P´olya-type estimates for the ﬁrst Robin eigenvalue of elliptic operators

. The aim of this paper is to obtain optimal estimates for the ﬁrst Robin eigenvalue of the anisotropic p -Laplace operator, namely:

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p-Laplace operator, namely: , where p Ps1, `8r, Ω is a bounded, convex domain in R N , ν Ω is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on R N .We show an upper bound for λ F pβ, Ωq in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on the volume and the anisotropic perimeter of Ω, in the spirit of the classical estimates of Pólya [Po] for the Euclidean Dirichlet Laplacian.
We will also provide a lower bound for the torsional rigidity τ p pβ, Ωq p´1 " max when β ą 0. The obtained results are new also in the case of the classical Euclidean Laplacian.
Key words and phrases: Nonlinear eigenvalue problems; Robin boundary conditions; Finsler norm; Optimal estimates.

Introduction
The purpose of this work is to provide optimal upper bounds for the first Robin eigenvalue of a Finsler p-Laplacian operator with Robin boundary conditions, as well as lower bounds for the corresponding torsional rigidity, in convex domains.This will be achieved by formulating these quantities in terms of a one-dimensional eigenvalue problem.The results presented here are novel, even in the well-established context of the Euclidean Robin-Laplacian operator.To be more specific, let us first define where p Ps1, `8r, Ω is a bounded Lipschitz domain in R N , ν Ω is its Euclidean outward normal, β is a real number, F : R N Ñ r0, `8r, N ě 2, is a convex, even, 1-homogeneous and C 2 pR N zt0uq function such that rF 2 s ξξ is positive definite in R N zt0u (we refer the reader to Section 2 for all the definitions, here and below used, related the Finsler metric F ).When β " 0, λ F pβ, Ωq " 0 corresponds to the first (trivial) Neumann eigenvalue, while, when β goes to `8, it reduces to the first Dirichlet eigenvalue λ D F pΩq.Let us observe that, by using ψ " 1 and ψ " u D , the first Dirichlet eigenfunction, in (1.1), it holds that λ F pβ, Ωq ď min where P F pΩq is the anisotropic perimeter of Ω and |Ω| is its Lebesgue measure.We get in particular that λ F pβ, Ωq Ñ ´8 as β Ñ ´8.
If u P W 1,p pΩq is a minimizer of (1.1), then it satisfies where In the simplest case where F pξq " Epξq :" ař i |ξ i | 2 is the Euclidean norm, then Q p " ∆ p u " divp|∇u| p´2 ∇uq is the standard p-Laplacian.
In the Dirichlet-Laplacian case, a sharp estimate of the first eigenvalue in terms of perimeter and volume can be given.Indeed where Ω is a bounded convex open set of R N and |Ω| and P F pΩq denote respectively the Lebesgue measure and the anisotropic perimeter of Ω, and (1.4) The estimate is optimal, being achieved asymptotically when the convex domain Ω goes to an infinite N ´dimensional slab s ´a, arˆR N ´1 (up to a suitable rotation).
In the planar, Euclidean, Laplacian case (N " p " 2 and F " E), this is a classical result obtained by Pólya in [Po].Then it was generalized, in any dimension, for any p Ps1, `8r and for any sufficiently smooth norm F in [DG].Being λ F pβ, Ωq ď λ D F pΩq, the estimate (1.3) is true also in the case of the first Robin eigenvalue, but it is no longer optimal.The main contribution of this paper lies in demonstrating that it is possible to refine the Pólya estimate when switching from Dirichlet to Robin boundary conditions.
where µ 1 pβ, s 0 q " min hPA ż s 0 0 r´h 1 psqs p ds `βh ps 0 q p ż s 0 0 rhpsqs p ds where s 0 " |Ω| P F pΩq , and A is the class of positive decreasing functions in W 1,p p0, s 0 q.The estimate is optimal, being achieved asymptotically when Ω goes to a suitable slab.
Remark 1.2.We draw attention to the fact that this result is new even for the Euclidean norm and for the case of p " 2.
Remark 1.4.In the case p " 2 (hence in the Laplacian case if F is the Euclidean norm), we get a fine estimate for µ 1 by using the expression of µ 1 and a Becker-Stark inequality [BS].So we have the following Corollary 1.5.Suppose that p " 2, and that the hypotheses of Theorem 1.1 are satisfied, with β ą 0. Then We observe that other kind of optimal lower bounds, obtained in terms of different geometrical quantities related to Ω (as the inradius), can be found for example in [Sp, Sa, LW] (in the Euclidean case) and in [DP2].
The second aim of the paper is also to provide a lower bound to the anisotropic ptorsional rigidity with Robin boundary conditions (β ą 0), namely the value τ F pβ, Ωq such that or, equivalently where u p P W 1,p pΩq is the unique solution of Let us remark that, by choosing ψ " 1, or ψ " u Ω the Dirichlet torsion function, it holds that where τ D F pΩq is the torsional rigidity with Dirichlet boundary conditions.The obtained lower bound for τ p is the following. . (1.9) When β " `8, we recover the Pólya estimate contained in [Po] for the Euclidean Laplacian and then in [DG] for the general case.
The structure of the paper is the following.In Section 2 we recall some useful properties of the Finsler norm, as well as some basic definitions of the anisotropic perimeter and of convex analysis.Moreover, in Section 3 we recall the main properties of the one dimensional nonlinear eigenvalue problem (1.6).Finally, in Section 4 we give the proof of the main results.
2 Notation and preliminaries 2.1 The Finsler norm be a convex, even, 1´homogeneous function, that is a convex function such that and such that for some constants 0 ă a ď b.Moreover, we suppose that F P C 2 pR N zt0uq, and that 3) The hypothesis (2.3) on F ensures that the operator Q p rus :" div ˆ1 p ∇ ξ rF p sp∇uq is elliptic, namely there exists a positive constant γ such that for some positive constant γ, for any η P R N zt0u and for any ξ P R N .The polar function of F is From (2.4) it holds that xξ, ηy ď F pξqF o pηq @ξ, η P R N . (2.5) The set is Wulff shape centered at the origin.We put κ N " |W|, where |W| denotes the Lebesgue measure of W.More generally, we denote with W r px 0 q the set rW `x0 , that is the Wulff shape centered at x 0 with measure κ N r N , and W r p0q " W r .The anisotropic distance of x P Ω to the boundary of a bounded domain Ω is the function We stress that when F " | ¨| then d F " d E , the Euclidean distance function from the boundary.
It is not difficult to prove that d F is a uniform Lipschitz function in Ω and F p∇d F pxqq " 1 a.e. in Ω.
Obviously, d F P W 1,8 0 pΩq.Finally, the anisotropic inradius of Ω is the quantity that is the radius of the largest Wulff shape W r pxq contained in Ω.
Let us finally recall the definition of anisotropic perimeter of a set K Ă R N in Ω: The following co-area formula for the anisotropic perimeter ´8 P F ptu ą su, Ωq ds, @u P W 1,1 pΩq (2.6) holds, moreover where H N ´1 is the pN ´1q´dimensional Hausdorff measure in R N , B ˚K is the reduced boundary of F and ν F is the outer normal to F .As usual, we will denote by P F pKq the perimeter in R N , that is P F pK, R N q.

A one dimensional p-Laplacian eigenvalue problem
Here we briefly summarize the definitions and some properties of the p-trigonometric functions.These functions are generalizations of the standard trigonometric functions, and they coincide with the standard trigonometric functions when p " 2. We refer the reader, for example, to [LE, Li].The function arccos p : r0, 1s Ñ R is defined as If zptq is the inverse function of arccos p , which is defined on the interval " 0, πp 2

‰
, where then, the p-cosine function cos p is the even function defined as the periodic extension of zptq: and extended periodically to all R, with period 2π p ; the extension is continuosly differentiable on R. If p " 2, then cos p x and arccos p x coincides with the standard trigonometric functions cos x and arccos x.
Let now also recall the definitions of the generalized hyperbolic cosine and arccosine functions.The function arccosh p is defined as arccosh p pxq " Its inverse function will be denoted by cosh p : t P r0, `8rÞ Ñ r1, `8r.This functio is strictly increasing in r0, `8r; it can be extended on all R as cosh p p´tq " cosh p ptq, t ą 0. If p " 2, cosh p and arccosh p are the standard hyperbolic functions.Now we consider the following eigenvalue problem in the unknown X " Xpsq: where s 0 is a given positive number.
The following result holds (see for example [DP2]).
Moreover, the corresponding eigenfunctions are unique up to a multiplicative constant and have constant sign.The first eigenvalue µ 1 pβ, s 0 q has the sign of β.
In the case β ą 0, the first eigenfunction is Xpsq " cos p ˜ˆµ 1 pβ, s 0 q p ´1 ˙1 p s ¸, s P p0, s 0 q; the eigenvalue µ 1 pβ, s 0 q is the first positive value that satisfies µ p ´1 " , and it holds that the eigenvalue µ 1 pβ, s 0 q is the unique negative value that satisfies and it holds that

Proof of the main results
Proof of Theorem 1.1.Let gptq " gpd F pxqq, where g is a nonnegative, increasing, sufficiently smooth function, where d F pxq is the distance of x P Ω from BΩ. Being F p∇d F q " 1, by the coarea formula it holds that We want to minimize the quantity in the right-hand side of equation (4.2).As matter of fact, such minimum does not depend on C so we can set C " s 0 " |Ω| P F pΩq .This gives us the following expression: where µ 1 pβ, s 0 q " min A ż s 0 0 r´h 1 psqs p ds `βh ps 0 q p ż s 0 0 rhpsqs p ds and A is the class of positive decreasing functions h P W 1,p p0, s 0 q.The optimality of the inequality follows by using a similar argument of [ where µ ℓ is the first eigenvalue of (3.1) in p0, s ℓ q, with s ℓ " |Ω ℓ | P F pΩ ℓ q .
Proof of Corollary 1.5.Given that P ptq is decreasing, we explicitly state that the second equality in (4.7) is followed by an integration by parts, observing that for

.
Last equality follows performing an integration by parts.Substituting gptq "where c is a constant to be chosen later, it follows that τ F pβ, Ωq p´1 ě ˜ż R c, it holds that the optimal choice is c "
DP2, Proposition 5.1].More precisely, what can be proved is the following.Proposition 4.1.Let Ω ℓ "s ´a 2 , a 2 rˆs ´ℓ 2 , ℓ 2 r N ´1.Then, up to a suitable rotation of Ω ℓ , it holds that We use the same notation and argument of the proof of Theorem 1.1.For a test function gptq " gpd F pxqq, with g nonnegative increasing sufficiently smooth function, we have that τ F pβ, Ωq p´1 ě ptq p P F ptqdt `βgp0q p P F pΩq p