On a reverse Kohler-Jobin inequality

We consider the shape optimization problems for the quantities $\lambda(\Omega)T^q(\Omega)$, where $\Omega$ varies among open sets of $\mathbb{R}^d$ with a prescribed Lebesgue measure. While the characterization of the infimum is completely clear, the same does not happen for the maximization in the case $q>1$. We prove that for $q$ large enough a maximizing domain exists among quasi-open sets and that the ball is optimal among {\it nearly spherical domains}.


Introduction
In the present paper we consider two well-known quantities that occur in the study of elliptic equations in the Euclidean space R d , d ≥ 2. The first one is usually called torsional rigidity and is defined, for every nonempty open set Ω ⊂ R d with finite Lebesgue measure (in the following a domain), as where w Ω is the unique solution of the PDE −∆u = 1 in Ω, u ∈ H 1 0 (Ω).Equivalently, we may define T (Ω) as In the integrals above and in the following we use the convention that integrals without the indicated domain are intended over the entire space R d .The quantity T (Ω) verifies the scaling property T (tΩ) = t d+2 T (Ω) for every t > 0; in addition, the maximum of T (Ω) among domains with prescribed measure is reached by the ball (Saint Venant inequality), which can be written in the scaling free formulation as for every domain Ω and for every ball B ⊂ R d .
The second quantity is the first eigenvalue λ(Ω) of the Dirichlet Laplacian, defined as the smallest λ such that the PDE −∆u = λu in Ω, u ∈ H 1 0 (Ω) admits a nonzero solution.Equivalently, λ(Ω) can be defined through the minimization of the Rayleigh quotient The quantity λ(Ω) verifies the scaling property λ(tΩ) = t −2 λ(Ω) for every t > 0; in addition, the minimum of λ(Ω) among domains with prescribed measure is reached by the ball (Faber-Krahn inequality), which can be written in the scaling free formulation as for every domain Ω and for every ball B ⊂ R d .
The study of relations between T (Ω) and λ(Ω) was performed in several papers (see for instance [1], [2], [3], [4], [5], [12], [13], [18], [21], [22], [23]), where some important inequalities were established.In particular: -the Kohler-Jobin inequality λ(Ω)T q (Ω) ≥ λ(B)T q (B), valid for every q ∈ [0, 2/(d + 2)] and for every domain Ω, where B is any ball in R d with |B| = |Ω|; -the Pólya inequality valid for every domain Ω of R d .In the present paper we consider the scaling free shape functional and the two quantities While the situation for m q is fully clear, and by Kohler-Jobin inequality, together with the Saint Venant inequality, we have the characterization of M q is not yet complete.The results available up to now are (see [1] and [3]): M q = ∞ for every q < 1; M q = 1 when q = 1, with the upper bound 1 not reached by any domain Ω; M q < ∞ for every q > 1.We investigate here this last case.The maximal expectation would be having the following result (reverse Kohler-Jobin inequality): -for every q > 1 the supremum M q is reached on an optimal domain Ω q ; -there exists a threshold q * > 1 such that for every q ≥ q * the supremum M q is reached by a ball.We are unable to prove the results in the strong form above, and we prove here the weaker results below: -for every q > 1 the supremum M q is reached on a capacitary measure µ q (Theorem 4.3); -there exists a threshold q 0 > 1 such that for every q ≥ q 0 the supremum M q is reached by a domain Ω q (Theorem 5.3); -there exists another threshold q 1 such that for every q ≥ q 1 the ball is a maximizer for the shape functional F q among nearly spherical domains (Theorem 6.2).
While finishing this paper we have been informed that similar problems are considered in the work in progress [11].

Capacitary measures
The concept of capacitary measure and the related properties is a very useful tool for our purposes.When dealing with sequences of PDEs of the form , a natural question is to establish if the sequence u n,f of solutions, or a subsequence of it, converges in L 2 to some function u f and to determine in this case the PDE that the function u f solves.Starting from the pioneering papers [15], [16] is now well understood that the right framework to treat such a kind of questions is that of capacitary measures.Below we recall the main results and definitions following [10] and [24].For further information we refer the reader to the monographs [8], [20] and references therein.Definition 2.1.We say that a nonnegative Borel regular measure µ, possibly taking the value ∞, is a capacitary measure if A property P (x) is said to hold quasi-everywhere (briefly q.e.) if the set where P (x) does not hold has zero capacity.A Borel set Ω ⊂ R d is said to be quasi-open if there exists a function u ∈ H 1 (R d ) such that Ω = {u > 0} up to a set of capacity zero.A function f : R d → R is said to be quasi-continuous if there is a sequence of open sets ω n ⊂ R d such that lim n→∞ cap(ω n ) = 0 and f is continuous when restricted to R d \ ω n .It is well known (see for instance [19]) that every Sobolev function has a quasi-continuous representative, and that two quasi-continuous representatives coincide quasi-everywhere.We then identify the space H 1 (R d ) with the space of quasi-continuous representatives.We recall that a sequence u n ∈ H 1 (R d ) that converges in norm to some u ∈ H 1 (R d ), converges quasi-everywhere (up to a subsequence) to u.
Given µ a capacitary measure we denote by H 1 µ the following space The space H 1 µ is an Hilbert space when endowed with u , where the quantity u L 2 µ (R d ) is well defined, being Sobolev functions defined up to a set of zero capacity.We always identify two capacitary measures µ, ν for which (2.1) If instead (2.1) holds with "≤" we say that µ ≤ ν, and in this case we have H 1 ν ⊆ H 1 µ .We can associate to any open set (or more generally to any quasi-open set) Ω ⊂ R d the capacitary measure I Ω defined as follows To extend the notion of torsional rigidity to a capacitary measure µ we need to carefully deal with the fact that the embedding H 1 µ ֒→ L 1 (R d ) can be noncompact and even noncontinuous.Nevertheless we can follow an approximation argument: for every R > 0, let w R be the solution to the following minimization problem The torsion function w µ and the torsional rigidity T (µ) of the capacitary measure µ are defined as: The Dirichlet eigenvalue of µ can be defined through the following Rayleigh-type quotient: Clearly, if µ = I Ω for some domain Ω ⊂ R d , we have T (µ) = T (Ω) and λ(µ) = λ(Ω) (we adopt this notation also if Ω is a quasi-open set).For a general capacitary measure µ, neither λ(µ) is necessarily attained by some function u ∈ H 1 µ nor T (µ) is necessarily finite.However, as shown in [9], it holds the following: For every capacitary measure µ with T (µ) < ∞ we define the set of finiteness A µ as the quasi-open set In the case when µ = I Ω , for some domain Ω ⊂ R d , we have A µ = Ω.The set of capacitary measures with finite torsion can be endowed with the following notion of distance.
Definition 2.2.Given two capacitary measures µ, ν such that We summarize the main properties of the γ−distance below: • The space ({µ : µ capacitary measure with • The map µ → |A µ |, or more generally integral functionals as Aµ f (x) dx with f ≥ 0 and measurable, are lower semicontinuous with respect to the γ-convergence.
• For a given capacitary measures µ with finite torsion we call resolvent of µ the linear compact and self-adjoint operator where w µ,f is the solution of the problem The γ-convergence of µ n to µ implies the norm convergence of R µn to R µ , i.e.
• If µ n is a sequence of capacitary measures whose set of finiteness have uniformly bounded measures |A µn |, then The classical concentration-compactness principle of P.L. Lions was extended to sequences of open sets in [7].Notably, the following result holds.
Theorem 2.3.Let Ω n be a sequence of open sets with uniformly bounded measures.Then there exists a subsequence (still denoted with the same indices n) such that one of the following situations occurs.
-Compactness: there exists a sequence x n ⊂ R d such that the sequence of capacitary measures The proof of the theorem above can be deduced by combining Theorem 2.2 of [7] and Theorem 3.5 of [10].

Relaxation of F q
In this section we characterize the relaxation of the functional F q to the set of capacitary measures.We define the set M ad of admissible capacitary measures as For µ ∈ M ad we define the relaxed form of our functional F q as Proof.Being the sequence Ω n ∩ A µ of uniformly bounded measure, by the properties of γconvergence seen above we have to show that , where we set µ n = I Ωn∩Aµ .
The "Γ-liminf" inequality readily follows by the fact that To prove the "Γ-limsup" inequality we can suppose without loss of generality that u ∈ H We denote respectively by u + n and u − n the positive and negative part of u n .Since we have and u n = u + n − u − n , by possibly passing to a subsequence (still indexed by n) we can suppose that lim sup We define µ and u n ∈ H 1 0 (Ω n ) we have u = 0 q.e. on A c µ and u n = 0 q.e. on Ω c n .This implies that both v + n and v − n vanish q.e. on (Ω n ∩ A µ ) c and consequently that v We have By lower semicontinuity we have lim inf Indeed, to show the inequality above, it is enough to write and to notice that This in turns implies that the set Furthermore, we can extend both Saint-Venant, Faber-Krahn and Pólya inequalities to any capacitary measure.That is and 0 for every measure µ ∈ M ad and every ball B ⊂ R d .
Proposition 3.3.Let µ ∈ M ad .Then we have The quantity |A µ | is then the relaxation, in the γ-convergence, of the Lebesgue measure |Ω|.
As a consequence, we have Proof.The inequality ≤ in (3.9) follows from the γ-lower semicontinuity of the map µ → |A µ | seen above.The opposite inequality follows at once by Remark 3.2.Since T (µ) and λ(µ) are γ-continuous, the proof of (3.10) is achieved by a similar argument.
The scaling properties of the shape functionals |Ω|, λ(Ω), T (Ω) and F q (Ω) extend to their relaxations |A µ |, λ(µ), T (µ) and F q (µ) in M ad .More precisely, setting for t > 0 we have 4. Existence of an optimal measure for q > 1 In [3] it is proved that the supremum M 1 = 1 is not attained in the class of domains.In the next proposition we point out that the same occurs even in the class M ad .Proposition 4.1 (Nonexistence for q = 1 of an optimal measure).Given a capacitary measure µ ∈ M ad the problem sup{F 1 (µ) : µ ∈ M ad } does not have a maximizer.
Proof.The proof follows at once by exploiting Theorem 1.1.of [3] which asserts that there exists a dimensional constant c d > 0 for which for every domain Ω.Then, for every µ ∈ M ad , by Remark 3.2 we can select a sequence Ω n γ → A µ for which Thus, using (4.1) with Ω = Ω n and passing to the limit as n → ∞, we get To prove the main result of this section we need the following elementary lemma.
Then, there exists β < 1 such that, for every a, b, c, d ∈ (c 1 , c 2 ) it holds Proof.Letting x = b/a and y = d/c, is enough to prove that Suppose that x ≤ y.Since x ≥ c 1 c 2 , it holds Eventually we achieve the thesis by letting and combining (4.2) and (4.3).
Theorem 4.3 (Existence for q > 1 of an optimal measure).For every q > 1 there exists a measure µ ⋆ ∈ M ad such that Proof.We select a sequence µ n ∈ M ad such that F q (µ n ) → M q , as n → ∞.By density, we can suppose that µ n = I Ωn , for some sequence of open sets Ω n .Further, being F q scaling free, we can also assume |Ω n | = 1.Hence, we can apply Theorem 2.3.If dichotomy occurs, then there exist two sequences of quasi-open sets Taking into account the Saint-Venant inequality and the fact that |Ω n | = 1, there exist constants c 1 , c 2 > 0, which depend only on the dimension, such that Since λ 1 is increasing with respect to set inclusion, we have Lemma 4.2 together with (4.4) gives By taking the limit for n → ∞ in the latter inequality we obtain the contradiction sup and hence dichotomy cannot occur.Now, the maximality condition on the sequence Ω n together with Pólya inequality gives that for n large enough where B is any ball of R d .In particular it cannot be lim n→∞ T (Ω n ) = 0, and this rules out the vanishing case.Therefore compactness holds and there exists a capacitary measure µ ⋆ and a sequence x n ∈ R d such that I xn+Ωn γ−converges to µ ⋆ .By (4.5) we deduce that T (µ ⋆ ) > 0 which by (3.7) implies |A µ ⋆ | > 0 and hence that µ ⋆ belongs to M ad .Clearly the measure µ ⋆ maximizes the functional F q on M ad and this concludes the proof.

Optimal measures are quasi-open sets for large q
We are now interested to prove that, when q is large enough, optimal measures µ coming from Theorem 4.3 can be represented as quasi-open sets.We begin by recalling the following result, see [17] and [24] Proposition 3.83.Theorem 5.1.Let µ be a capacitary measure with finite torsion.Then the eigenfunctions u ∈ L 2 (R d ) of the operator −∆ + µ with unitary L 2 norm are in L ∞ (R d ) and satisfy We also use the following lemma.Lemma 5.2.For every q > 1 let µ q ∈ M ad be a maximal measure for the functional F q , such that Proof.Let q n be a diverging sequence and B ⊂ R d be a ball of unitary measure.
Hence we deduce lim inf n→∞ which implies that it cannot be T (Ω n ) → 0, as n → ∞.Therefore compactness holds true and the sequence Ω n has a subsequence (still denoted by the same indices) that γ-converges to some µ ∈ M ad up to translations.By the maximality of µ qn it holds ) 1/qn and we deduce, passing to the limit as n → ∞ Since the sequence q n was arbitrary we obtain the conclusion.
Theorem 5.3.Let µ ∈ M ad be an optimal measure for F q with q > 1.There exists q 0 > 1 such that for q > q 0 we have µ = I Aµ .In particular the optimal measure can be represented by a quasi-open set.
Proof.Since F q is scaling free, we can suppose that |A µ | = 1.Let ε > 0 be a small parameter and let µ ε be the capacitary measure defined by Being A µ = A µε we have µ ε ∈ M ad .We assume by contradiction that µ = I Aµ (notice that this implies µ ε = µ).For the sake of brevity, we denote respectively by w and w ε the torsion functions of µ and µ ε .It is easy to verify that, as ε → 0, and therefore we have µ ε γ → µ and w ε → w in L 1 (R d ), as ε → 0. Let us denote by t(ε), l(ε) and f q (ε) the real functions and by t ′ + (0), l ′ + (0), (f q ) ′ + (0) the limits for ε → 0 of the respective different quotients.By writing w ε = w + εξ ε for some ξ ε ∈ L 1 (R d ) and using the fact that w, w ε respectively weakly solve the PDEs: ) we deduce that ξ ε weakly solves the PDE − ∆ξ ε + ξ ε µ ε = wµ. ( This allows us to compute the derivative where we test (5.2) with ξ ε and we use (5.3) tested with w ε .Since, as ε → 0, We can treat with a similar argument the eigenvalue.Let u, u ε be the first eigenfunctions (with unitary L 2 norm) respectively of the operator −∆ + µ ε and −∆ + µ and let By testing the PDE above with u ∈ H 1 µ and since u 2 dx = 1, we obtain λ By taking the limit as ε → 0 and exploiting the fact that u ε → u weakly in H 1 µ and λ(µ ε ) → λ(µ) we get (5.5) By combining (5.4) and (5.5) we get Now, the optimality condition on µ implies (f q ) ′ (0) ≤ 0 and hence that We claim that u 2 λ(µ) − q w 2 T (µ) < 0 q.e on R d (5.7) for q large enough.Indeed, by an application of Theorem 5.1 together with a comparison principle, we have u ≤ e 1/(8π) λ d/4+1 (µ)w q.e on R d , and so by the Pólya inequality The latter implies that Therefore, for every q such that sup µ∈M ad e 1/(4π) λ d/2 (µ) < q, (5.7) is verified.Notice that the supremum in the inequality above is finite as a consequence of Lemma 5.2 combined again with Pólya inequality.
To conclude it is now enough to notice that (5.7) contradicts (5.6).

Optimality for nearly spherical domains
In the following we consider the classes S δ,γ of nearly spherical domains.Let B 1 be the unitary ball of R We recall the following result.for suitable constants C 1 and C 2 depending only on the dimension d.
Proof.The inequality for the torsional rigidity follows from Theorem 3.3 in [6] while the inequality for the eigenvalue follows by combining Theorem 1.2 and Lemma 2.8 of [14].
Hence, if q is such that q ≥ 2 d+2 C 2 C 1 T (B 1 ), we obtain λ(B 1 )T q (B 1 ) ≥ λ(Ω)T q (Ω) and this concludes the proof.Remark 6.3.Although for large q we expect the ball to be optimal for the functional F q , it is easy to see that this does not occur when q approaches 1.Indeed, if the ball maximizes F q for every q > 1, passing to the limit as q → 1, this would happen also for q = 1, which is not true, even in the class of convex domains.To see this it is enough to notice that By a standard diagonal argument we can select a sequence Ω n ⊂ R d of open sets such that |Ω n | = 1 for every n and |F qn (Ω n ) − F qn (µ qn )| = o(T qn (B)) as n → ∞. (5.1)Then we can apply Theorem 2.3 to the sequence Ω n .Dichotomy can be ruled out by the same argument as in the proof of Theorem 4.3 once noticed that a combination of (3.7) and (3.8) implies F 1/qn qn (µ) ≤ T (qn−1)/qn (B) → T (B) as n → ∞.The vanishing case can be excluded too by following again the proof of Theorem 4.3.Indeed, for n large enough, Pólya inequality and (5.1) imply d .A domain Ω such that |Ω| = |B 1 |, Ω xdx = 0, belongs to the class S δ,γ if there exists φ ∈ C 2,γ (∂B 1 ) with φ L ∞ (∂B 1 ) ≤ 1/2 and such that ∂Ω = {x ∈ R d : x = (1 + φ(y))y, y ∈ ∂B 1 }, φ C 2,γ (∂B 1 ) ≤ δ.