Characterization of the subdifferential and minimizers for the anisotropic p-capacity

We obtain existence of minimizers for the $p$-capacity functional defined with respect to a centrally symmetric anisotropy for $1<p<\infty$, including the case of a crystalline norm in $\mathbb R^N$. The result is obtained by a characterization of the corresponding subdifferential and it applies for unbounded domains of the form $\mathbb R^N \setminus \overline{\Omega}$ under mild regularity assumptions (Lipschitz-continuous boundary) and no convexity requirements on the bounded domain $\Omega$. If we further assume an interior ball condition (where the Wulff shape plays the role of a ball), then any minimizer is shown to be Lipschitz continuous.


Introduction and statement of main results
1.1.The p-capacity and its anisotropic version.For N ≥ 2, the pcapacity of a given compact set K ⊂ R N is defined as where • denotes the Euclidean norm and C ∞ c are smooth functions with compact support.The relevance of this quantity comes from its geometric meaning, since for p > 1 the usual geometric functionals like area or volume do not provide a satisfactory depiction of properties of domains.Here is where inequalities involving the p-capacity naturally arise.
From the viewpoint of partial differential equations, the search for minimizers of the p-capacity, called p-capacitary functions or equilibrium potentials, has attracted a lot of attention due to its relation to the p-Laplace equation.In fact, when 1 < p < N and under suitable smoothness of the boundary of K, a unique p-capacitary function, u ∈ L pN N −p (R N ) and ∇u ∈ L p (R N ; R N ), exists and it satisfies the Euler-Lagrange equation The purpose of this paper is to obtain existence of p-capacitary functions within anisotropic media, with regularity assumptions so mild as to include the case of a crystalline anisotropy.It is crucial to move out of classical Euclidean isotropic settings to allow the possibility of environments where properties differ with the direction, since this extra flexibility will be key for applications to crystal growth or noise removal.
Accordingly, replacing the Euclidean norm in the definition of the pcapacity functional by a generic norm F in R N , leads to the anisotropic p-capacity functional: In the range 1 < p < N , existence and uniqueness of minimizers (called anisotropic p-capacitary functions) have been shown under different conditions on the anisotropy F (most of the results deal with C ∞ (R N \ {0}) uniformly elliptic norms) and the domain K.Moreover, in these cases the minimizer is the unique solution to the following PDE: Our main goal is to find a similar characterization of minimizers of the anisotropic p-capacity by relaxing the assumptions on regularity of F and K as much as possible, and allowing a non-necessarily constant boundary condition.To state our main result in this direction, we need to introduce some definitions and notation.For a continuous norm F in R N , we consider the energy functional defined by +∞, otherwise. (1.2) Notice that, since the domain of the above functional is bounded and F p is convex and coercive, [18, Theorem 5 in section 8.2.4] ensures the existence of a minimizer.However, since F (and therefore F p ) is not required to be strictly convex, uniqueness is not anymore guaranteed.
Our objective is to characterize the minimizers of this energy functional by means of the corresponding Euler-Lagrange equation (notice that here we are not allowed to write ∇F as in (1.1)).Since F Ω,ϕ is a proper and convex functional, we have that u ∈ arg min {F Ω,ϕ } if, and only if, 0 ∈ ∂F Ω,ϕ (u), (1.3) where the latter stands for the subdifferential of the energy functional, whose exact expression (and accordingly the concrete Euler-Lagrange equation for this problem) is unknown and does not follow from standard arguments, due to the non smoothness of the integrand.Therefore, our first goal is to characterize the subdifferential; more precisely, we get Here F • denotes the dual of the norm F .
The strategy and methods carried out to prove Theorem 1.1, suitably adapted to work with homogeneous Sobolev spaces, permit to obtain the corresponding characterization for unbounded exterior domains of the form R N \ Ω (see Theorem 4.1), where Ω ⊂ R N is again a bounded domain with Lipschitz boundary.As a byproduct of this result, we get a characterization for all minimizers of Cap F p (Ω) for bounded Lipschitz domains Ω.
then any minimizer u of the energy functional F D,ϕ is a weak solution of Here the equality on the boundary is understood in the sense of traces.In particular, if ϕ ≡ 1, then and hence anisotropic p-capacitary functions are solutions to (1.4).
Up to our knowledge, the most general result in this spirit can be found in [11] (see also [2]).The authors show existence, uniqueness and regularity of a p-capacitary function u ∈ C 2,α (R N \ K) in the case that K is convex and with boundary of class C 2,α for norms F ∈ C 2,α (R N \ {0}) such that F p is twice continuously differentiable in R N \ {0} with a strictly positive definite Hessian matrix.
Here we generalize the results in the previous literature in several directions.Indeed, regarding the anisotropy, we only require it to be a norm, without any extra assumptions on smoothness or uniform ellipticity.In particular, we allow norms whose dual unit balls have corners and/or straight segments, including crystalline cases as the ∞ or 1 norms in R N .
Secondly, we study the Dirichlet problem both in bounded domains with Lipschitz boundary and in exterior domains, as well as for a generic Dirichlet boundary constraint.Moreover, we do not require that the set K is convex and, about its regularity, we just ask for Lipschitz continuity of the boundary, instead of the traditional C 2,α much stronger constraint.Lipschitz cannot be further weakened because it is the milder condition that guarantees e.g. that the trace operator is surjective on the fractional Sobolev space W In short, we will be working in an unfriendly setting in the sense that we cannot perform any argument that involves second derivatives of the functions u nor principal curvatures (even in weak sense) of ∂K.To overcome these additional technical difficulties, the proof of Theorem 1.1 strongly relies on the theory of maximal monotone operators in Banach spaces as in the case of p = 1, previously studied in [30].
First, we associate a possibly multivalued operator A ϕ : L p (Ω) → L p (Ω) to item (b) in Theorem 1.1.In order to ensure that this operator coincides with ∂F Ω,ϕ , we show that A ϕ ⊆ ∂F Ω,ϕ and that both are maximal monotone (see Theorem 3.10).The maximal monotonicity of A ϕ will be proved by verifying that the range condition L p (Ω) = R(J L p + A ϕ ) holds (cf.Proposition 3.8), with J L p being the duality mapping.The latter, in turn, needs an approximation process with a sequence of coercive, monotone and weakly continuous operators defined on W 1,p ϕ (Ω).For the continuity, we will approximate the norm F by its Moreau-Yosida approximation while for the coercivity, we will add a p-Laplacian term.Suitable a-priori estimates and the use of the Minty-Browder technique (see Lemma 3.7) will permit to pass to the limit, first in the Yosida regularization and then in the p-Laplacian to finally achieve that the range condition holds.1.3.Construction of minimizers with extra properties.We also show that there exists one p-capacitary function which is trapped between two explicit solutions of the Euler-Lagrange equations.By translation invariance, we can suppose that 0 ∈ Ω and, since Ω is bounded, we can find 0 < r 1 < r 2 such that W r 1 ⊂ Ω ⊂ W r 2 , with W r being the Wulff shape with radius r; i.e., the ball of radius r with respect to the dual norm of F .Then, there are three p-capacitary functions u, u r 1 , u r 2 , minimizers of Cap F p (Ω), Cap F p (W r 1 ) and Cap F p (W r 2 ), respectively, such that u r 1 ≤ u ≤ u r 2 .Moreover, u r 1 and u r 2 are explicitly given as follows: (c) If F is required to be strictly convex, then the minimizer is unique.
In order to prove the above result, we need to approximate p-capacitary functions with minimizers of the relative capacity with respect to a ball.Recall that, in the anisotropic framework, Wulff shapes play the role of balls.Therefore, for Ω ⊂ W R , we consider which is also known as the anisotropic p-capacity of the condenser (Ω; W R ) or the condenser anisotropic p-capacity of the obstacle Ω in the bounded domain W R .Notice that the unique minimizer clearly satisfies that u| Ω ≡ 1, so we actually work within the annular domain Ω R := W R \ Ω, and try to let R → ∞, as in the classical isotropic setting [23].
If we assume that F is strictly convex, a comparison argument directly yields two barriers v r 1 ,R (lower) and v r 2 ,R (upper) into which the minimizer u R to Cap F p (Ω; W R ) is trapped.Since u R are shown to be increasing with respect to R, we can pass to the limit when R → ∞ and we get the result.
In the case that F is not a strictly convex norm, we need to approximate it with a sequence of strictly convex norms in a uniform way; to obtain uniform bounds on the barriers and then finish the argument as in the strictly convex case.Notice that, under this generality, one does not expect to get uniqueness, as this does not happen in the extremal case p = 1 (see Appendix A for an example with infinitely many minimizers for p = 1).
1.4.Regularity of minimizers.We complete the paper with the study of the regularity of minimizers.We show that all minimizers both of the relative p-capacity with respect to a Wulff shape and of the anisotropic p-capacity are Lipschitz continuous, provided that the domain is regular enough in the following sense: Definition 1.4 (Uniform interior ball condition).Let r > 0. We say that Ω satisfies the W r -condition if, for any x ∈ ∂Ω, there exists This condition is milder than F -regularity for non-convex domains (see e.g. the discussion in Lemma 2.8 and Remark 2.9 of [14]).In this setting, we conclude Theorem 1.5.Let r > 0 and suppose that Ω satisfies the W r -condition.
Then any minimizer of Cap F p (Ω; W R ) is Lipschitz continuous.Moreover, any minimizer of the energy functional F R N \Ω,1 is also Lipschitz continuous.
We point out that this is the best expected regularity since the explicit solutions u r in Theorem 1.3 are only Lipschitz continuous in the case that F does not have any extra regularity assumption.The proof requires an application of our comparison arguments, which is trickier than for the previous theorem, as we need that the upper and lower barrier coincide on the boundary, in order to exploit a regularity result from [25].
1.5.Geometric and physical meaning of p-capacity.As it is the main character of this paper, let us talk briefly about the physical and geometric relevance of p-capacity.Physically speaking, Cap 2 (K) measures the total electric charge flowing into R N \ K across the boundary ∂K.But this interpretation does not restrict to electric charges, it can also be applied to heat transfer or even fluid flow through a porous medium.Indeed, the problems studied above can be interpreted as the steady states of such flows.In the classical case (p = 2), Ohm's law says that the electric current is driven by the field J = −c∇u, where u is the corresponding p-capacitary potential, and c denotes the conductivity.But all the physical laws (Ohm, Fourier or Darcy) governing the aforementioned flows are empirical and linearity is just a simplifying assumption, hence the next level of complexity should consider flows driven by J = −c ∇u p−2 ∇u for p > 1, which has already been studied in the context of turbulent flows and deformation plasticity (pioneering works in this direction are [31,5]).
As suggested by Pólya in [32], the thermal analogy works as a source of geometric intuition: take a body (e.g. a cat) within a uniform infinite medium whose temperature vanishes at infinity, while the skin of the cat is kept at a constant temperature (that we normalize to 1).Then the thermal conductance (quantity of outgoing heat per time unit) is, up to a constant which depends on the nature of the ambient, equal to the electrostatic capacity of the cat.In addition, we have all noticed that, to protect themselves from the cold, cats tend to curl up in a ball; this happens in order to minimize the thermal conductance or, equivalently, their capacity.This can be formalized by means of isocapacitary inequalities (see [26]), telling that, among all sets with fixed volume, balls minimize p-capacity, i.e., where r is such that the Lebesgue measure of B r coincides with that of K. Equality holds if, and only if, K = B r , up to a set of zero p-capacity.There are further interesting characterizations of balls as the equality case of Minkowski type inequalities, which relate suitable powers of the p-capacity and integrals involving a p-power of the mean curvature of ∂K (cf.[1]).
Anisotropic generalizations of the above interpretations come naturally by considering bodies embedded in non-uniform media.For the corresponding anisotropic inequalities, see [24] and the references therein.
1.6.Structure of the paper.The paper is organized as follows.We first introduce in section 2 the basic background material about anisotropies, maximal monotone theory in Banach spaces and Yosida regularization in Hilbert ambients, while section 3 gathers all the approximation arguments needed to prove Theorem 1.1.These include estimates for the Moreau-Yosida approximation of F p and the Yosida approximation of ∂F p (Lemma 3.5), as well as a result in the spirit of Minty-Browder (Lemma 3.7) giving a sufficient condition for elements to belong to ∂F p .Then in section 4 we introduce the technical machinery of homogeneous Sobolev spaces to extend the characterization of the subdifferential to unbounded exterior domains (Theorem 4.1).Section 5 includes a comparison result (Lemma 5.1) for strictly convex norms; the obtaining of minimizers u R for the relative pcapacity Cap F p (Ω; W R ) as limits of minimizers of energy functionals, which involve strictly convex anisotropies approximating a generic norm F (Lemma 5.5); and the proof that u R are trapped between explicit solutions of the corresponding Euler-Lagrange problem within annular domains (Proposition 5.6).Then the next natural step is to carry over these barrier arguments in bigger and bigger rings to reach Theorem 1.3 as the outer boundary tends to infinity, which stands for the content of section 6; in turn, Theorem 1.5 is proven in section 7. Finally, we include an appendix with an explicit example to justify the lack of uniqueness in the case p = 1.

Notation and background material
2.1.Anisotropies and Wulff shape.A continuous function F : R N → [0, ∞) is said to be an anisotropy if it is convex, positively 1-homogeneous (i.e., F (λx) = λF (x) for all λ > 0 and all x ∈ R N ) and coercive.We will always consider additionally that F is even, that is, a norm.In particular, as all norms in R N are equivalent, there exist constants 0 We define the dual or polar function for every ξ ∈ R N .It can be verified that F • is convex, lower semi-continuous and 1-positively homogeneous.Moreover, (2.1) leads to From the definition of F • one gets a Cauchy-Schwarz-type inequality of the form 3) The Wulff shape W R of F is defined by As we are dealing with even anisotropies, W R is a centrally symmetric convex body.We say that F is crystalline if, furthermore, W R is a convex polytope.

2.2.
Maximal monotone operators on Banach spaces.Let X be a reflexive Banach space with dual X , and denote by • , • the pairing between X and X.Let A : X → 2 X be a multivalued operator on X (equivalently, we write A ⊂ X × X for its graph).Hereafter, D(A) and R(A) mean the domain and range of A, respectively.
Moreover, A is called maximal monotone if there exists no other monotone multivalued map whose graph strictly contains the graph of A.
Let Φ : X → R ∪ {∞} be lower semicontinuous, proper and convex; its subdifferential ∂Φ : X → 2 X is a multivalued operator given as follows: This is a well-known example of a maximal monotone operator from X to X (cf.[33, Theorem A]).
In turn, the duality mapping J X : X → 2 X is defined as In particular, the duality mapping J L p : L p (Ω) → L p (Ω) is single-valued and has the following explicit expression: and for all u ∈ L p (Ω).
This will be a crucial characterization (see [7, Theorem 2.2]): Theorem 2.2 (Minty [28] and Browder [10]).Let A : L p (Ω) → L p (Ω) be a monotone functional and let J L p : L p (Ω) → L p (Ω) be the duality mapping of L p .Then A is maximal monotone if, and only if, R(A Therefore, we need results that ensure that the above range condition holds in order to deduce that an operator is maximal monotone.In particular, we will apply the following statement (see [20,Corollary 1.8]): Theorem 2.3 (Hartman and Stampacchia [19]).Let K = ∅ be a closed convex subset of a reflexive Banach space X, and B : K → X be monotone, coercive and weakly continuous on finite dimensional subspaces.Then B is surjective.
Typically the literature refers to the hypotheses in the above theorem as the classical Leray-Lions assumptions (cf.[22]).
2.3.Yosida approximation of maps in Hilbert spaces.Now consider I the identity map on a Hilbert space H. Then the resolvent of A : H → 2 H is given by The Yosida approximation of a maximal monotone map A is defined as and hence it satisfies In addition, we have the following properties (see [6, Theorem 3.5.9]): Lemma 2.4.(a) A λ is a single-valued maximal monotone map which is Lipschitz with constant 1/λ.
(b) A λ (ξ) converges to A 0 (ξ) as λ → 0, where A 0 denotes the minimal section of A given by Recall that given a proper, convex and lower semicontinuous function g : R N → [0, ∞), for every λ > 0 the Moreau-Yosida approximation of g is defined by for all ξ ∈ R N .The minimum in (2.8) is attained at J ∂g,λ (ξ), where ∂g denotes the subdifferential of g.Furthermore, g λ is convex and Fréchet differentiable with gradient Thus the Yosida approximation of the subdifferential is equal to the gradient of the Moreau-Yosida approximation.We also have that g λ (ξ) for every ξ ∈ R N (see [12], [36, Proposition 1.8] or [6, Theorem 6.5.7]).
Recall that the subdifferential ∂F Ω,ϕ of F Ω,ϕ is the multivalued operator from L p (Ω) to L p (Ω) given by ϕ (Ω), and consequently in L p (Ω). Recall that u belongs to the effective domain of To get the sought characterization of ∂F Ω,ϕ , we first introduce an auxiliary functional A ϕ .Definition 3.2.We define The goal is to show that both operators A ϕ and ∂F Ω,ϕ coincide (cf.Theorem 3.10).Once we have established this result, we can conclude that 0 ∈ ∂F Ω,ϕ (u) is equivalent to u satisfying the following: in a weak sense, where z ∈ ∂F (∇u) and we have used the chainrule for subdifferentials (see [8,Corollary 16.72] or [16, Theorem 2.3.9 (ii)]).
The first inclusion is quite straightforward: In particular, the functional A ϕ is monotone.
Proof.Let (u, v) ∈ A ϕ and z as in (A3) above.In particular, by definition of subdifferential for every w ∈ W 1,p ϕ (Ω).Consequently, for every v ∈ L p (Ω). Hence the statement follows by definition of ∂F Ω,ϕ . 2

3.2.
Properties of the subdifferential of F p .To establish the remaining inclusion, we need some properties of ∂F p that will be used repeatedly.We start by writing the elements of ∂F p in terms of those in ∂F because, as F is positively 1-homogeneous, its subdifferential has a useful well-known characterization (obtained by choosing η = 0 and η = 2ζ in (2.4)): Notice that the above inequality is equivalent to F • (δ) ≤ 1.In particular, for η = δ δ it follows by means of (2.1) that δ ≤ C for all δ ∈ ∂F (ζ). (3. 3) The latter bound, combined with the chainrule for subdifferentials and again (2.1), leads directly to Lemma 3.4.Every z ∈ ∂F p (ξ) can be written as and we can estimate z ≤ p C p ξ p−1 .
Next we see that the same estimate holds for the corresponding Yosida approximation.
Lemma 3.5.The Moreau-Yosida approximation of F p and the Yosida approximation of ∂F p satisfy: Here c and C are the constants coming from (2.1).
Proof.(a) Note that, using (2.1) in the definition given by (2.8), one has Hence we are done in the case that η p ≥ ξ p /2. Otherwise, if η < ξ /2 1/p , by the reverse triangle inequality we get where the last inequality holds, if and only if, (b) By (2.7) with A = ∂F p and the chainrule for subdifferentials, we can write In particular, for η = ξ, we get F p (J p,λ (ξ)) ≤ F p (ξ). Raising the inequality to 1 p and using (2.1) leads to This, combined with (2.9) (with g = F p ) and (3.4), implies for every δ ∈ ∂F (J p,λ (ξ)).Note that we used (2.1) and (3.3) for the last inequality.2 The existence of minimizers follows from the above properties.
Proposition 3.6.There exists at least one function u which is a minimizer of the functional F Ω,ϕ defined in (1.2) for any domain Ω ⊂ R N .Moreover, F Ω,ϕ is lower semicontinuous with respect to the weak convergence in W 1,p (Ω).
Proof.As by (2.1) the functional F Ω,ϕ is bounded from below, there exists a minimizing sequence u n .By definition of the functional and the Poincaré inequality, {u n } n∈N is uniformly bounded in W 1,p (Ω) and hence it converges subsequentially and weakly in W 1,p (Ω) to u ∈ W 1,p (Ω).
Now let z ∈ ∂F p (∇u), then z ∈ L p (Ω) and z = pF p−1 (∇u)δ with δ ∈ ∂F (∇u) by Lemma 3.4.By (3.2) and the weak convergence, we have where the latter follows by application of Hölder's inequality.Thus It remains to show that such a minimizer is a weak solution of (3.1), which will be a direct consequence of the characterization of ∂F Ω,ϕ .
Throughout this paper, we will perform several approximation arguments whose outcome is a suitable weak limit z ∈ L p (Ω).The following result, in the spirit of the Minty-Browder technique (see [29]), gives a practical criterion to guarantee that z is indeed in the appropriate subdifferential.For brevity, here and in the sequel we will use the notation (3.5) Lemma 3.7.Given u ∈ W 1,p (Ω), z ∈ L p (Ω) and α ∈ R, suppose that the following inequality holds for a.e.x ∈ Ω and every ξ ∈ R N : Then one can conclude that z ∈ ∂F p (∇u) + αΨ(∇u).The same conclusion holds if we only require that (3.6) holds for z = (∂F p ) 0 (ξ).
Proof.Notice that it is enough to prove the second claim, but we keep the stronger hypothesis (3.6) because of the applications afterwards.Therefore, there exists a null set Then, for a.e.0 = η ∈ R N , {t : ∇u(x) + tη ∈ N x } is null and we may take a sequence {ε n } n∈N with ε n 0 and {∇u(x) + ε n η} n∈N ⊂ R N \ N x .If we are in this case, we can write Now, since (∂F p ) 0 (∇u(x) + ε n η) is bounded we may suppose that, up to a subsequence, (∂F p ) 0 (∇u(x) + ε n η) → z ∈ ∂F p (∇u(x)) as n → ∞ (recall that ∂F p is closed since it is maximal monotone).Hence, by the continuity of Ψ, taking limits in the previous equation we get that Since this holds for a.e.η ∈ R N , we conclude that z(x) − αΨ(∇u(x)) = z ∈ ∂F p (∇u(x)). 2

3.3.
A range condition for the auxiliary functional.To prove Theorem 3.10 we need to show that the operator A ϕ introduced in Definition 3.2 satisfies a range condition, which involves the duality mapping of L p (Ω) given by (2.6).
Proof.Given f ∈ L p (Ω), we aim to prove that there exists u ∈ W 1,p ϕ (Ω) such that (u, f − J L p u) ∈ A ϕ .With this purpose, let us consider , which is a sequence of operators approximating A ϕ and defined as follows: in the weak sense.In particular, with the notation from (3.5) it holds which hold for every ξ, η ∈ R N .This, combined with Poincaré inequality, easily leads to coercivity when p ≥ 2; in fact, for any u, w ∈ W 1,p ϕ (Ω) we get To establish the coercivity of A n,m ϕ + J L p in the remaining case 1 < p < 2, again by Poincaré inequality, it is enough to show that, given a fixed w ∈ W where we applied Hölder and the reverse triangle inequality.Notice that, as p ∈ (1, 2), the right hand side tends to 0 + ∇w p−1 L p as ∇u L p → ∞, which is a bounded limit, as desired. 2 Now, since W 1,p ϕ (Ω) is closed and convex in W 1,p (Ω), by Theorem 2.3 we deduce that Thus there exist u n,m ∈ W 1,p ϕ (Ω) such that for every w ∈ W 1,p 0 (Ω).Claim 2. {u n,m } n∈N is uniformly bounded in W 1,p (Ω).
Proof of Claim 2. Given w ∈ W 1,p ϕ (Ω), letting w − u n,m as a test function in (3.9) yields First, the integral on the left hand side, by means of the definition of the duality mapping in (2.5), can be written as Rearranging terms, we obtain Now, by (2.10) for g = F p and by Hölder's inequality, we can estimate On the other hand, by Cauchy-Schwarz for the inner product in R N , Lemma 3.5 (b) and Young's inequality ab ≤ εa p + C(ε)b p , where where C denotes hereafter any constant which depends only on p and the constants from (2.1), whose concrete meaning may change from line to line (and we use C whenever two of such constants appear on the same line).
Bringing these bounds together, using J L p (u n,m ) L p = u n,m L p , Cauchy's inequality ab ≤ a 2 2 + b 2 2 and Young's inequality with ε = 1/4, we get In short, we reach from which the claim follows. 2 Accordingly, up to a subsequence, we may assume that u n,m converges in L p (Ω) and a.e. to some u m ∈ L p (Ω) as n → ∞; and ∇u n,m ∇u m weakly in L p (Ω) as n → ∞.The continuity of the trace operator with respect to the weak convergence in W 1,p (Ω) (see [21,Corollary 18.4]) guarantees that u m ∈ W 1,p ϕ (Ω).Claim 3.There exists z m ∈ ∂F p (∇u m ) such that for every ψ ∈ W 1,p 0 (Ω).
Proof of Claim 3. First, Lemma 3.5 (b) ensures that Consequently, we may assume that In turn, we also have that Ψ(∇u n,m ) = ∇u n,m p−1 , and hence Therefore, taking ψ ∈ W 1,p 0 (Ω) as test function in (3.9) and letting n → ∞ we get in particular, for every g ∈ W 1,p (Ω).
Proof of Claim 4. We proceed as before, but now taking w − u m as test function in (3.10) with w ∈ W 1,p ϕ (Ω).Then, as z m = pF p−1 (∇u m )z m with zm ∈ ∂F (∇u m ), by means of (3.2) we can estimate Rearranging terms, using (2.5) and applying Hölder's inequality, we have where c, C are the constants coming from (3.2).By definition of the duality map, Cauchy's inequality and Young's inequality with ε = 1 2 where . Accordingly, we reach Therefore, u m converges subsequentially in L p (Ω) and a.e. to some u ∈ L p (Ω) as m → ∞; and ∇u m ∇u weakly in L p (Ω) as m → ∞.The continuity of the trace operator with respect to the weak convergence in W 1,p (Ω) ensures that u ∈ W 1,p ϕ (Ω).Next, by Lemma 3.4, as z m ∈ ∂F p (∇u m ), we have This implies that {z m } is uniformly bounded in L p (Ω), and hence z m z weakly in L p (Ω) as m → ∞.On the other hand, by Claim 4, {Ψ(∇u m )} m∈N is bounded in L p (Ω) so Thus, taking limits as m → ∞ in (3.10), we get that It remains to see that z ∈ ∂F p (∇u), which can be obtained after establishing (3.6).Therefore, let ξ ∈ R N , z ∈ ∂F p (ξ) and take ω : Ω → R defined by ω(η) := ξ • η so that ∇ω = ξ.Then, letting φ ∈ C ∞ 0 (Ω) such that φ ≥ 0, by the monotonicity of ∂F p + 1 m Ψ and Claim 3 we have, for every m ∈ N, taking the infimum limit as m → ∞, by means of (3.16) we get where the last equality comes from (3.17).Consequently, since φ ∈ C ∞ 0 (Ω) is arbitrary and ∇ω = ξ, we conclude (3.6) with α = 0, as desired.Finally, Lemma 3.7 yields z ∈ ∂F p (∇u), which leads to (u, f − J L p (u)) ∈ A ϕ , which finishes the proof. 2 Remark 3.9.Let us point out that in the singular case, that is, when 1 < p < 2, there is a significative shortcut of the above proof, as one can exploit Lemma 3.5 (a) to prove that the operators A n ϕ (u) := −div ∇(F p ) 1 n (∇u) are coercive, and then the result follows by performing only one limiting argument.The drawback of this shorter proof is that some bounds depend explicitly on the measure of Ω, and in the next section we plan to extend the results (with minor changes) to unbounded domains.

3.4.
Proof of the characterization.After all the machinery developed in the previous subsections, the following result is almost straightforward.Proof.Recall that, by Lemma 3.3, A ϕ ⊆ ∂F Ω,ϕ .Next by Proposition 3.8 we have that thus, by Theorem 2.2, we have that A ϕ is a maximal monotone operator, and hence the claim follows. 2 By Definition 3.2, Lemma 3.4 and (3.2), this finishes the proof of Theorem 1.1.

Characterization of the subdifferential in unbounded exterior domains
After suitable adaptations that we will sketch here, we can reproduce the proof of the previous section and make it work for unbounded domains of the form R N \ Ω, where Ω is a bounded domain with Lipschitz boundary.The main difference is that we work with the homogeneous Sobolev space Ẇ 1,p (R N ), defined as the completion of see e.g.[9].Setting p * = N p N −p , Ẇ 1,p (R N ) can be identified with the following space of functions: By means of the Gagliardo-Nirenberg-Sobolev inequality, this ensures that and hence on these spaces we have a Poincaré type inequality, which is exactly what we need for our estimates to work also in unbounded settings.This also tells us that the norm (4.1) is equivalent to With these considerations, Ẇ 1,p (R N ) is a reflexive Banach space (see [17,Theorem 12.2.3])and, therefore, Theorem 2.3 can be applied in this setting.
Given [21,Theorem 18.40] ensures that the trace operator is surjective on sets with Lipschitz continuous boundary, thus we can find Φ ∈ W 1,p (Ω) whose trace on ∂Ω is ϕ.Then, given u ∈ L p * (R N \ Ω) with ∇u ∈ L p (R N \ Ω) such that u = ϕ on ∂Ω, we will consider the extension Hence we work with functions on the space

Now we consider the energy functional
and observe that it is a proper and convex functional.To characterize its subdifferential, we proceed exactly as before, that is, we introduce an auxiliary operator , then the following are equivalent: where We notice that Theorem 4.1 gives a characterization for all minimizers of Cap F p (Ω) for bounded Lipschitz domains Ω since in the weak sense for some z i ∈ ∂F (∇u i ), i = 1, 2, then u 1 ≤ u 2 in Ω.
On the other hand, as pF p−1 (∇u i )z i ∈ ∂F p (∇u i ) and the latter is monotone, we reach Therefore, the integrand on left hand side vanishes a.e. in Ω + .Accordingly, by means of (3.2), we get meaning that F (∇u 1 ) = F (∇u 2 ) a.e. in Ω + .Hence the strict convexity of F ensures that ∇u 1 = ∇u 2 a.e. in Ω + .As u 1 = u 2 on ∂Ω + , we conclude that Ω + has null measure and thus the claim follows. 2 Remark 5.2.We observe that this result gives uniqueness of minimizers of F Ω,ϕ (and solutions to (3.1)) if F is a strictly convex norm.
We also need the following auxiliary result: Lemma 5.3.Let F be a strictly convex norm in R N , then for every x = 0 Proof.The proof of the first equality is exactly as in [37, Proposition 1.3], since it only uses the differentiability of F • , which follows from the strict convexity of F .For the second claim, from the 1-homogeneity of F • (recall (3.2)), we can write By (2.3) this implies that ∇F • (x) minimizes g(ξ) meaning that x ∈ F • (x)∂F (∇F • (x)), as desired.In this setting, we aim to obtain a minimizer of F R and then apply suitable comparison results to take limits as R → ∞.However, in order to exploit Lemma 5.1 in general, that is, for non-necessarily strictly convex norms, we will approximate the anisotropy F by a sequence of strictly convex anisotropies {F ε = F + ε • } ε>0 , and consider the subsequent energy functional , where, by simplicity, we denote (F ε ) p by F p ε .
Lemma 5.5.Let {u ε } ε>0 be a sequence of minimizers of F ε R .Then there exists a subsequence that converges weakly in W 1,p Proof.Let u ε be a minimizer of F ε R and let w ∈ W 1,p ψ (Ω R ).By the uniform convergence of {F ε } ε>0 to F as ε 0 and (2.1), there exist k > 0 not depending on ε such that Accordingly, by Poincaré inequality and up to a subsequence, we may suppose that u ε u R ∈ W 1,p (Ω R ) weakly in W 1,p (Ω R ) and strongly in L p (Ω R ).Now, since F R is lower semicontinuous with respect to the weak convergence in W 1,p (Ω R ) (recall Proposition 3.6), we can write for every w ∈ W 1,p (Ω R ).Therefore, we get that u R is a minimizer of F R .By the continuity of traces, u R = 1 on ∂Ω and equals zero on ∂W R . 2 5.3.Construction of explicit barriers trapping a minimizer.
Proposition 5.6.Let 0 < r 1 < r 2 < R be such that W r 1 ⊆ Ω ⊆ W r 2 , and let u R be a minimizer of F R which arises as a subsequential limit of minimizers of for a.e.x ∈ Ω R .
Proof.Given r < R, it is a routine computation to check that Here W ε r is the Wulff shape of radius r associated to the anisotropy F ε .Notice that the first equality follows after checking that ∂F ε (∇v ε r,R ) = ∂F (∇F • ε ) and using Lemma 5.3.It can be easily checked, as in [14, Lemma 5.3] that W ε r = W r + B • (εr). (5.2) On the other hand, as for a.e.x ∈ Ω R . (5.4) Finally, given ε > 0, we set So both R 1 and R 2 converge to R and, taking limits in (5.3) and (5.4), we conclude (5.1), as desired. 2

Existence of minimizers with double side bounds
Proof of Theorem 1.3.Recall that for any R > 0 one can actually construct a minimizer u R , by means of Lemma 5.5, for which we have upper and lower barriers as stated in Proposition 5.6.Now the aim is to show that these estimates pass to the limit as R → ∞.
For each big enough R, let u R be a minimizer of F R which arises as a limit of minimizers of {F ε R } ε>0 and extend u R by zero to R N \ Ω (without renaming).Then by Lemma 5.1 we have that {u R } is increasing in R and, by Proposition 5.6, we get Consequently, there exists a measurable function u : R N \ Ω → R such that, up to a subsequence, u R (x) u(x) as R → ∞ for a.e.x ∈ R N \ Ω and In particular, u → 0 as x → ∞.On the other hand, notice that the bounding functions in (6.1) belong to L q (R N \ Ω) if q > N (p−1) N −p , and this ensures by dominated convergence that u R → u in L p * (R N \ Ω).Now, let ϕ ∈ C ∞ c (R N ) with ϕ = 1 on ∂Ω, and consider R large enough so that supp ϕ ⊂ W R .Since u R is a minimizer of F R , we have that Therefore, { ∇u R p } R is bounded thus u ∈ W 1,p (R N \ Ω) and ∇u R ∇u weakly in L p (R N \ Ω); in particular, u = 1 on ∂Ω.Moreover, since 0 ∈ ∂F R (u R ), by Theorem 1.1 we have that there exists z R ∈ ∂F (∇u R ) (which we extend by zero in R N \ W R ) such that R N \Ω pF p−1 (∇u R )z R • ∇w = 0 (6.2) for every w ∈ W 1,p 0 (W R \ Ω).Now, by (3.3), we have z R ∞ ≤ C so, up to a subsequence, pF p−1 (∇u R )z R z in the weak topology of L p (R N \ Ω). (6.3) Taking limits as R → ∞ in (6.2) we get for every w ∈ C ∞ c (R N \Ω) (thus, by approximation, for any w ∈ W 1,p 0 (R N \Ω) that is, −div(z) = 0 in the weak sense.(6.4) It remains to show that z ∈ ∂F p (∇u), which will be deduced as usual from Lemma 3.7.Therefore, it suffices to get (3.6) for z = z and α = 0.With this goal, let ξ ∈ R N and take ω : R N \ Ω → R defined by ω(η) := ξ • η so that ∇ω = ξ.Then, letting φ ∈ C ∞ c (R N \ Ω) such that φ ≥ 0, by the monotonicity of ∂F p and (6.2) we have, for every R > 0 and any z ∈ ∂F p (ξ), Since by (6.1) we have that u R → u strongly in L p (K) for every compact set K ⊂ R N \ Ω, letting R → ∞, by (6.3) and (6.If we further assume that Ω satisfies the condition stated in Definition 1.4, by a careful application of our comparison results, we will show that a minimizer constructed as in Lemma 5.5 is bounded above and below by two Lipschitz solutions which coincide on the boundary, and accordingly we will deduce the Lipschitz regularity of the minimizer by application of the following result (see [25,Corollary 4.2] for a more general version): Working as above we get that η is L 2 -Lipschitz for some constant L 2 which decreases with R. In short, we have shown that u R ∈ W 1,p η,η (Ω R ).Now, by Theorem 7.1, there exists at least one minimizer of F R within W 1,p η,η (Ω R ) which is Lipschitz continuous with constant L ≤ max{L 1 , L 2 }.If we denote this minimizer by v R , notice that Next consider any minimizer w R of F R (not necessarily constructed by a limiting procedure or trapped between η and η ).Therefore, As the reverse inequality holds because w R ∈ arg min F R , we conclude In this appendix, we show an example of non-uniqueness for relative 1capacitary functions; i.e., minimizers of the relative 1-capacity with respect to a ball W R .We take N = 2 and F (ξ) = ξ 1 = |ξ 1 |+|ξ 2 | as the anisotropy.
Let Ω := B 1 be the (Euclidean) unit ball in R 2 centered at the origin and set R = 2, thus
by doing the obvious modifications in Definition 3.2, and repeat the arguments from the previous section for a sequence of approximating operators A n,mΦ : Ẇ 1,p Φ (R N ) → Ẇ 1,p (R N )given again by the expression (3.7).The aforementioned properties of homogeneous Sobolev spaces allow us to repeat almost verbatim the steps in the previous section to conclude the following result:

Theorem 4 . 1 (
Characterization of the subdifferential on R N \ Ω).Let 1 < p < N , Ω ⊂ R N an open bounded set with Lipschitz boundary and Φ

5. 2 .Notation 5 . 4 .
Existence of minimizers within annular domains.For any R > 0 and Ω a bounded domain with Lipschitzcontinuous boundary in R N such that Ω ⊂ W R , we consider the annular region Ω R := W R \ Ω and the function ψ defined on ∂Ω R by ψ = 1 on ∂Ω and ψ = 0 on ∂W R .Now it makes sense to work with the functional F R := F Ω R ,ψ given by (1.2).

R ) 2 a 2 ≤F
.e. in Ω R .Now, if F (∇w R ) = F (∇v R ), by the strict convexity of t → |t| p and the convexity of F , we have that (a.e. in Ω R )F (∇w R ) + F (∇v R ) 2 p < F p (∇w R ) + F p (∇v R ) 2 = F p ∇(w R + v R ) (∇w R ) + F (∇v R ) 2 p , which is a contradiction.Therefore, F (∇w R ) = F (∇v R ) a.e. in Ω R thus w R is C c L-Lipschitz (as v R is L-Lipschitz), where c and C come from (2.1).Now, since the u R are uniformly Lipschitz and u R u by the proof of Theorem 1.3, we get that u is Lipschitz continuous (with Lipschitz constant C c L). Now take w an arbitrary minimizer of F R N \Ω,1 and redo the previous argument with u and w playing the role of v R and w R , respectively.Then we conclude that w is also Lipschitz continuous with constant C 2 c 2 L. 2Appendix A. Non-uniqueness of minimizers in the case p = 1 [11,comparison principle for strictly convex norms and a barrier argument 5.1.Results for strictly convex anisotropies.We start by obtaining a comparison principle for problem (3.1) with a strictly convex even anisotropy F .It is well-known (cf.[34,Corollary 1.7.3]) that strict convexity of F is closely related to differentiability of F • .More precisely,F • ∈ C 1 (R N \ {0}) if and only if B F (1) = {ξ : F (ξ) < 1} is strictly convex.In the above conditions, it may happen that F ∈ C 1 (R N \ {0}), as shown by the examples in[13, Example A.1.19]or[15,SectionA].In this spirit, the following result generalizes[11, Lemma 2.3] by relaxing the regularity requirements for F and for the functions u i involved.Lemma 5.1.Let Ω ⊂ R N be a bounded domain with Lipschitz continuous boundary and F a strictly convex norm in R N