Related papers: Faber-Krahn inequality for Robin problem involving…

The goal of this paper is to investigate the minimisation of the first eigenvalue of the (vectorial) incompressible Dirichlet-Stokes operator. After providing an existence result, we investigate optimality conditions and we prove the…

Let $\Omega$ be a bounded $C^{2,\alpha}$ domain in $\R^n$ ($n\geq 1$, $0<\alpha<1$), $\Omega^{\ast}$ be the open Euclidean ball centered at 0 having the same Lebesgue measure as $\Omega$, $\tau\geq 0$ and $v\in L^{\infty}(\Omega,\R^n)$ with…

The main result of the paper shows that the regular $n$-gon is a local minimizer for the first Dirichlet-Laplace eigenvalue among $n$-gons having fixed area for $n \in \{5,6\}$. The eigenvalue is seen as a function of the coordinates of the…

Let $\Omega \subset \mathbb{R}^d$ with $d\geq 2$ be a bounded domain of class $\mathcal{C}^{1,\beta }$ for some $\beta \in (0,1)$. For $p\in (1, \infty )$ and $s\in (0,1)$, let $\Lambda ^s_{p}(\Omega )$ be the first eigenvalue of the mixed…

In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals $((p-1)/p)^p$ whenever Dirichlet boundary…

The second eigenvalue of the Robin Laplacian is shown to be maximal for the ball among domains of fixed volume, for negative values of the Robin parameter $\alpha$ in the regime connecting the first nontrivial Neumann and Steklov…

In this paper we study the $p$-Poisson equation with Robin boundary conditions, where the Robin parameter is a function. By means of some weighted isoperimetric inequalities, we provide various sharp bounds for the solutions to the problems…

In this paper we prove a sharp lower bound for the first nontrivial Neumann eigenvalue $\mu_1(\Omega)$ for the $p$-Laplace operator in a Lipschitz, bounded domain $\Omega$ in $\R^n$. Our estimate does not require any convexity assumption on…

In this paper, we obtain sharp Faber-Krahn inequalities for the first Dirichlet eigenvalue of the combinatorial $p$-Laplacian on connected graphs with a fixed number of vertices or with a fixed number of edges. More precisely, we show that…

We consider the eigenvalue problem for the Laplace operator in a planar domain which can be decomposed into a bounded domain of arbitrary shape and elongated \branches" of variable cross-sectional profiles. When the eigenvalue is smaller…

We present a fractional counterpart of a generalized Kohler-Jobin inequality, showing that, among all bounded, open sets $\Omega\subset \mathbb{R}^N$ with Lipschitz boundary, having the same fractional torsional rigidity, the first…

In this paper, we obtain the Bossel-Daners inequality for the first eigenvalue of the p-Laplacian with Robin boundary conditions on complete Riemannian manifolds with lower Ricci curvature bounds. Furthermore, we demonstrate that the…

We exploit the connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians. First, we find a graphical relation between their smallest positive eigenvalues, which allows us to deduce a recipe for translating…

We consider the well-known following shape optimization problem: $$\lambda_1(\Omega^*)=\min_{\stackrel{|\Omega|=a} {\Omega\subset{D}}} \lambda_1(\Omega), $$ where $\lambda_1$ denotes the first eigenvalue of the Laplace operator with…

In this paper we address the problem of the minimization of the $k$-th Robin eigenvalue $\lambda_{k,\beta}$ with parameter $\beta>0$ among bounded open Lipschitz sets with prescribed perimeter. The perimeter constraint allows us to…

We study the eigenvalue problem for the $p$-Laplacian on K\"ahler manifolds. Our first result is a lower bound for the first nonzero eigenvalue of the $p$-Laplacian on compact K\"ahler manifolds in terms of dimension, diameter, and lower…

We prove a local Faber-Krahn inequality for solutions $u$ to the Dirichlet problem for $\Delta + V$ on an arbitrary domain $\Omega$ in $\mathbb{R}^n$. Suppose a solution $u$ assumes a global maximum at some point $x_0 \in \Omega$ and…

Lieb has shown a lower bound on the smallest Dirichlet eigenvalue of the Laplace operator in terms of a generalized inradius. We derive similar bounds for Robin eigenvalues, for eigenvalues of the polyharmonic operator and the sub-Laplacian…

This paper investigates the first Dirichlet eigenvalue for the $p$-Laplacian in Riemannian manifolds. Firstly, we establish a lower bound for this eigenvalue under the condition that the domain includes a specific function which fulfills…

In this paper we prove a reverse Faber-Krahn inequality for the principal eigenvalue $\mu_1(\Omega)$ of the fully nonlinear eigenvalue problem \[ \label{eq} \left\{\begin{array}{r c l l} -\lambda_N(D^2 u) & = & \mu u & \text{in }\Omega, \\…