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

Theorem 1 (Characterization of the subdifferential in Ω bounded).

Let 1 < p < ∞ , Ω ⊂ R N a bounded domain with Lipschitz-continuous boundary and φ ∈ W 1 - 1 p , p ⁢ ( ∂ ⁡ Ω ) . If u ∈ W φ 1 , p ⁢ ( Ω ) and v ∈ L p ′ ⁢ ( Ω ) , with p ′ = p p - 1 , then the following are equivalent:

1. v ∈ ∂ ⁡ ℱ Ω , φ ⁢ ( u ) .

2. There exists z ∈ L ∞ ⁢ ( Ω ; ℝ N ) , with F ∘ ⁢ ( z ) ≤ 1 and z ⋅ ∇ ⁡ u = F ⁢ ( ∇ ⁡ u ) , such that v = - div ⁡ ( p ⁢ F p - 1 ⁢ ( ∇ ⁡ u ) ⁢ z ) in the weak sense, that is,

   ∫ Ω v ⁢ w = ∫ Ω p ⁢ F p - 1 ⁢ ( ∇ ⁡ u ) ⁢ z ⋅ ∇ ⁡ w   for every  ⁢ w ∈ W 0 1 , p ⁢ ( Ω ) .

Here F ∘ denotes the dual of the norm F.

Corollary 2.

Let 1 < p < ∞ , Ω ⊂ R N a bounded domain with Lipschitz boundary and φ ∈ W 1 - 1 p , p ⁢ ( ∂ ⁡ Ω ) . Consider either D = Ω or D = R N ∖ Ω ¯ . Then any minimizer u of the energy functional F D , φ is a weak solution of

for some z ∈ L ∞ ⁢ ( D ; R N ) , with F ∘ ⁢ ( z ) ≤ 1 and z ⋅ ∇ ⁡ u = F ⁢ ( ∇ ⁡ u ) . 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).

Theorem 3.

If 1 < p < N and Ω ⊂ R N is a bounded domain with Lipschitz boundary, there exists a minimizer u of Cap p F ⁢ ( Ω ¯ ) such that:

1. u ∈ L p ∗ ⁢ ( ℝ N ∖ Ω ¯ ) is a weak solution of ( 1.4 ) with D = ℝ N ∖ Ω ¯ , p ∗ := N ⁢ p N - p and ∇ ⁡ u ∈ L p ⁢ ( ℝ N ∖ Ω ¯ ; ℝ N ) .

2. We can find constants 0 < r 1 < r 2 with 𝒲 r 1 ⊂ Ω ⊂ 𝒲 r 2 so that

   u r 1 ≤ u ≤ u r 2    with  ⁢ u r i = ( F ∘ ) p - N p - 1 ⁢ ( ⋅ r i ) , i = 1 , 2 .

3. If F is required to be strictly convex, then the minimizer is unique.

Definition 4 (Uniform interior ball condition).

Let r > 0 . We say that Ω satisfies the W r -condition if, for any x ∈ ∂ ⁡ Ω , there exists y ∈ R N such that

Theorem 5.

Let r > 0 and suppose that Ω satisfies the W r -condition. Then any minimizer of Cap p F ⁢ ( Ω ¯ ; W R ) is Lipschitz continuous. Moreover, any minimizer of the energy functional F R N ∖ Ω ¯ , 1 is also Lipschitz continuous.

Definition 1.

The operator A is said to be monotone if

Moreover, A is called maximal monotone if there exists no other monotone multivalued map whose graph strictly contains the graph of A .

Theorem 2 (Minty [29], Browder [13]).

Let A : L p ⁢ ( Ω ) → L p ′ ⁢ ( Ω ) be a monotone operator and J L p : L p ⁢ ( Ω ) → L p ′ ⁢ ( Ω ) the duality mapping of L p . Then A is maximal monotone if, and only if, R ⁢ ( A + J L p ) = L p ′ ⁢ ( Ω ) .

Theorem 3 (Hartman and Stampacchia [21]).

Let K ≠ ∅ be a closed convex subset of a reflexive Banach space X, and let B : K → X ′ be monotone, coercive and weakly continuous on finite-dimensional subspaces. Then B is surjective.

Lemma 4.

The following statements hold:

1. 𝒜 λ is a single-valued maximal monotone map which is Lipschitz with constant 1 λ .

2. 𝒜 λ ⁢ ( ξ ) converges to 𝒜 0 ⁢ ( ξ ) as λ → 0 , where 𝒜 0 denotes the minimal section of 𝒜 given by

   𝒜 0 ⁢ ( ξ ) := { η ∈ 𝒜 ⁢ ( ξ ) : η ⁢ has minimal norm in  ⁢ 𝒜 ⁢ ( ξ ) } .

Remark 1.

It is a well-known result (see, e.g., [8, Proposition 1.6]) that the effective domain 𝒟 ⁢ ( ∂ ⁡ ℱ Ω , φ ) is dense in 𝒟 ⁢ ( ℱ Ω , φ ) = W φ 1 , p ⁢ ( Ω ) , and consequently in L p ⁢ ( Ω ) . Recall that u belongs to the effective domain of 𝒜 if ∥ 𝒜 ⁢ ( u ) ∥ X ′ < ∞ .

Definition 2.

We define A φ ⊂ L p ⁢ ( Ω ) × L p ′ ⁢ ( Ω ) as follows: ( u , v ) ∈ A φ if

1. u ∈ W φ 1 , p ⁢ ( Ω ) ,

2. v ∈ L p ′ ⁢ ( Ω ) ,

3. there exists z ∈ ∂ ⁡ F p ⁢ ( ∇ ⁡ u ) such that v = - div ⁡ ( z ) weakly, meaning that

   ∫ Ω v ⁢ w = ∫ Ω z ⋅ ∇ ⁡ w   for every  ⁢ w ∈ W 0 1 , p ⁢ ( Ω ) .

Lemma 3.

We have A φ ⊆ ∂ ⁡ F Ω , φ . In particular, 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 φ 1 , p ⁢ ( Ω ) . Consequently,

for every v ∈ L p ′ ⁢ ( Ω ) . Hence the statement follows by definition of ∂ ⁡ ℱ Ω , φ . ∎

Lemma 4.

Every z ∈ ∂ ⁡ F p ⁢ ( ξ ) can be written as

and we can estimate

Lemma 5.

The Moreau–Yosida approximation of F p and the Yosida approximation of ∂ ⁡ F p satisfy:

1. If 1 < p < 2 , there exists a positive constant K λ = K λ ⁢ ( c , p , λ ) with lim λ ↘ 0 ⁡ K λ = 0 so that

   ( F p ) λ ⁢ ( ξ ) ≥ c p 2 ⁢ ∥ ξ ∥ p   if  ⁢ ∥ ξ ∥ ≥ K λ .

2. ∥ ( ∂ F p ) λ ( ξ ) ∥ = ∥ ( ∇ ( F p ) λ ) ( ξ ) ∥ ≤ p C p ∥ ξ ∥ p - 1 .

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

where J p , λ ⁢ ( ξ ) = J ∂ ⁡ F p , λ ⁢ ( ξ ) satisfies

In particular, for η = ξ , we get

Raising the inequality to 1 p and using (2.1) leads to

This, combined with (2.9) (with g = F p ) and (3.5), implies

for every δ ∈ ∂ ⁡ F ⁢ ( J p , λ ⁢ ( ξ ) ) . Note that we used (2.1) and (3.3) for the last inequality. ∎

Proposition 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 the 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 = p ⁢ F p - 1 ⁢ ( ∇ ⁡ u ) ⁢ δ with δ ∈ ∂ ⁡ F ⁢ ( ∇ ⁡ u ) by Lemma 4. By (3.2) and the weak convergence, we have

where the latter follows by application of Hölder’s inequality. Thus

as desired. ∎

Lemma 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 ) + α ⁢ Ψ ⁢ ( ∇ ⁡ u ) . The same conclusion holds if we only require that (3.8) holds for z ′ = ( ∂ ⁡ F p ) 0 ⁢ ( ξ ) .

Proof.

Notice that it is enough to prove the second claim, but we keep the stronger hypothesis (3.8) because of the applications afterwards. Therefore, there exists a null set N 1 ⊂ ℝ N such that

for every x ∈ Ω ∖ N 1 and ξ ∈ ℝ N ∖ N x , where N x ⊂ ℝ N is null. Let x ∈ Ω ∖ N 1 . Then, for a.e. 0 ≠ η ∈ ℝ N , the set { t : ∇ ⁡ u ⁢ ( x ) + t ⁢ η ∈ 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

for every n ∈ ℕ .

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 ) - α ⁢ Ψ ⁢ ( ∇ ⁡ u ⁢ ( x ) ) = z ~ ∈ ∂ ⁡ F p ⁢ ( ∇ ⁡ u ⁢ ( x ) ) . ∎

Proposition 8.

If φ ∈ W 1 - 1 p , p ⁢ ( ∂ ⁡ Ω ) , then L p ′ ⁢ ( Ω ) = R ⁢ ( J L p + A φ ) .