Optimal shapes for general integral functionals

We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain $\Omega$ that varies over all subdomains of a given bounded domain $D$ of ${\bf R}^d$. We show in a rather elementary way the existence of a solution that is in general a quasi open set. Under very mild conditions we show that the optimal domain is actually open and with finite perimeter. Some counterexamples show that in general this does not occur.


Introduction
In this paper we consider a shape optimization problem for a general integral functional of the form Here p > 1 is a fixed real number, D is a given bounded domain of R d , and the function f : R d × R × R d →] − ∞, +∞] is a general integrand satisfying suitable rather mild assumptions. Note that in problem (1.2) the volume constraint can be incorporated into the cost functional F by means of a Lagrange multiplier of the form λ|Ω| or more generally Ω λ(x) dx. For a detailed presentation of shape optimization problems we refer the interested reader to the books [BB05] and [HM05].
The first result is Theorem 2.1, which gives the existence of an optimal domain Ω opt . This optimal domain belongs to the class of p-quasi open sets, defined as the sets {u > 0} for some function u ∈ W 1, p 0 (D). As a consequence, if p > d these optimal sets are actually open, but if p d this fact does not occur any more under the very general assumptions we made, see Example 4.3.
The existence of optimal sets Ω opt could have been obtained through a generalization of a result in [BDM93] to the case p > 1, making use of a γ p -convergence on the class of p-quasi open sets. However, we have preferred to give an independent proof that, in the particular case of integral functionals of the form (1.1), is much simpler.
In order to obtain that the optimal sets Ω opt are open, we need slightly stronger assumptions: this is the goal of Theorem 3.4, in which we use the Hölder continuity result of [GG84,Giu03] on the minimizers of general integral functionals.
Finally, in Theorem 5.1 we prove, under rather general assumptions on the integrand f , that Ω opt has a finite perimeter. This result is obtained by adapting a previous result of [Buc12] to the general case of an integrand f with a p-growth.

Setting of the problem and existence result
We recall here some well-known notions from the Sobolev spaces theory; for all details we refer to [BB05] and to [Maz11]. In all the paper p > 1 will be a fixed real number.
For every set E ⊂ R d the p-capacity of E is defined as It is well known (see for instance Ziemer [Zie89]) that every function u of the Sobolev space W 1,p (D) has a p-quasi continuous representativeũ, which is uniquely defined up to a set of capacity zero. The functionũ is given bỹ in the sense that the limit above exists for p-quasi every x ∈ D. In the following we always identify the function u with its p-quasi continuous representativeũ, so that pointwise conditions can be imposed on u(x) for p-quasi every x ∈ D. Again, when p > d p-quasi continuous functions are continuous, because points have a positive p capacity.
If Ω is a p-quasi open set we may define the Sobolev space W 1, p 0 (Ω) as W 1, p 0 (Ω) = u ∈ W 1, p (R d ) : u = 0 q.e. on R d \ Ω . We notice that this definition coincides, in the case when Ω is open, with the usual one obtained as the closure of the class of smooth functions compactly supported in Ω with respect to the W 1,p (R d ) norm It is also important to stress that two sets Ω 1 and Ω 2 which are equivalent in the Lebesgue sense, that is |Ω 1 Ω 2 | = 0, where denotes the symmetric difference for sets, may produce very different Sobolev spaces. For instance, in R 2 if Ω 1 is the unit disk in R 2 and Ω 2 is the unit disk minus a radius S, the Sobolev spaces W 1, p 0 (Ω 1 ) and W 1, p 0 (Ω 2 ) differ a lot: the functions in the second one vanish on the radius S, which is not the case for functions in the first space. Similarly, in R d an open set Ω and Ω without a k-dimensional manifold S provide very different Sobolev spaces In the following we fix a bounded domain D of R d and we consider the admissible class For every Ω ∈ A and u ∈ W 1,p 0 (Ω) we define the integral functional where the integrand f is assumed to verify the following conditions: being λ 1,p (D) the first Dirichlet eigenvalue of the p-Laplacian on D, defined as (f3) f (x, 0, 0) 0. It is well-known (see for instance [But89]) that under conditions (f1) and (f2) for every Ω ∈ A the functional F (·, Ω) defined in (2.1) is lower semicontinuous with respect to the weak convergence in W 1,p 0 (Ω) and that the minimum problem admits a solution. Let us denote by F(Ω) the minimum value in (2.2). The shape optimization problem we deal with is In the following theorem we prove that the shape optimization problem above admits a solution. For the proof we could use the general theory of γ-convergence and weak γ-convergence (see [BB05]), and the fact that the shape functional F has some monotonicity properties with respect to the set inclusion; however, in our case a simpler proof is available and we report this one.
Proof. -Consider the auxiliary minimum problem and, thanks to assumptions (f1) and (f2), it verifies the lower semicontinuity and coercivity properties that guarantee it admits a solutionū. We claim that the p-quasi open set Ω 0 = {ū = 0} solves the shape optimization Problem (2.3). Indeed, let Ω ∈ A and let u Ω be the solution of the minimum Problem (2.2); then we have where the last inequality follows from the definition of Ω 0 and from assumption (f3).
Remark 2.2. -Notice that, when f (x, 0, 0) = 0, the functional F(Ω) is decreasing with respect to the set inclusion. Indeed, in this case we have for every u ∈ W 1,p 0 (Ω) Therefore, when f (x, 0, 0) = 0, if Ω 0 is a solution of the shape optimization problem (2.3), then every Ω ⊃ Ω 0 is also a solution. In particular, the whole set D is a solution of (2.3).

Existence of open optimal domains
In the present section we show that, under mild additional assumptions on the integrand f , the optimal domain Ω 0 of problem (2.3), obtained in Theorem 2.1 is actually an open set. To do this we show that the solutionū of the auxiliary minimum problem (2.4) is a continuous function. This follows by means of a well-known result of Giaquinta and Giusti in [GG84] (see also [Giu03]), that we summarize here below for the sake of completeness.
Remark 3.2. -In the paper [GG84] the integrand h above was assumed of Carathéodory type, but in fact condition (f1) is still enough, provided condition (3.1) is satisfied. Actually, as the authors say, even the convexity of h with respect to z is not needed, if we assume that a solutionū exists. We can now apply Theorem 3.1 to obtain that in a large number of situations the optimal set Ω 0 obtained in Theorem 2.1 is actually an open set. Proof. -Since Ω 0 = {ū = 0} whereū is a solution of the auxiliary problem (2.4), it is enough to show that the functionū is continuous on D. We have for every satisfies the conditions of Theorem 3.1. Then the Hölder continuity ofū follows.

Optimal domains that are not open
As we have seen in Theorem 3.4 quite mild assumptions on the integrand f imply the existence of open optimal domains Ω opt . In this section we show that, when these assumptions are not satisfied, there may exist optimal domains which are not better than quasi open sets, even in very simple cases as the Dirichlet energy We start by a preliminary result. Using the optimality ofū and the fact that f (x, 0, 0) = 0 we obtain

ANNALES HENRI LEBESGUE
which implies (4.1). We consider here two shape optimization problems for the Dirichlet energy; we set (4.2) F(Ω) = min The first problem, that we may call penalized problem, has the form with λ > 0, while the second one, that we may call constrained problem has the form with m > 0. From what seen in the previous sections both the shape optimization problems (P λ ) and (Q m ) admit a solution.
Proposition 4.2. -Let λ > 0 and let Ω 0 be an optimal domain for the shape optimization problem (P λ ). Then there exists m > 0 such that Ω 0 solves the shape optimization problem (Q m ).
Proof. -Take m = |Ω 0 | and let Ω ∈ A with |Ω| m. By the optimality of Ω 0 for (P λ ) we have which proves that Ω 0 solves the shape optimization problem (Q m ) too.

TOME 3 (2020)
While Proposition 4.2 is rather simple, the opposite implication, showing that an optimal domain for (Q m ) also solves (P λ ) for some λ, is a very delicate issue, which has been studied in [Bri04]. In particular, a rigorous proof of the equivalence of the two formulations is not available in full generality, see for instance [Bri04], [BHP05] and [Hay99] for a discussion on this matter.
Here we show that problem (Q m ) may have optimal domains that are not open, when the summability of the datum g is not strong enough. We consider the unconstrained problem  w(x) > 0 for q.e. x ∈ Ω 0 , so that we have Ω 0 = {w > 0}. We claim that the function g = −∆ p (w p ) is in W −1, p (D) ∩ L 1 (D). Indeed, by the maximum principle w is bounded and, since ∇(w p ) = p w p − 1 ∇w, we get that w p ∈ W 1, p 0 (D) and so g ∈ W −1, p (D). Moreover, for every ψ ∈ C ∞ c (D) we have Since w ∈ L ∞ (D) and w ∈ W 1,p 0 (D) we obtain that g ∈ L 1 (D). Now, we apply Proposition 4.1 with g as above; since the functional in the minimization problem (4.2) is strictly convex, its minimizer is unique and so this implies that the functionū coincides with w p . Hence Ω opt is the set {w p = 0}, which coincides with Ω 0 .
Remark 4.4. -We have seen that if p > d or if the function g in (4.2) is in L q (D) with q > d/p then the optimal set Ω opt is open. On the contrary, if q = 1 we can construct a counterexample showing that Ω opt is merely a p-quasi open set, and actually every p-quasi open set Ω 0 can be optimal for some g ∈ L 1 (D). This picture is sharp when p = d in the sense that in this case q = 1 is the borderline situation and q 1 gives a counterexample, while q > 1 gives that Ω opt is open. When p < d we do not know if similar counterexamples hold in the case 1 < q d/p. In addition, it would be interesting to provide counterexamples showing that also the optimal domains for the penalized problem (P λ ) may be not open if the data are not summable enough.

Cases when optimal domains have finite perimeter
In this section we show that, under some assumptions slightly stronger than (f1), (f2), (f3) the optimal set Ω 0 obtained in Section 2 has a finite perimeter. We adapt the proof contained in [Buc12] to our general case. The assumptions we need are: (f2") there exist c > 0 and α < λ 1,p (D) such that for every x, s, z c |z| p − α|s| p + 1 f (x, s, z); (f3") there exist K > 0 and a ∈ L 1 (D) such that for every x, s, t, z f (x, s, z) − f (x, t, z) K|s − t| a(x) + |s| p * + |t| p * + |z| p , where p * = dp/(d−p) (with p * any positive number if p = d and |·| p * replaced by any continuous function if p > d) is the Sobolev exponent associated to p.