1 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

$$\begin{aligned} \lambda _{F}(\beta ,\Omega )= \min _{\psi \in W^{1,p}(\Omega ){\setminus }\{0\} } \frac{\displaystyle \int _\Omega F(\nabla \psi )^p dx +\beta \int _{\partial \Omega }|\psi |^pF(\nu _{\Omega }) d{\mathcal {H}}^{N-1} }{\displaystyle \int _\Omega |\psi |^p dx}, \end{aligned}$$
(1.1)

where \(p\in ]1,+\infty [,\) \(\Omega \) is a bounded Lipschitz domain in \({\mathbb {R}}^{N},\) \(\nu _{\Omega }\) is its Euclidean outward normal, \(\beta \) is a real number, \(F : {\mathbb {R}}^N \rightarrow [0, +\infty [,\) \(N\ge 2,\) is a convex, even, 1-homogeneous, and \(C^{2}({\mathbb {R}}^N{\setminus } \{0\})\) function such that \([F^{2}]_{\xi \xi }\) is positive definite in \({\mathbb {R}}^{N}{\setminus }\{0\}\) (we refer the reader to Section 2 for all the definitions, here and below used, related to the Finsler metric F). When \(\beta =0,\) \(\lambda _F(\beta ,\Omega )=0\) corresponds to the first (trivial) Neumann eigenvalue, while, when \(\beta \) goes to \(+\infty ,\) it reduces to the first Dirichlet eigenvalue \(\lambda _{F}^{D}(\Omega ).\) Let us observe that, by using \(\psi =1\) and \(\psi =u_{D},\) the first Dirichlet eigenfunction, in (1.1), it holds that

$$\begin{aligned} \lambda _{F}(\beta ,\Omega )\le \min \left\{ \beta \frac{P_{F}(\Omega )}{|\Omega |},\lambda _{F}^{D}(\Omega )\right\} , \end{aligned}$$

where \(P_{F}(\Omega )\) is the anisotropic perimeter of \(\Omega \) and \(|\Omega |\) is its Lebesgue measure. We get in particular that \(\lambda _{F}(\beta ,\Omega )\rightarrow -\infty \) as \(\beta \rightarrow -\infty .\)

If \(u\in W^{1,p}(\Omega )\) is a minimizer of (1.1), then it satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} -{\mathcal {Q}}_p u =\lambda _{F}(\beta ,\Omega )\left| u\right| ^{p-2}u &{}\text {in}\ \Omega ,\\ F(\nabla u)^{p-1}F_{\xi }(\nabla u)\cdot \nu _{\Omega }+\beta F(\nu _{\Omega }) |u|^{p-2}u= 0 &{} \text {on}\ \partial \Omega , \end{array}\right. } \end{aligned}$$
(1.2)

where

$$\begin{aligned} {\mathcal {Q}}_p u={{\,\textrm{div}\,}}(F^{p-1}(\nabla u)F_\xi (\nabla u)). \end{aligned}$$

In the simplest case where \(F(\xi )={\mathcal {E}}(\xi ):=\sqrt{\sum _{i}|\xi _{i}|^{2}}\) is the Euclidean norm, then \({\mathcal {Q}}_{p}=\Delta _{p}u={{\,\textrm{div}\,}}(|\nabla u|^{p-2}\nabla u)\) 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,

$$\begin{aligned} \lambda _F^{D}(\Omega ) \le (p-1) \left( \frac{\pi _p}{2} \right) ^p \frac{P_F(\Omega )^p}{|\Omega |^p}, \end{aligned}$$
(1.3)

where \(\Omega \) is a bounded convex open set of \({\mathbb {R}}^N\) and \(|\Omega |\) and \(P_{F}(\Omega )\) denote respectively the Lebesgue measure and the anisotropic perimeter of \(\Omega ,\) and

$$\begin{aligned} \pi _{p}=2\int \limits _{0}^{1}(1-t^{p})^{-\frac{1}{p}}dt=\frac{2\pi }{p\sin \frac{\pi }{p}}. \end{aligned}$$
(1.4)

The estimate is optimal, being achieved asymptotically when the convex domain \(\Omega \) goes to an infinite N-dimensional slab \(]-a,a[\times {\mathbb {R}}^{N-1}\) (up to a suitable rotation).

In the planar, Euclidean, Laplacian case (\(N=p=2\) and \(F={\mathcal {E}}\)), this is a classical result obtained by Pólya in [11]. Then it was generalized, in any dimension, for any \(p\in ]1,+\infty [\) and for any sufficiently smooth norm F in [4]. Since \(\lambda _{F}(\beta ,\Omega )\le \lambda _F^{D}(\Omega ),\) 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.

Theorem 1.1

Let \(\Omega \) be a bounded,  convex,  open set of \({\mathbb {R}}^N.\) Then, 

$$\begin{aligned} \lambda _F(\beta ,\Omega ) \le \mu _{1}(\beta ,s_{0}) \end{aligned}$$
(1.5)

where

$$\begin{aligned} \mu _{1}(\beta ,s_{0})=\min _{h\in W^{1,p}(0,s_{0})} \frac{\displaystyle \int _0^{s_{0}} |h'(s)|^p ds+ \beta |h(s_{0})|^{p}}{\displaystyle \int _0^{s_{0}} |h(s)|^p ds} \end{aligned}$$

with \(s_{0}=\frac{|\Omega |}{P_{F}(\Omega )}.\) The estimate is optimal,  being achieved asymptotically when \(\Omega \) 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 1.3

The minimum \(\mu _{1}\) is the first eigenvalue of the nonlinear one-dimensional problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (|X'|^{p-2}X')'+\mu |X|^{p-2}X=0 \quad \text {in}\ (0,s_0),\\ X'(0)=0, \\ |X'(s_{0})|^{p-2}X'(s_{0})+ \beta |X(s_{0})|^{p-2}X(s_{0})=0. \end{array}\right. } \end{aligned}$$
(1.6)

This value can be written in terms of the generalized trigonometric functions (when \(\beta >0\)) or generalized hyperbolic functions (when \(\beta <0\)) (see Section 3). In particular:

  • if \(\beta >0,\) it holds that (see Theorem 3.1 and Remark 3.2 below)

    $$\begin{aligned} \mu _{1}(\beta ,s_{0}) < (p-1) \left( \frac{\pi _p}{2} \right) ^p \frac{P_F(\Omega )^p}{|\Omega |^p} \end{aligned}$$

    and

    $$\begin{aligned} \lim _{\beta \rightarrow +\infty }\mu _{1}(\beta ,s_{0})=(p-1) \left( \frac{\pi _p}{2} \right) ^p \frac{P_F(\Omega )^p}{|\Omega |^p}. \end{aligned}$$

  • If \(\beta <0,\) it is well-known that (see, for example, [5,6,7])

    $$\begin{aligned} \lambda _{F}(\beta ,\Omega )\le -(p-1)|\beta |^{p'}. \end{aligned}$$
    (1.7)

    Also in this case the estimate in (1.5) is sharper. Indeed, the analogous one-dimensional inequality \(\mu _{1}(\beta ,s_{0})\le -(p-1)|\beta |^{p'}\) (see Theorem 3.1 below) and (1.5) give (1.7).

  • It holds that (see Remark 3.3)

    $$\begin{aligned} \lim _{\beta \rightarrow 0}\frac{\mu _{1}(\beta ,s_{0})}{\beta }=\frac{1}{s_{0}}=\frac{P_{F}(\Omega )}{|\Omega |}. \end{aligned}$$

    Since \(\lambda _{F}(\beta ,\Omega )= \frac{P_{F}(\Omega )}{|\Omega |}\beta +o(\beta )\) as \(\beta \rightarrow 0\) (see [2, 6] in the Euclidean Laplacian and p-Laplacian case), the estimate (1.5) is optimal up to the first order as \(\beta \rightarrow 0.\)

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 \(\mu _{1}\) by using the expression of \(\mu _{1}\) and a Becker–Stark inequality [3]. So we have the following:

Corollary 1.5

Suppose that \(p=2,\) and that the hypotheses of Theorem 1.1 are satisfied,  with \(\beta >0.\) Then

$$\begin{aligned} \lambda _{F}(\beta ,\Omega ) \le \frac{\pi ^{2}}{4} \frac{P_{F}^{2}(\Omega )}{|\Omega |^{2}} \frac{1}{1+\frac{2P_{F}(\Omega )}{\beta |\Omega |}}. \end{aligned}$$

We observe that other kinds of optimal lower bounds, obtained in terms of different geometrical quantities related to \(\Omega \) (as the inradius), can be found, for example, in [9, 12, 13] (in the Euclidean case) and in [5].

The second aim of the paper is also to provide a lower bound to the anisotropic p-torsional rigidity with Robin boundary conditions \((\beta >0),\) namely the value \(\tau _{F}(\beta ,\Omega )\) such that

$$\begin{aligned} \tau _{F}(\beta ,\Omega )^{p-1} = \max _{\begin{array}{c} \psi \in W^{1,p}(\Omega ){\setminus }\{0\} \end{array}} \dfrac{\left( \displaystyle \int _\Omega |\psi | \, dx\right) ^p}{\displaystyle \int _\Omega F(\nabla \psi )^p dx+\beta \int _{\partial \Omega }|\psi |^{p}F(\nu )d{\mathcal {H}}^{N-1}},\qquad \end{aligned}$$
(1.8)

or, equivalently

$$\begin{aligned} \tau _{F}(\beta ,\Omega )=\int \limits _\Omega F(\nabla u_p)^p dx +\beta \int \limits _{\partial \Omega }|u_{p}|^{p}F(\nu )d{\mathcal {H}}^{N-1} = \int \limits _\Omega u_p dx, \end{aligned}$$

where \(u_p\in W^{1,p}(\Omega )\) is the unique solution of

$$\begin{aligned} \left\{ \begin{array}{ll} -{\mathcal {Q}}_p u = 1 &{}\quad \text {in }\Omega ,\\ F(\nabla u)^{p-1}F_{\xi }(\nabla u)\cdot \nu +\beta |u|^{p-2}u=0 &{}\quad \text {on }\partial \Omega . \end{array} \right. \end{aligned}$$

Let us remark that, by choosing \(\psi =1,\) or \(\psi =u_{\Omega }\) the Dirichlet torsion function, it holds that

$$\begin{aligned} \tau _{F}(\beta ,\Omega )^{p-1} \ge \max \left\{ \frac{|\Omega |^{p}}{\beta P_{F}(\Omega )},\tau _{F}^{D}(\Omega )^{p-1}\right\} , \end{aligned}$$

where \(\tau _{F}^{D}(\Omega )\) is the torsional rigidity with Dirichlet boundary conditions. The obtained lower bound for \(\tau _{p}\) is the following.

Theorem 1.6

Let \(1<p<+\infty ,\) \(\beta >0,\) and \(\Omega \) be a bounded convex open set in \({\mathbb {R}}^{N}.\) Then, 

$$\begin{aligned} \tau _{F}(\beta ,\Omega ) \ge \left( \frac{p-1}{2p-1}|\Omega | +\frac{1}{\beta ^{\frac{1}{p-1}}} \right) \left( \frac{|\Omega |}{P_{F}(\Omega )}\right) ^{\frac{p}{p-1}}. \end{aligned}$$
(1.9)

When \(\beta =+\infty ,\) we recover the Pólya estimate contained in [11] for the Euclidean Laplacian and then in [4] 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

Let

$$\begin{aligned} \xi \in {\mathbb {R}}^{N}\mapsto F(\xi )\in [0,+\infty [ \end{aligned}$$

be a convex, even, 1-homogeneous function, that is, a convex function such that

$$\begin{aligned} F(t\xi )=|t|F(\xi ), \quad t\in {\mathbb {R}},\,\xi \in {\mathbb {R}}^{N}, \end{aligned}$$
(2.1)

and such that

$$\begin{aligned} a|\xi | \le F(\xi )\le b|\xi |,\quad \xi \in {\mathbb {R}}^{N}, \end{aligned}$$
(2.2)

for some constants \(0<a\le b.\) Moreover, we suppose that \(F\in C^{2}({\mathbb {R}}^N{\setminus } \{0\}),\) and that

$$\begin{aligned} \nabla ^{2}_{\xi }[F^{2}](\xi )\text { is positive definite in }{\mathbb {R}}^{N}{\setminus }\{0\}. \end{aligned}$$
(2.3)

The hypothesis (2.3) on F ensures that the operator

$$\begin{aligned} \mathcal {Q}_p[u]:= {{\,\textrm{div}\,}}\left( \frac{1}{p}\nabla _{\xi }[F^{p}](\nabla u)\right) \end{aligned}$$

is elliptic, namely there exists a positive constant \(\gamma \) such that

$$\begin{aligned} \sum _{i,j=1}^{n}{\nabla ^{2}_{\xi _{i}\xi _{j}}[F^{p}](\eta ) \xi _i\xi _j}\ge \gamma |\eta |^{p-2} |\xi |^2 \end{aligned}$$

for some positive constant \(\gamma ,\) for any \(\eta \in {\mathbb {R}}^N{\setminus }\{0\}\), and for any \(\xi \in {\mathbb {R}}^N.\)

The polar function of F is

$$\begin{aligned} F^o(v)=\sup _{\xi \ne 0} \frac{\langle \xi , v\rangle }{F(\xi )}, \quad v\in {\mathbb {R}}^{N}. \end{aligned}$$
(2.4)

It holds that \(F^o\) is a convex function which satisfies properties (2.1) and (2.2) (with different constants). Furthermore,

$$\begin{aligned} F(v)=\sup _{\xi \ne 0} \frac{\langle \xi , v\rangle }{F^o(\xi )}. \end{aligned}$$

From (2.4), it holds that

$$\begin{aligned} \langle \xi , \eta \rangle \le F(\xi ) F^{o}(\eta ) \qquad \forall \xi , \eta \in {\mathbb {R}}^{N}. \end{aligned}$$
(2.5)

The set

$$\begin{aligned} {\mathcal {W}} = \{ \xi \in {\mathbb {R}}^N :F^o(\xi )< 1 \} \end{aligned}$$

is the Wulff shape centered at the origin. We put \(\kappa _N=|{\mathcal {W}}|,\) where \(|{\mathcal {W}}|\) denotes the Lebesgue measure of \({\mathcal {W}}.\) More generally, we denote by \({\mathcal {W}}_r(x_0)\) the set \(r{\mathcal {W}}+x_0,\) that is, the Wulff shape centered at \(x_0\) with measure \(\kappa _Nr^N,\) and \({\mathcal {W}}_r(0)={\mathcal {W}}_r.\)

The anisotropic distance of \(x\in {\overline{\Omega }}\) to the boundary of a bounded domain \(\Omega \) is the function

$$\begin{aligned} d_{F}(x)= \inf _{y\in \partial \Omega } F^o(x-y), \quad x\in {\overline{\Omega }}. \end{aligned}$$

We stress that when \(F=|\cdot |,\) then \(d_F=d_{\mathcal {E}},\) the Euclidean distance function from the boundary.

It is not difficult to prove that \(d_{F}\) is a uniform Lipschitz function in \({\overline{\Omega }}\) and

$$\begin{aligned} F(\nabla d_F(x))=1 \quad \text {a.e. in }\Omega . \end{aligned}$$

Obviously, \(d_F\in W_{0}^{1,\infty }(\Omega ).\) Finally, the anisotropic inradius of \(\Omega \) is the quantity

$$\begin{aligned} R_{F}(\Omega )=\max \{d_{F}(x),\; x\in {\overline{\Omega }}\}, \end{aligned}$$

that is, the radius of the largest Wulff shape \({\mathcal {W}}_{r}(x)\) contained in \(\Omega .\)

Let us finally recall the definition of the anisotropic perimeter of a set \(K\subset {\mathbb {R}}^N\) in \(\Omega \) (see also, for example, [1]):

$$\begin{aligned} P_F(K,\Omega ) = \sup \left\{ \int \limits _K {{\,\textrm{div}\,}}\sigma dx:\sigma \in C_0^1(\Omega ;{\mathbb {R}}^N),\; F^o(\sigma )\le 1 \right\} . \end{aligned}$$

The following coarea formula for the anisotropic perimeter

$$\begin{aligned} \int \limits _\Omega F(\nabla u) dx = \int \limits _{-\infty }^{+\infty } P_F (\{u>s\},\Omega )\, ds,\quad \forall u\in W^{1,1}(\Omega ), \end{aligned}$$
(2.6)

holds [1]; moreover

$$\begin{aligned} P_F(K;\Omega )= \int \limits _{\Omega \cap \partial ^*K} F(\nu _K) d{\mathcal {H}}^{N-1} \end{aligned}$$

where \({\mathcal {H}}^{N-1}\) is the \((N-1)\)-dimensional Hausdorff measure in \({{\mathbb {R}}}^N,\) \(\partial ^*K\) is the reduced boundary of F,  and \(\nu _F\) is the outer normal to F. As usual, we will denote by \(P_{F}(K)\) the perimeter in \({\mathbb {R}}^{N},\) that is, \(P_{F}(K,{\mathbb {R}}^{N}).\)

3 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 [8, 10].

The function \(\arccos _p:[0, 1]\rightarrow {\mathbb {R}}\) is defined as

$$\begin{aligned} \arccos _{p}(x)=\int \limits _x^1\frac{dt}{\left( 1-{t^p}\right) ^\frac{1}{p}}. \end{aligned}$$

If z(t) is the inverse function of \(\arccos _{p},\) which is defined on the interval \(\left[ 0,\frac{\pi _p}{2}\right] ,\) where

$$\begin{aligned} \pi _p=2\int \limits _{ 0}^{1}\frac{dt}{\left( 1-{t^p}\right) ^\frac{1}{p}} = \frac{2\pi }{p\sin \frac{\pi }{p}}, \end{aligned}$$

then, the p-cosine function \(\cos _p\) is the even function defined as the periodic extension of z(t):

$$\begin{aligned} \cos _p(t)=\left\{ \begin{array}{ll} z(t) &{}\quad \text {if} \ t \in \left[ 0,\frac{\pi _p}{2}\right] ,\\ -z(\pi _p-t) &{}\quad \text {if} \ t\in \left[ \frac{\pi _p}{2}, \pi _p\right] , \\ \cos _p(-t)&{}\quad \text {if} \ t\in \left[ -\pi _p, 0\right] , \end{array} \right. \end{aligned}$$

and extended periodically to all of \({\mathbb {R}},\) with period \(2\pi _p;\) the extension is continuously differentiable on \({\mathbb {R}}.\) If \(p=2,\) then \(\cos _{p}x\) and \(\arccos _{p}x\) coincide with the standard trigonometric functions \(\cos x\) and \(\arccos x.\)

Let us now also recall the definitions of the generalized hyperbolic cosine and arccosine functions. The function \({{\,\textrm{arccosh}\,}}_{p}\) is defined as

$$\begin{aligned} {{\,\textrm{arccosh}\,}}_p(x)=\int \limits _1^x\frac{1}{(t^p-1)^\frac{1}{p}}dt,\ x\in [1,+\infty [. \end{aligned}$$

Its inverse function will be denoted by \(\cosh _{p} :t\in [0,+\infty [\mapsto [1,+\infty [.\) This function is strictly increasing in \([0,+\infty [;\) it can be extended to all of \({\mathbb {R}}\) as \(\cosh _{p}(-t)=\cosh _{p}(t),\) \(t>0.\) If \(p=2,\) \(\cosh _{p}\) and \({{\,\textrm{arccosh}\,}}_{p}\) are the standard hyperbolic functions.

Now we consider the following eigenvalue problem in the unknown \(X=X(s)\):

$$\begin{aligned} {\left\{ \begin{array}{ll} (|X'|^{p-2}X')'+\mu |X|^{p-2}X=0 \quad \text {in}\ (0,s_0),\\ X'(0)=0, \\ |X'(s_{0})|^{p-2}X'(s_{0})+\beta |X(s_{0})|^{p-2}X(s_{0})=0, \end{array}\right. } \end{aligned}$$
(3.1)

where \(s_{0}\) is a given positive number.

The following result holds (see, for example, [5]).

Theorem 3.1

Let \(1<p<+\infty ,\) \(s_{0}>0,\) and \(\beta \in {\mathbb {R}}.\) Then there exists the smallest eigenvalue \(\mu \) of (3.1), which has the following variational characterization : 

$$\begin{aligned} \mu _{1}(\beta ,s_0)=\inf _{v\in W^{1,p}(0,s_{0})} \frac{\displaystyle \int \nolimits _{0}^{s_{0}}\left| v'(s)\right| ^{p}ds+\beta |v(s_{0})|^{p}}{\displaystyle \int \nolimits _{0}^{s_{0}}\left| v(s)\right| ^{p}ds}. \end{aligned}$$

Moreover, the corresponding eigenfunctions are unique up to a multiplicative constant and have constant sign. The first eigenvalue \(\mu _{1}(\beta ,s_0)\) has the sign of \(\beta .\)

In the case \(\beta >0,\) the first eigenfunction is

$$\begin{aligned} X(s)=\cos _p\left( \left( \frac{\mu _{1}(\beta ,s_0)}{p-1}\right) ^\frac{1}{p}s \right) ,\quad s\in (0,s_{0}); \end{aligned}$$
(3.2)

the eigenvalue \(\mu _{1}(\beta ,s_0)\) is the first positive value that satisfies

$$\begin{aligned} \frac{\mu }{p-1}=\frac{\beta ^{p'}}{\cos _p^{-p}\left( \left( \frac{\mu }{p-1}\right) ^\frac{1}{p} s_{0} \right) -1}, \end{aligned}$$

and it holds that

$$\begin{aligned} \mu _{1}(\beta ,s_0) < (p-1)\left( \frac{\pi _p}{2s_{0}}\right) ^{p}. \end{aligned}$$
(3.3)

If \(\beta <0,\) the first eigenfunction is

$$\begin{aligned} X(s)=\cosh _p\left( \left( \frac{-\mu _{1}(\beta ,s_0)}{p-1}\right) ^\frac{1}{p}s \right) ,\quad s\in (0,s_{0}); \end{aligned}$$

the eigenvalue \(\mu _{1}(\beta ,s_0)\) is the unique negative value that satisfies

$$\begin{aligned} -\frac{\mu }{p-1} =\frac{|\beta |^{p'}}{1-\cosh _p^{-p}\left( \left( \frac{-\mu }{p-1}\right) ^\frac{1}{p} s_{0} \right) }, \end{aligned}$$

and it holds that

$$\begin{aligned} \mu _{1}(\beta ,s_0) \le -(p-1)|\beta |^{p'}. \end{aligned}$$
(3.4)

Remark 3.2

The inequality in (3.3) can be easily seen using the variational characterization of \(\mu _{1}\):

$$\begin{aligned} \mu _{1}(\beta ,s_{0}) < \inf _{v\in W^{1,p}(0,s_{0}),\; v(s_{0})=0} \frac{\displaystyle \int _{0}^{s_{0}}\left| v'(s)\right| ^{p}ds}{\displaystyle \int _{0}^{s_{0}} \left| v(s)\right| ^{p}ds}= (p-1)\left( \frac{\pi _{p}}{2s_{0}}\right) ^{p} \end{aligned}$$

(see, for example, [4] for the last equality). Actually, by definition, \(\mu _{1}(\beta ,s_{0})\) is increasing in \(\beta ;\) it holds that

$$\begin{aligned} \lim _{\beta \rightarrow +\infty }\mu _{1}(\beta ,s_{0})= (p-1)\left( \frac{\pi _{p}}{2s_{0}}\right) ^{p}. \end{aligned}$$
(3.5)

The limit (3.5) can be obtained in the following way: let us take the sequence of positive minimizers \(X_{\beta }(s),\) given by (3.2), of \(\mu _{1}(\beta ,s_{0}),\) with \(X_{\beta }(0)=1.\) By (3.3), \(\mu _{1}(\beta ,s_{0})\) is bounded as \(\beta \rightarrow +\infty \) and hence, using the variational characterization of \(\mu _{1},\) it holds that \(X_{\beta }(s_{0})\rightarrow 0\) as \(\beta \rightarrow +\infty .\) Then

$$\begin{aligned} 0=\lim _{\beta \rightarrow +\infty } X_{\beta }(s_{0})= \cos _p\left( \left( \frac{\ell }{p-1}\right) ^\frac{1}{p}s_{0} \right) ,\quad \ell =\lim _{\beta \rightarrow +\infty }\mu _{1}(\beta ,s_{0}). \end{aligned}$$

In conclusion, \(\left( \frac{\ell }{p-1}\right) ^\frac{1}{p}s_{0}=\frac{\pi _{p}}{2},\) which is the first positive zero of \(\cos _{p}(s),\) and that corresponds to (3.5).

Remark 3.3

If as before we denote, for any \(\beta \in {\mathbb {R}},\) the solution given in Theorem 3.1 by \(X_{\beta }(s),\) \(s\in [0,s_{0}],\) it holds that \(\mu (\beta ,s_{0})\rightarrow 0\) and \(X_{\beta }(s)\rightarrow 1\) uniformly as \(\beta \rightarrow 0.\)

Let us also observe that by choosing constant test functions,

$$\begin{aligned} \mu _{1}(\beta ,s_{0})< \frac{\beta }{s_{0}},\quad \beta \ne 0, \end{aligned}$$

with equality if and only if \(\beta =0.\) Actually

$$\begin{aligned} \lim _{\beta \rightarrow 0} \frac{\mu _{1}(\beta ,s_{0})}{\beta }=\frac{1}{s_{0}}. \end{aligned}$$
(3.6)

Indeed,

$$\begin{aligned} \frac{\beta }{s_{0}}>\mu _{1}(\beta ,s_{0})=\frac{\displaystyle \int \nolimits _{0}^{s_{0}} |X_{\beta }'(s)|^{p}ds+\beta X_{\beta }^{p}(s_{0})}{\displaystyle \int \nolimits _{0}^{s_{0}}X_{\beta }^{p}(s)ds}> \frac{\beta X_{\beta }^{p}(s_{0})}{\displaystyle \int \nolimits _{0}^{s_{0}}X_{\beta }^{p}(s)ds}. \end{aligned}$$

Dividing by \(\beta \) and passing to the limit, we have (3.6).

4 Proof of the main results

Proof of Theorem 1.1

Let \(g(t)=g(d_F(x)),\) where g is a nonnegative, increasing, sufficiently smooth function, where \(d_{F}(x)\) is the distance of \(x\in \Omega \) from \(\partial \Omega .\) Since \(F(\nabla d_F)=1,\) by the coarea formula, it holds that

$$\begin{aligned} \int \limits _\Omega F(\nabla g(d_F(x)))^p dx = \int \limits _0^{R_F(\Omega )} g'(t)^p dt \int \limits _{\{d_F=t\}} \frac{1}{|\nabla d_F|} d{\mathcal {H}}^{N-1} = \int \limits _0^{R_F(\Omega )} g'(t)^p P(t)dt, \end{aligned}$$

where \(P(t)=P_F(\{x\in \Omega :d_F>t\}),\) and \(R_F(\Omega )=\sup _{x\in \Omega } d_{F}(x);\) similarly,

$$\begin{aligned} \int \limits _\Omega g(d_F(x))^p dx = \int \limits _0^{R_F(\Omega )} g(t)^p P(t)dt, \quad \int \limits _{\partial \Omega } g(d_{F}(x))F(\nu ) d{\mathcal {H}}^{N-1}= g(0) P_{F}(\Omega ). \end{aligned}$$

Using \(g(d_F(x))\) as test function in the Rayleigh quotient of (1.2), we have

$$\begin{aligned} \lambda _F(\beta ,\Omega ) \le \frac{\displaystyle \int _0^{R_F(\Omega )} g'(t)^p P(t)\,dt + \beta P_{F}(\Omega )g(0)^{p}}{\displaystyle \int _0^{R_F(\Omega )} g(t)^p P(t)\,dt}. \end{aligned}$$
(4.1)

Now, we perform the change of variable

$$\begin{aligned} s = \frac{C}{|\Omega |} A(t), \end{aligned}$$

where we have denoted \(A(t)=|\{x\in \Omega :d_F(x)>t\}|,\) and C is a positive constant which will be chosen later. Observe that \(A(0)=|\Omega |.\) Let h(s) be the function such that

$$\begin{aligned} h(s)=g(t). \end{aligned}$$

We stress that h(s(t)) is decreasing. Then, substituting h in (4.1), it follows that

$$\begin{aligned} \lambda _F(\beta ,\Omega ) \le \frac{C^p}{|\Omega |^p} \frac{\displaystyle \int _0^{R_{F}(\Omega )} \left[ -h'\left( \frac{C}{|\Omega |}A(t)\right) \right] ^p P(t)^p \left[ \frac{C}{|\Omega |} P(t) \right] dt + \beta \frac{|\Omega |^{p-1}}{C^{p-1}} P_{F}(\Omega ) h(C)^{p} }{\displaystyle \int _0^{R_F(\Omega )} \left[ h \left( \frac{C}{|\Omega |}A(t)\right) \right] ^p \left[ \frac{C}{|\Omega |} P(t) \right] dt} \end{aligned}$$

and then, by the monotonicity of P(t),  we get

$$\begin{aligned} \lambda _F(\beta ,\Omega ) \le \frac{P_F(\Omega )^p}{|\Omega |^p} C^p \frac{\displaystyle \int _0^{C} [-h'(s)]^p ds+\beta \frac{|\Omega |^{p-1}}{C^{p-1} P_{F}(\Omega )^{p-1}} h(C)^{p}}{\displaystyle \int _0^{C} [h(s)]^p ds}. \end{aligned}$$
(4.2)

We want to minimize in h the quantity in the right-hand side of Equation (4.2). We set \(C=s_{0}=\frac{|\Omega |}{P_{F}(\Omega )}.\) This gives us the following expression:

$$\begin{aligned} \lambda _F(\beta ,\Omega ) \le \mu _{1}(\beta ,s_{0}), \end{aligned}$$
(4.3)

where

$$\begin{aligned} \mu _{1}(\beta ,s_{0})=\min _{{\mathcal {A}}} \frac{\displaystyle \int _0^{s_{0}} [-h'(s)]^p ds+\beta h\left( s_{0}\right) ^{p}}{\displaystyle \int _0^{s_{0}} [h(s)]^p ds} \end{aligned}$$

and \({\mathcal {A}}\) is the class of positive decreasing functions \(h\in W^{1,p}(0,s_{0}).\) \(\square \)

The optimality of the inequality follows by using a similar argument as in [5, Proposition 5.1]. More precisely, what can be proved is the following.

Proposition 4.1

Let \(\Omega _\ell =]-\frac{a}{2},\frac{a}{2}[\times ]-\frac{\ell }{2},\frac{\ell }{2}[^{N-1}.\) Then,  up to a suitable rotation of \(\Omega _{\ell },\) it holds that

$$\begin{aligned} \lim _{\ell \rightarrow +\infty }\frac{\lambda _F(\beta ,\Omega _\ell )}{\mu _\ell }=1, \end{aligned}$$

where \(\mu _{\ell }\) is the first eigenvalue of (3.1) in \((0,s_{\ell }),\) with \(s_{\ell }=\frac{|\Omega _{\ell }|}{P_{F}(\Omega _{\ell })}.\)

Proof of Corollary 1.5

The one-dimensional eigenvalue in this case satisfies

$$\begin{aligned} \tan \left( \sqrt{\mu }_{1}s_{0}\right) = \frac{\beta }{\sqrt{\mu _{1}}}, \end{aligned}$$

where \(s_{0}=\frac{|\Omega |}{P_{F}(\Omega )}.\) Recalling the Becker–Stark [3] inequality

$$\begin{aligned} \frac{2t}{\frac{\pi ^{2}}{4}-t^{2}} \le \tan t, \qquad 0\le t < \frac{\pi }{2}, \end{aligned}$$
(4.4)

it holds that

$$\begin{aligned} \frac{\beta }{\sqrt{\mu _{1}}} \ge \frac{2s_{0}\sqrt{\mu }_{1}}{\frac{\pi ^{2}}{4}-s_{0}^{2}\mu _{1}}. \end{aligned}$$

Hence, rearranging and recalling Theorem 1.1, we get

$$\begin{aligned} \lambda _{F}(\beta ,\Omega ) \le \mu _{1} \le \frac{\pi ^{2}}{4} \frac{P_{F}^{2}(\Omega )}{|\Omega |^{2}} \frac{1}{1+\frac{2P_{F}(\Omega )}{\beta |\Omega |}}. \end{aligned}$$

\(\square \)

Proof of Theorem 1.6

We use the same notation and argument as in the proof of Theorem 1.1. For a test function \(g(t)=g(d_F(x)),\) with g nonnegative, increasing, sufficiently smooth function, we have that

$$\begin{aligned} \tau _{F}(\beta ,\Omega )^{p-1}\ge & {} \dfrac{\left( \displaystyle \int _0^{R_F(\Omega )}g(t) P_F(t)dt\right) ^p}{\displaystyle \int _0^{R_F(\Omega )} g'(t)^p P_F(t)dt +\beta g(0)^{p }P_{F}(\Omega )^{p}} \\= & {} \dfrac{\left( \displaystyle \int _0^{R_F(\Omega )}g'(t) A(t)dt+g(0)|\Omega |\right) ^p}{\displaystyle \int _0^{R_F(\Omega )} g'(t)^p P_F(t)dt+\beta g(0)^{p }P_{F}(\Omega )^{p}}. \end{aligned}$$

The last equality follows by performing an integration by parts. Substituting

$$\begin{aligned} g(t)=\int \limits _0^{t} \left( \frac{A(s)}{P_F(s)}\right) ^{1/(p-1)}ds + c, \end{aligned}$$

where c is a constant to be chosen later, it follows that

$$\begin{aligned} \tau _{F}(\beta ,\Omega )^{p-1} \ge \dfrac{\left( \displaystyle \int _0^{R_F(\Omega )} \frac{A(t)^{\frac{p}{p-1}}}{P(t)^{\frac{1}{p-1}}} dt+c|\Omega |\right) ^{p}}{\displaystyle \int _0^{R_F(\Omega )} \frac{A(t)^{\frac{p}{p-1}}}{P(t)^{\frac{1}{p-1}}} dt+\beta c^{p }P_{F}(\Omega )^{p}}. \end{aligned}$$
(4.5)

Maximizing in c,  it holds that the optimal choice is \(c=\frac{|\Omega |^{\frac{1}{p-1}}}{\beta ^{\frac{1}{p-1}}P_{F}(\Omega )^{\frac{p}{p-1}}}\) and then

$$\begin{aligned} \tau _{F}(\beta ,\Omega ) \ge \int \limits _0^{R_F(\Omega )} \frac{A(t)^{\frac{p}{p-1}}}{P(t)^{\frac{1}{p-1}}} dt + \frac{1}{\beta ^{\frac{1}{p-1}}} \left( \frac{|\Omega |}{P_{F}(\Omega )}\right) ^{\frac{p}{p-1}}. \end{aligned}$$
(4.6)

Now, let us observe that

$$\begin{aligned} \int \limits _0^{R_F(\Omega )} \frac{A(t)^{\frac{p}{p-1}}}{P(t)^{\frac{1}{p-1}}} dt= & {} \int \limits _0^{R_F(\Omega )} \frac{A(t)^{\frac{p}{p-1}}[-A'(t)]}{P(t)^{\frac{p}{p-1}}} dt\nonumber \\= & {} \frac{p-1}{2p-1} \frac{|\Omega |^{\frac{2p-1}{p-1}}}{P_F(\Omega )^{\frac{p}{p-1}}}\nonumber \\{} & {} + \frac{p}{2p-1} \int \limits _0^{R_F(\Omega )} \left( \frac{A(t)}{P(t)}\right) ^{\frac{2p-1}{p-1}} [-P'(t)] dt \nonumber \\\ge & {} \frac{p-1}{2p-1} \frac{|\Omega |^{\frac{2p-1}{p-1}}}{P_F(\Omega )^{\frac{p}{p-1}}}. \end{aligned}$$
(4.7)

Given that P(t) is decreasing, we explicitly state that the second equality in (4.7) follows by an integration by parts, observing that for

$$\begin{aligned} A(t) =\int \limits _t^{R_F(\Omega )} P(t) dt \le P(t)(R_F(\Omega )-t), \end{aligned}$$

we have

$$\begin{aligned} \frac{|A(t)|^{\frac{2p-1}{p-1}}}{P(t)^{\frac{p}{p-1}}}\rightarrow 0\quad \text {as }t\rightarrow R_F(\Omega ). \end{aligned}$$

In conclusion, by (4.7) and (4.6), we have

$$\begin{aligned} \tau _{F}(\beta ,\Omega ) \ge \left( \frac{p-1}{2p-1}|\Omega | +\frac{1}{\beta ^{\frac{1}{p-1}}} \right) \left( \frac{|\Omega |}{P_{F}(\Omega )}\right) ^{\frac{p}{p-1}} \end{aligned}$$

and this concludes the proof. \(\square \)