An optimal bound for nonlinear eigenvalues and torsional rigidity on domains with holes

In this paper we prove an optimal upper bound for the first eigenvalue of a Robin-Neumann boundary value problem for the p-Laplacian operator in domains with convex holes. An analogous estimate is obtained for the corresponding torsional rigidity problem.


Introduction
Let β ą 0, and Ω and D be two bounded open sets, with D convex, Ω Lipschitz, connected, and D Ť Ω. We will denote by Σ " ΩzD.
Let us consider the following problem: where β is a positive constant and 1 ă p ă`8. A minimizer u P W 1,p pΣq of (1.1) Bν`β |u| p´2 u " 0 on BD. (1.2) In this paper we obtain an optimal upper bound on λpβ, Σq, when β ą 0 is fixed and Σ " ΩzD varies among all domains such that the volume of Σ and the pn´1q-quermassintegral of D are given (see Section 2.3 for the precise definition). If n " 2, the geometrical constraint on D corresponds to fix its perimeter. In particular, we show that λpβ, Σq is maximized by the spherical shell. The first main result is the following.
In the case p " n " 2, Theorem 1.1 recovers a result proved by Hersch in [18]. Our result generalizes it to the p-Laplacian operator and to any dimension, giving an answer to a question posed by Henrot in [17, Chapter 3, Open problem 5]. Actually, we are not able to prove or disprove that the first eigenvalue λpβ, Σq is maximized on the spherical shell when the perimeter of D, and not the W n´1 , is fixed. We stress that a related result has been recently proved for an optimal insulating problem (see [14]).
Optimal estimates for eigenvalues with different Robin or Robin-Neumann boundary conditions have been proved by several authors. For example, if D " H, and a Robin condition on BΩ is given, with β ą 0, a Faber-Krahn inequality has been proved in [3,9] for p " 2 and in [5,8] for p Ps1,`8r, stating that the first eigenvalue is minimum on the ball of the same volume of Ω. Otherwise, if β is negative the problem is still open; in this direction, in [15] the authors showed that, among planar domains of the same measure, the disk is a maximizer only for small value of the parameter. On the other hand, if the perimeter rather than the volume is fixed, the ball maximizes the first eigenvalue among all open, bounded, convex, smooth enough sets (see [2,6]).
In this framework, the case of Robin eigenvalue problems when β ě 0 is not a constant has been considered in [16], and, for example, in [4,13,11] where optimization with respect to β is considered.
Coming back to the problem with mixed boundary conditions, we recall that the case of a Neumann condition on BD and Robin condition on BΩ has been considered in [22,20].
Another problem we deal with is the p-torsional rigidity with the same boundary conditions, namely with β ą 0. The unique function u Σ ą 0 such that is a maximizer of (1.3), and Moreover, any maximizer of (1.3) is proportional to u Σ . We prove that the spherical shell minimizes T pβ, Σq among all domains such that the volume of Σ and the pn´1q-quermassintegral of D are given.
is a ball centered at the origin with radius R i , i " 1, 2. Suppose that |A| " |Σ|, and W n´1 pB R 1 q " W n´1 pDq. Then, Regarding optimal estimates related to the torsional rigidity when D " H and a Robin boundary condition is given on BΩ, a Saint-Venant inequality can be proved: we refer the reader to [1,7].
Finally, we recall that in this context, estimates have been obtained also for a more general class of problems, involving the so called Finsler operator. We refer the reader, for example, to [10,12,21].
The structure of the paper is the following. In the second section, we prove some properties of the first eigenvalue λpβ, Σq and of the torsional rigidity T pβ, Σq, as well as we recall some basic tool of convex analysis. In the third section, we prove the main results.

Notation and preliminaries 2.1 Eigenvalue problem
The following result can be proved by a standard argument of Calculus of Variations.
• There exists a positive minimizer u P W 1,p pΣq of (1.1). Moreover, u is a solution of (1.2).
• The eigenvalue λpβ, Σq is simple, that is all the associated eigenfunctions are scalar multiple of each other.
Another simple upper bound for λpβ, Σq is given by that is the first eigenvalue of the corresponding Dirichlet-Neumann problem.

It holds that
3. The first eigenvalue λpβ, Σq is concave with respect to β.
Proof. Let us denote by Let u β , u β`h be two positive minimizers of (1.1) associated to β and β`h, respectively. Moreover, suppose On the other hand, Finally, Then, being tu β`h u h bounded in W 1,p pΩq, there exists a subsequence, still denoted by u β`h such that u β`h Ñ u β strongly in L p and almost everywhere and ∇u β á ∇u β weakly in L p pΩq. As a consequence, by the compactness of the trace operator (see for example [19,Cor. 18.4]), u β`h converges strongly to u β in L p pBDq.
Taking into account the boundary conditions ψ 1 pR 2 q " 0, it follows by integrating that ψ 1 prq ą 0, and this concludes the proof.

The p-torsional rigidity
Similarly to the case of the first eigenvalue, the following results hold.

Quermassintegrals and the Aleksandrov-Fenchel inequalities
Here we list some basic properties of convex analysis which will be useful in the following. For an extended discussion on the subject we refer the reader to [23].
Let K be a bounded convex open set, and B 1 " tx : |x| ă 1u. The outer parallel body of K at distance ρ ą 0 is the Minkowski sum K`ρB 1 " tx`ρy P R n | x P K, y P B 1 u.
The Steiner formulas assert that The coefficients W i pKq are known as the quermassintegrals of K. In particular, it holds that W 0 pKq " |K| , nW 1 pKq " P pKq, W n pKq " ω n .
The Aleksandrov-Fenchel inequalities state that where the inequality is replaced by an equality if and only if K is a ball.
In what follows, we use the Aleksandrov-Fenchel inequalities for particular values of i and j. When i " 0 and j " 1, we have the classical isoperimetric inequality: Let us denote by K˚a ball such that W n´1 pKq " W n´1 pK˚q. Then by Aleksandrov-Fenchel inequalities (2.4), for 0 ď i ă n´1

Proof of main results
In this Section we prove the main results (Theorem 1.1 and Theorem 1.2).
Proof of Theorem 1.1. Let vpxq " ψp|x|q be a positive radial solution of problem (1.1) on A, and denote by v m " ψpR 1 q and v M " ψpR 2 q be the minimum and maximum of v, respectively. For x P Σ, let us denote by dpxq the distance of x from D, dpxq " inft|x´y| , y P Du, and consider as test function where G is defined as with gptq " |Dv| v"t . We observe that vpxq " Gp|x|´R 1 q and w satisfies the following properties: Since R 2´R1 " G´1pv M q, to prove that the maximum value of w is v M , we need to verify that G´1pv M q " ş v M vm 1 gpτ q dτ . Indeed since vpxq " ψp|x|q, then Let us denote by Let us observe that E t Ď tx P R n : dpxq ă G´1ptqu :"Ẽ t , F t " x P R n : |x| ă R 1`G´1 ptq ( . By Steiner formula and the Aleksandrov-Fenchel inequalities, we get, as ρ " G´1ptq, that H n´1 pBE t X Σq ď P pẼ t q " P pD`ρB 1 q " n n´1 ÿ k"0ˆn´1 k˙W k`1 pDqρ k ď n n´1 ÿ k"0ˆn´1 k˙W k`1 pB R 1 qρ k " P pB R 1`ρ Bq " P pF t q. (3.1) Using now the coarea formula and (3.1): Since, by construction, wpxq " w m " v m on BD, then ż BD w p dH n´1 " w p m P pDq ď v p m P pB r 1 q " Now, we define µptq " |E t X Σ| and ηptq " |F t | and using again coarea formula, we obtain, for v m ď t ă v M , µ 1 ptq " ż tw"tuXΣ 1 |∇wpxq| dH n´1 " H n´1 pBE t X Σq gptq ď P pẼ t q gptq ď P pF t q gptq " ż tv"tu 1 |∇vpxq| dH n´1 " η 1 ptq.
This inequality is trivially true also if 0 ă t ă v m . Since µp0q " ηp0q " 0, by integrating from 0 to t ă v M , we have: On the other hand, we have