Related papers: Upper and lower bounds for the first Robin eigenva…

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 study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general…

This paper deals with the Neumann eigenvalue problem for the Hermite operator defined in a convex, possibly unbounded, planar domain $\Omega$, having one axis of symmetry passing through the origin. We prove a sharp lower bound for the…

In this work we present a Lyapunov inequality for linear and quasilinear elliptic differential operators in $N-$dimensional domains $\Omega$. We also consider singular and degenerate elliptic problems with $A_p$ coefficients involving the…

We consider the first eigenvalue $\lambda_1(\Omega,\sigma)$ of the Laplacian with Robin boundary conditions on a compact Riemannian manifold $\Omega$ with smooth boundary, $\sigma\in\bf R$ being the Robin boundary parameter. When $\sigma>0$…

Let $\Omega\subset \mathbb{R}^n$ be a bounded $C^1$ domain and $p>1$. For $\alpha>0$, define the quantity \[ \Lambda(\alpha)=\inf_{u\in W^{1,p}(\Omega),\, u\not\equiv 0} \Big(\int_\Omega |\nabla u|^p\,\mathrm{d}x - \alpha…

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of…

In this paper we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue $\lambda_{F}(p,\Omega)$ of the anisotropic $p$-Laplacian, $1<p<+\infty$. Our aim is to enhance how, by means of the $\mathcal…

For a bounded domain $\Omega$ with a piecewise smooth boundary in an $n$-dimensional Euclidean space $\mathbf{R}^{n}$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for…

We study the spectrum of the Robin Laplacian with a complex Robin parameter $\alpha$ on a bounded Lipschitz domain $\Omega$. We start by establishing a number of properties of the corresponding operator, such as generation properties, local…

In this paper we present an approximation result concerning the first eigenvalue of the 1-Laplacian operator. More precisely, for $\Omega$ a bounded regular open domain, we consider a minimisation of the functional ${\ds \int_\Omega}|\nabla…

We give a new lower bound for the first gap $\lambda_2 - \lambda_1$ of the Dirichlet eigenvalues of the Schr{\"o}dinger operator on a bounded convex domain $\Omega$ in R$^n$ or S$^n$ and greatly sharpens the previous estimates. The new…

In this paper we study the maximum principle, the existence of eigenvalue and the existence of solution for the Dirichlet problem for operators which are fully-nonlinear, elliptic but presenting some singularity or degeneracy which are…

This paper addresses the geometric optimization problem of the first Robin eigenvalue in exterior domains, specifically the lowest point of the spectrum of the Laplace operator under Robin boundary conditions in the complement of a bounded…

We compute estimates for eigenvalues of a class of linear second-order elliptic differential operators in divergence form (with Dirichlet boundary condition) on a bounded domain in a complete Riemannian manifold. Our estimates are based…

We consider the eigenvalue problem for certain classes of elliptic operators, namely inhomogeneous membrane operators $ L = \tfrac{1}{ \rho } ( -\Delta + V ) $ and divergence form operators $ L = -\operatorname{div} A \nabla $, on bounded…

We consider the Robin boundary value problem $\mathrm{div} (A \nabla u) = \mathrm{div} \mathbf{f}+F$ in $\Omega$, $\mathcal{C}^1$ domain, with $(A \nabla u - \mathbf{f})\cdot \mathbf{n} + \alpha u = g$ on $\Gamma$, where the matrix $A$…

We consider the torsional rigidity and the principal eigenvalue related to the Laplace operator with Dirichlet and Robin boundary conditions. The goal is to find upper and lower bounds to products of suitable powers of the quantities above…

We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for a class of very degenerate elliptic operators, with the aim to show that, at least for square type domains having fixed volume, the symmetry of the domain…

We derive a priori bounds for positive supersolutions of $ - \Delta_{p} u = \rho(x) f(u) $, where $p>1$ and $\Delta_{p}$ is the $p$-Laplace operator, in a smooth bounded domain of $R^{N}$ with zero Dirichlet boundary conditions. We apply…