Ωn)2≤Cd|Ωn|2/d∫Ωn|∇un|2dx=Cd|Ωn|2/d∫Ωnunfdx≤Cd|Ωn|2/d‖f‖L2(Rd)‖un‖L2(Rd).

Thus,‖un‖L2(Ωn)≤Cd|Ωn|2/d‖f‖L2(Rd),

and so (since we can suppose that J~(Ωn)≤J~(∅)=0), we get0≥J~(Ωn)=∫Ωn(-g(x)un+Q(x))dx≥-Cd|Ωn|2/d‖f‖L2(Rd)‖g‖L2(Rd)+cQ|Ωn|,

which implies that the sequence of measures |Ωn| is bounded9.8 |Ωn|d-2d≤CdcQ‖f‖L2(Rd)‖g‖L2(Rd).

If d=2, to obtain a bound on the measure analogous to (9.8), we need to use a generalized Poincaré-Sobolev (or generalized Faber–Krahn) inequality (see for example [3, equation (1.2)]) instead of the classical one, namely, for 2<q<+∞, there exists a dimensional constant C~2 such that (using also Hölder inequality)‖un‖Lq(Ωn)2≤C~2|Ωn|1+2-qq∫Ωn|∇un|2dx=C~2|Ωn|1+2-qq∫Ωnunfdx≤C~2|Ωn|1+2-qq‖f‖L′(R2)‖un‖Lq(Ωn).

We immediately deduce that9.9 ‖un‖Lq(Ωn)≦C~2|Ωn|1+2-qq‖f‖L′(R2),

and again supposing that J~(Ωn)≤J~(∅)=0, we obtain, using (9.9)0≥J~(Ωn)=∫Ωn(-g(x)un+Q(x))dx≥-C~2|Ωn|1+2-qq‖f‖L′(R2)‖g‖L′(R2)+cQ|Ωn|,

which, since q>2 implies that the sequence of measures |Ωn| is bounded, namely|Ωn|q-2q≤C~2cQ‖f‖L′(R2)‖g‖L′(R2).

Now the proof continues in the same way both for d=2 and d≧3. Using again the equation for un we get that the sequence un is bounded in H1(Rd). Thus, up to pass to a subsequence, we have that un converges weakly in Hloc1(Rd) and strongly in Lloc2(Rd) (and, up to another subsequence, also pointwise a.e. in Rd) to a certain u∈Hloc1(Rd). Now, for all R>0|{u≠0}∩BR|≤lim infn→∞|{un≠0}∩BR|≤lim infn→∞|Ωn∩BR|≤lim infn→∞|Ωn|.

Taking a supremum over R>0, we obtain that|{u≠0}|≦lim infn→∞|Ωn|,

and as a consequence (using Fatou Lemma),∫{u≠0}Q(x)dx≦lim infn→+∞∫ΩnQ(x)dx,

Then one can prove also that u∈H1(Rd)∩H~01({u≠0}), as, for all R>0,∫BR|∇u|2dx≦lim infn→+∞∫BR∩Ωn|∇un|2dx≦lim infn→+∞∫Ωn|∇un|2dx,

and then we can take the supremum over R>0. We note that from the pointwise a.e. convergence and the fact that un≧0 (since f≧0), we deduce u≧0. As a consequence, using also that g≧0, for all R>0 we have-∫{u≠0}gudx≦-∫{u≠0}∩BRgudx≦lim infn→+∞-∫Ωn∩BRgundx.

We then conclude that Ω={u≠0} is a solution to (9.7). □

Lemma 9.3

Let d≥2, f,g∈L1(Rd)∩L∞(Rd) be non-negative, and let Q:Rd→R be a measurable function bounded from below by a positive constant cQ>0. If the measurable set Ω⊂Rd, |Ω|<+∞, is a solution to (9.7), then Ω is bounded.

Proof

For any set E⊂Rd of finite measure and any function h∈L2(E), we will denote by R~E(h) the unique solution to the problem-Δu=hinEu∈H~01(E).

We have that R~E is linear, R~E(h1+h2)=R~E(h1)+R~E(h2) and positive: if h≥0, then R~E(h)≥0. Moreover, by the weak maximum principle, if E1⊂E2 and h≥0, then R~E2(h)≥R~E1(h).

Let ω be any measurable set contained in Ω. Then, the minimality of Ω gives that∫Ω(-g(x)R~Ω(f)+Q(x))dx≤∫ω(-g(x)R~ω(f)+Q(x))dx.

So, rearranging the terms and using the positivity of f and g, and the inequality0≤R~Ω(f)-R~ω(f)≤‖f‖L∞(R~Ω(1)-R~ω(1)),

we getcQ|Ω\ω|≤∫ΩQ(x)dx-∫ωQ(x)dx≤∫Ωg(x)(R~Ω(f)-R~ω(f))dx≤‖f‖L∞∫Ωg(x)(R~Ω(1)-R~ω(1))dx=‖f‖L∞∫Ω(R~Ω(g)-R~ω(g))dx≤‖f‖L∞‖g‖L∞∫Ω(R~Ω(1)-R~ω(1))dx.

Finally, rearranging the terms again, we get that-12∫ΩR~Ω(1)dx+cQ2‖f‖L∞‖g‖L∞|Ω|≤-12∫ωR~ω(1)dx+cQ2‖f‖L∞‖g‖L∞|ω|,

so the set Ω is inwards minimizing (or, in terms of [9], a shape subsolution) for the functionalE(Ω)=-12∫ΩR~Ω(1)dx+cQ2‖f‖L∞‖g‖L∞|Ω|.

Thus, applying [9, Theorem 3.13], for every0<η≤cQ2‖f‖L∞‖g‖L∞,

the set Ω is contained in an open set A⊂Rd obtained as a finite union of N balls Bρ(xi), i=1,⋯,N, with N and ρ depending only on the dimension d and η. In particular, A is bounded and the diameter of any connected component of A is at most Nρ. □

Proof of Theorem 1.6

The existence of an optimal set is a consequence of Proposition 9.4 below, while the regularity follows from Theorem 1.2.

Proposition 9.4

In Rd, d≥2, let f,g,Q:Rd→R be non-negative functions. Suppose that: f,g∈L∞(Rd)∩L1(Rd)∩C(Rd) and that f>0 and g>0 on Rd ;

there is a constant cQ>0 such that cQ≤Q on Rd.

Then, there exists a solution Ω⊂Rd to the shape optimization problem (1.4). Moreover, every solution to (1.4) is also a solution to (9.7).

Proof

By Lemma 9.2, there is a measurable set Ω (of finite measure) that minimizes (9.7). By Lemma 9.3, Ω is contained in some ball BR⊂Rd. Since f and g are continuous and strictly positive on B¯R, we can find positive constants C1,C2 such that C1g≤f≤C2g on B¯R. Thus, reasoning as in Proposition 3.1 and Corollary 3.3 we get that the setA:={u~Ω>0}

is open. Now, the optimality of Ω gives that |ΩΔA|=0, so we have u~A=u~Ω. Moreover, by Proposition 3.5, A satisfies the exterior density estimate from Lemma 9.1. Thus, uA=u~A. In order to check that the open set A minimizes (1.4), we notice that, for any open set E⊂Rd,∫A(-g(x)uA+Q(x))dx=∫A(-g(x)u~A+Q(x))dx≤∫E(-g(x)u~E+Q(x))dx≤∫E(-g(x)uE+Q(x))dx.

Moreover, by the same chain of inequalities, we obtain that if the open set E is a solution to (1.4), then it also minimizes (9.7). □

Appendix A. An Optimization Problem in Heat Conduction

As a consequence of our analysis from Sects. 2, 4 and 7, we obtain an estimate on the dimension of the singular part of the boundary of the optimal sets arising in a heat conduction problem studied by Aguilera, Caffarelli and Spruck in [1].

A.1 Statement of the Problem and Known Results

Let D be a smooth bounded open set in Rd. Let ϕ:∂D→R be a smooth function on ∂D bounded from below and above by positive constants 0<c≤C. For every open set Ω⊂D, for which the set K:=D\Ω is a compact subset of D, we consider the state function uΩ∈H1(D) solution to the problemA.1 ΔuΩ=0inΩ,uΩ=ϕon∂D,uΩ≡0onK=D\Ω,

and we define the functionalF(Ω)=∫∂D∂uΩ∂νdHd-1+Λ|Ω|

where Λ>0 is a real number and ν is the outer unit normal to ∂D. In [1] it was shown that there is a solution to the following shape optimization problem:A.2 min{F(Ω)Ω⊂D;Ω-open;D\Ω-compact subset of\ D},

and that for every solution Ω to (A.2) the following holds.The state function uΩ is Lipschitz continuous in D and smooth in a neighborhood of the fixed boundary ∂D.

uΩ is non-degenerate in the sense that there is a constant η>0, for which 1rd-1∫∂Br(x)uΩdHd-1≥ηrwheneverx∈Ω¯andBr(x)⊂D.

There is ε>0 such that, for every x on the boundary of the set K:=D\Ω, we have: ε0|Br|≤|Ω∩Br(x)|≤(1-ε0)|Br|wheneverBr(x)⊂D.

There is a constant M~>0 such that, for every B2r(x0)⊂D, we have the bound 0≤|∫B2r(x0)∇uΩ·∇φdx|≤M~‖φ‖L∞(B2r(x0))for everyφ∈Cc0,1(B2r(x0)),

where Cc0,1(B2r(x0)) is the space of Lipschitz functions with compact support in B2r(x0).

Then, they showed that the reduced boundary ∂∗Ω∩D is C∞ smooth and thatA.3 |∇uΩ||∇vΩ|=Λon∂∗Ω∩D,

where vΩ is the solution to the PDEA.4 ΔvΩ=0inΩ,vΩ=1on∂D,vΩ≡0onK=D\Ω.

In Theorem A.4, we will improve this result in low dimension (d≥4) by showing that the whole free boundary ∂Ω∩D is smooth.

Remark A.1

In the problem originally considered by Aguilera, Caffarelli and Spruck in [1], the minimization of F is among all domains with prescribed measure. Thus, all the results from [1] apply to (A.2) with the only difference that the constant Λ from (A.3) is an unknown positive Lagrange multiplier. In this last section of the present paper, we choose to work with the penalized version (A.2) since it allows to apply all the results from Sects. 2, 7 and 8 directly, without the need to add technical details related to the Lagrange multiplier.

A.2 First and Second Variation

By an integration by parts, we can rewrite F(Ω) asF(Ω):=∫Ω∇uΩ·∇vΩdx+Λ|Ω|,

where uΩ and vΩ are given by (A.1) and (A.4). Now, let ξ∈Cc∞(D;Rd) be a smooth compactly supported vector field in D; let Φt:D→D, t∈R, be the flow associated to ξ and let Ωt:=Φt(Ω). Reasoning as in Sect. 7, we get that∂∂t|t=0F(Ωt)=∫Ω(∇uΩ·(δA)∇vΩ+Λdivξ)dx,12∂2∂t2|t=0F(Ωt)=∫Ω(∇uΩ·(δ2A)∇vΩ-∇(δu)·∇(δv)+Λ2((divξ)2+ξ·∇(divξ)))dx,

where δA and δ2A are given by (2.6), and where δu and δv are the solutions to the PDEs-Δ(δu)=div((δA)∇uΩ)inΩ,δu∈H01(Ω);-Δ(δv)=div((δA)∇vΩ)inΩ,δv∈H01(Ω).

Using the blow-up analysis from Sect. 4 and the argument from Lemma 8.2, we get

Lemma A.2

Let Ω be an optimal set for (A.2) and let u:=uΩ and v:=vΩ be the state functions from (A.1) and (A.4). Suppose that x0∈∂Ω∩D and that rk→0 are such thatlimk→∞ux0,rk=u0andlimk→∞vx0,rk=v0

where u0,v0 are non-negative Lipschitz functions on Rd and both limits are locally uniform in Rd, and where, as usual, we set:ux0,rk(x):=1rku(x0+rkx)andvx0,rk(x):=1rkv(x0+rkx).

Then, u0 and v0 are proportional and there is a constant λ>0 such that λu0 is a global stable solution of the one-phase Bernoulli problem in the sense of Definition 7.3.

Proof

By the analysis of the blow-up sequences from Sect. 4, we get that ux0,rk and vx0,rk converge strongly in Hloc1(Rd) respectively to u0 and v0, and also that the sets 1rk(Ω-x0) converge (in the local Hausdorff distance) to Ω0={u0>0}={v0>0}. Then, as in Lemma 8.2, for every smooth compactly supported vector field ξ∈Cc∞(Rd;Rd), the first and the second variation of F pass to the limit, that is,A.5 ∫Ω0(∇u0·(δA)∇v0+Λdivξ)dx=0,∫Ω0(∇u0·(δ2A)∇v0-∇(δu0)·∇(δv0)+Λ2((divξ)2+ξ·∇(divξ)))dx≧0,

where δA and δ2A are as in (2.6), and where δu0 and δv0 solve the PDE (see Sect. 7.1)-Δ(δu0)=div((δA)∇u0)inΩ0,δu0∈H˙01(Ω0);-Δ(δv0)=div((δA)∇v0)inΩ0,δv0∈H˙01(Ω0).

On the other hand, since by [1] the Boundary Harnack Principle holds on the optimal set Ω, we get that the blow-up limits are proportional, that is, v0=Cu0 for some C>0. Finally, we notice that, up to multiplying u0 by a constant, the inequalities (A.5) correspond precisely to the stability condition (7.8) in Definition 7.3, which concludes the proof of the lemma. □

A.3 Main Theorem

Let now Ω be an optimal set for (A.2). We define the regular and the singular parts of the free boundary ∂Ω∩D as follows. The regular part, Reg(∂Ω), is the set of points x0∈∂Ω∩D at which there exists a blow up limit u0,v0:Rd→R of the formu0(x)=α(x·ν)+andv0(x)=β(x·ν)+,

for some unit vector ν∈Rd and some constants α>0 and β>0 such that αβ=Λ. The remaining part of the free boundary is the singular set:Sing(∂Ω):=(∂Ω∩D)\Reg(∂Ω).

Then, in terms of the decomposition into ∂Ω∩D=Reg(∂Ω)∪Sing(∂Ω), we can rewrite the regularity theorem of Aguilera–Caffarelli–Spruck (for the penalized problem) as follows:

Theorem A.3

(Aguilera–Caffarelli–Spruck [1]) If Ω is a solution to (A.2) in a domain D⊂Rd, then Reg(∂Ω) is a relatively open subset of ∂Ω∩D and (locally) a C∞ manifold.

For what concerns the singular set, by reasoning as in Theorem 8.1 and by applying Theorems 7.9 and 7.8, we obtain the following result, which in particular implies that in the physically relevant dimension d=3 (and also in d=2 and d=4) the free boundary ∂Ω∩D of a solution Ω to (A.2) is C∞ smooth.

Theorem A.4

(Dimension of the singular set for solutions to (A.2)) If Ω is a solution to (A.2) in a domain D⊂Rd, then the following holds: (i) if d<d∗, then Sing(∂Ω)=∅;

(ii) if d≥d∗, then the Hausdorff dimension of Sing(∂Ω) is at most d-d∗;

where d∗ is the critical dimension from Definition 7.7.

Acknowledgements

G.T. and B.V. are supported by the European Research Council’s (ERC) project n.853404 ERC VaReg - Variational approach to the regularity of the free boundaries, financed by the program Horizon 2020. G.B., F.P.M., D.M. and G.T. are members of INDAM-GNAMPA. D.M. has been partially supported by the MIUR-PRIN Grant 2020F3NCPX “Mathematics for industry 4.0 (Math4I4)”. D.M. and B.V. have been partially supported by the MIUR-PRIN Grant 2022R537CS “NO3” granted by the European Union - Next Generation EU. D.M. and G.T. have been partially supported by the INDAM-GNAMPA project CUP_E53C22001930001.

Funding

Open access funding provided by UniversitÃ di Pisa within the CRUI-CARE Agreement.

Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Declarations

Conflict of interest

The authors have no conflict of interest to declare.

1 Precisely, since G(u)=∞ as soon as u has positivity set of infinite measure, we will work with a localized version of G, in which the integration is over compact sets.

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==== Refs
References

1. Aguilera NE Caffarelli LA Spruck J An optimization problem in heat conduction Ann. Scuola Norm. Sup. Pisa Cl. Sci. 1987 14 3 355 387
Aguilera, N.E., Caffarelli, L.A., Spruck, J.: An optimization problem in heat conduction. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14(3), 355–387, 1987
2. Alt HW Caffarelli LA Existence and regularity for a minimum problem with free boundary J. Reine Angew. Math. 1981 325 105 144
Alt, H.W., Caffarelli, L.A.: Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105–144, 1981
3. Brasco L Mazzoleni D On principal frequencies, volume and inradius in convex sets NoDEA Nonlinear Differ. Equ. Appl. 2020 27 2 26 10.1007/s00030-019-0614-2
Brasco, L., Mazzoleni, D.: On principal frequencies, volume and inradius in convex sets. NoDEA Nonlinear Differ. Equ. Appl. 27(2), 26, 202010.1007/s00030-019-0614-2
4. Briançon T Lamboley J Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints Ann. Inst. H. Poincaré Anal. Non Linéaire 2009 26 4 1149 1163 10.1016/j.anihpc.2008.07.003
Briançon, T., Lamboley, J.: Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(4), 1149–1163, 200910.1016/j.anihpc.2008.07.003
5. Bucur D Minimization of the k-th eigenvalue of the Dirichlet Laplacian Arch. Ration. Mech. Anal. 2012 206 3 1073 1083 10.1007/s00205-012-0561-0
Bucur, D.: Minimization of the -th eigenvalue of the Dirichlet Laplacian. Arch. Ration. Mech. Anal. 206(3), 1073–1083, 201210.1007/s00205-012-0561-0
6. Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations, vol. 65. Birkhäuser Verlag, Basel (2005)
7. Bucur, D., Lamboley, J., Nahon, M., Prunier, R.: Sharp quantitative stability of the Dirichlet spectrum near the ball. arXiv:2304.10916.
8. Bucur D Mazzoleni D Pratelli A Velichkov B Lipschitz regularity of the eigenfunctions on optimal domains Arch. Ration. Mech. Anal. 2015 216 1 117 151 10.1007/s00205-014-0801-6
Bucur, D., Mazzoleni, D., Pratelli, A., Velichkov, B.: Lipschitz regularity of the eigenfunctions on optimal domains. Arch. Ration. Mech. Anal. 216(1), 117–151, 201510.1007/s00205-014-0801-6
9. Bucur D Velichkov B Multiphase shape optimization problems SIAM J. Control Optim. 2014 52 6 3556 3591 10.1137/130917272
Bucur, D., Velichkov, B.: Multiphase shape optimization problems. SIAM J. Control Optim. 52(6), 3556–3591, 201410.1137/130917272
10. Buttazzo G Dal Maso G An existence result for a class of shape optimization problems Arch. Ration. Mech. Anal. 1993 122 183 195 10.1007/BF00378167
Buttazzo, G., Dal Maso, G.: An existence result for a class of shape optimization problems. Arch. Ration. Mech. Anal. 122, 183–195, 199310.1007/BF00378167
11. Buttazzo G Maiale FP Velichkov B Shape optimization problems in control form Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 2021 32 3 413 435 10.4171/rlm/942
Buttazzo, G., Maiale, F.P., Velichkov, B.: Shape optimization problems in control form. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32(3), 413–435, 202110.4171/rlm/942
12. Buttazzo G Shrivastava H Optimal shapes for general integral functionals Ann. H. Lebesgue 2020 3 261 272 10.5802/ahl.31
Buttazzo, G., Shrivastava, H.: Optimal shapes for general integral functionals. Ann. H. Lebesgue 3, 261–272, 202010.5802/ahl.31
13. Buttazzo G Velichkov B A shape optimal control problem with changing sign data SIAM J. Math. Anal. 2018 50 3 2608 2627 10.1137/17M1126564
Buttazzo, G., Velichkov, B.: A shape optimal control problem with changing sign data. SIAM J. Math. Anal. 50(3), 2608–2627, 201810.1137/17M1126564
14. Caffarelli LA Jerison D Kenig CE Global energy minimizers for free boundary problems and full regularity in three dimensions Contemp. Math. 2004 350 83 97 10.1090/conm/350/06339
Caffarelli, L.A., Jerison, D., Kenig, C.E.: Global energy minimizers for free boundary problems and full regularity in three dimensions. Contemp. Math. 350, 83–97, 200410.1090/conm/350/06339
15. Caffarelli LA Shahgholian H Yeressian K A minimization problem with free boundary related to a cooperative system Duke Math. J. 2018 167 10 1825 1882 10.1215/00127094-2018-0007
Caffarelli, L.A., Shahgholian, H., Yeressian, K.: A minimization problem with free boundary related to a cooperative system. Duke Math. J. 167(10), 1825–1882, 201810.1215/00127094-2018-0007
16. De Philippis G Spolaor L Velichkov B Regularity of the free boundary for the two-phase Bernoulli problem Invent. Math. 2021 225 347 394 10.1007/s00222-021-01031-7
De Philippis, G., Spolaor, L., Velichkov, B.: Regularity of the free boundary for the two-phase Bernoulli problem. Invent. Math. 225, 347–394, 202110.1007/s00222-021-01031-7
17. De Philippis G Velichkov B Existence and regularity of minimizers for some spectral optimization problems with perimeter constraint Appl. Math. Optim. 2014 69 2 199 231 10.1007/s00245-013-9222-4
De Philippis, G., Velichkov, B.: Existence and regularity of minimizers for some spectral optimization problems with perimeter constraint. Appl. Math. Optim. 69(2), 199–231, 201410.1007/s00245-013-9222-4
18. De Silva D Free boundary regularity for a problem with right hand side Interfaces Free Bound. 2011 13 2 223 238 10.4171/ifb/255
De Silva, D.: Free boundary regularity for a problem with right hand side. Interfaces Free Bound. 13(2), 223–238, 201110.4171/ifb/255
19. De Silva D Jerison D A singular energy minimizing free boundary J. Reine Angew. Math. 2009 635 1 21 10.1515/CRELLE.2009.074
De Silva, D., Jerison, D.: A singular energy minimizing free boundary. J. Reine Angew. Math. 635, 1–21, 200910.1515/CRELLE.2009.074
20. Gustafsson B Shahgholian H Existence and geometric properties of solutions of a free boundary problem in potential theory J. Reine Angew. Math. 1995 473 137 180
Gustafsson, B., Shahgholian, H.: Existence and geometric properties of solutions of a free boundary problem in potential theory. J. Reine Angew. Math. 473, 137–180, 1995
21. Henrot A Pierre M Shape Variation and Optimization. A Geometrical Analysis. EMS Tracts in Mathematics 2018 Zürich European Mathematical Society (EMS)
Henrot, A., Pierre, M.: Shape Variation and Optimization. A Geometrical Analysis. EMS Tracts in Mathematics, vol. 28. European Mathematical Society (EMS), Zürich (2018)
22. Jerison D Savin O Some remarks on stability of cones for the one phase free boundary problem Geom. Funct. Anal. 2015 25 1240 1257 10.1007/s00039-015-0335-6
Jerison, D., Savin, O.: Some remarks on stability of cones for the one phase free boundary problem. Geom. Funct. Anal. 25, 1240–1257, 201510.1007/s00039-015-0335-6
23. Kamburov N Nondegeneracy and stability in the limit of a one-phase singular perturbation problem Discrete Contin. Dyn. Syst. Ser. A 2023 43 12 4328 4360 10.3934/dcds.2023089
Kamburov, N.: Nondegeneracy and stability in the limit of a one-phase singular perturbation problem. Discrete Contin. Dyn. Syst. Ser. A 43(12), 4328–4360, 202310.3934/dcds.2023089
24. Kinderlehrer D Nirenberg L Regularity in free boundary problems Ann. Scuola Norm. Sup. Pisa 1977 4 373 391
Kinderlehrer, D., Nirenberg, L.: Regularity in free boundary problems. Ann. Scuola Norm. Sup. Pisa 4, 373–391, 1977
25. Kriventsov D Lin F Regularity for shape optimizers: the degenerate case Commun. Pure Appl. Math. 2019 72 8 1678 1721 10.1002/cpa.21810
Kriventsov, D., Lin, F.: Regularity for shape optimizers: the degenerate case. Commun. Pure Appl. Math. 72(8), 1678–1721, 201910.1002/cpa.21810
26. Kriventsov D Lin F Regularity for shape optimizers: the nondegenerate case Commun. Pure Appl. Math. 2018 71 8 1535 1596 10.1002/cpa.21743
Kriventsov, D., Lin, F.: Regularity for shape optimizers: the nondegenerate case. Commun. Pure Appl. Math. 71(8), 1535–1596, 201810.1002/cpa.21743
27. Kukuljan T Higher order parabolic boundary Harnack inequality in C1 and Ck,α domains Discrete Contin. Dyn. Syst. 2022 42 6 2667 2698 10.3934/dcds.2021207
Kukuljan, T.: Higher order parabolic boundary Harnack inequality in and domains. Discrete Contin. Dyn. Syst. 42(6), 2667–2698, 202210.3934/dcds.2021207
28. Maiale FP Tortone G Velichkov B Epsilon-regularity for the solutions of a free boundary system Rev. Mat. Iberoam. 2023 39 5 1947 1972 10.4171/rmi/1430
Maiale, F.P., Tortone, G., Velichkov, B.: Epsilon-regularity for the solutions of a free boundary system. Rev. Mat. Iberoam. 39(5), 1947–1972, 202310.4171/rmi/1430
29. Maiale, F.P., Tortone, G., Velichkov, B.: The boundary Harnack principle on optimal domains. Ann. Sc. Norm. Super. Pisa Cl. Sci., 25 (1), 127-149, 2024
30. Mazzoleni D Pratelli A Existence of minimizers for spectral problems J. Math. Pures Appl. 2013 100 3 433 453 10.1016/j.matpur.2013.01.008
Mazzoleni, D., Pratelli, A.: Existence of minimizers for spectral problems. J. Math. Pures Appl. 100(3), 433–453, 201310.1016/j.matpur.2013.01.008
31. Mazzoleni D Terracini S Velichkov B Regularity of the optimal sets for some spectral functionals Geom. Funct. Anal. 2017 27 373 426 10.1007/s00039-017-0402-2
Mazzoleni, D., Terracini, S., Velichkov, B.: Regularity of the optimal sets for some spectral functionals. Geom. Funct. Anal. 27, 373–426, 201710.1007/s00039-017-0402-2
32. Mazzoleni D Trey B Velichkov B Regularity of the optimal sets for the second Dirichlet eigenvalue Ann. Inst. H. Poincaré Anal. Non Linéaire 2022 39 3 529 573 10.4171/aihpc/14
Mazzoleni, D., Trey, B., Velichkov, B.: Regularity of the optimal sets for the second Dirichlet eigenvalue. Ann. Inst. H. Poincaré Anal. Non Linéaire 39(3), 529–573, 202210.4171/aihpc/14
33. Mazzoleni, D., Tortone, G., Velichkov, B.: On the dimension of the singular set in optimization problems with measure constraint. J. Convex Anal. 31(2), 689-708, 2024
34. Russ E Trey B Velichkov B Existence and regularity of optimal shapes for elliptic operator with drift Calc. Var. Partial Differ. Equ. 2019 58 6 199 10.1007/s00526-019-1653-6
Russ, E., Trey, B., Velichkov, B.: Existence and regularity of optimal shapes for elliptic operator with drift. Calc. Var. Partial Differ. Equ. 58(6), 199, 201910.1007/s00526-019-1653-6
35. Spolaor L Velichkov B An epiperimetric inequality for the regularity of some free boundary problems: the 2-dimensional case Commun. Pure Appl. Math. 2019 72 2 375 421 10.1002/cpa.21785
Spolaor, L., Velichkov, B.: An epiperimetric inequality for the regularity of some free boundary problems: the -dimensional case. Commun. Pure Appl. Math. 72(2), 375–421, 201910.1002/cpa.21785
36. Terracini, S., Tortone, G., Vita, S.: Higher order boundary Harnack principle via degenerate equations. Arch. Ration. Mech. Anal., 248(2), 29 2024.
37. Velichkov, B.: Existence and regularity results for some shape optimization problems. Edizioni della Normale 19, Pisa, 2015
38. Velichkov, B.: Regularity of the one-phase free boundaries. Lecture Notes of the Unione Matematica Italiana, Springer Cham, 2023. 10.1007/978-3-031-13238-4
39. Weiss GS Partial regularity for a minimum problem with free boundary J. Geom. Anal. 1999 9 2 317 326 10.1007/BF02921941
Weiss, G.S.: Partial regularity for a minimum problem with free boundary. J. Geom. Anal. 9(2), 317–326, 199910.1007/BF02921941
