Related papers: First Dirichlet eigenvalue and exit time moment sp…

Let $m$ be a bounded function and $\alpha$ a nonnegative parameter. This article is concerned with the first eigenvalue $\lambda\_\alpha(m)$ of the drifted Laplacian type operator $\mathcal L\_m$ given by $\mathcal L\_m(u)=…

Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with nonpostive curvature, then we shall give improved estimates for the $L^2$-norms of the restrictions of eigenfunctions to unit-length geodesics, compared to the…

In this paper, we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domain $\Omega$ in $\mathbb{R}^n$. It is well known that the $k$-th Dirichlet eigenvalue $\lambda_k$ obeys the…

In 1954, G. Polya conjectured that the counting function $N(\Omega,\Lambda)$ of the eigenvalues of the Laplace operator of the Dirichlet (resp. Neumann) boundary value problem in a bounded set $\Omega\subset R^d$ is lesser (resp. greater)…

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 prove that, if $\Omega$ is an open bounded domain with smooth and connected boundary, for every $p \in (1, + \infty)$ the first Dirichlet eigenvalue of the normalized $p$-Laplacian is simple in the sense that two positive eigenfunctions…

This paper is dedicated to the spectral optimization problem $$ \mathrm{min}\left\{\lambda_1^s(\Omega)+\cdots+\lambda_m^s(\Omega) + \Lambda \mathcal{L}_n(\Omega)\colon \Omega\subset D \mbox{ s-quasi-open}\right\} $$ where $\Lambda>0,…

First, we obtain a new formula for Bremermann type upper envelopes, that arise frequently in convex analysis and pluripotential theory, in terms of the Legendre transform of the convex- or plurisubharmonic-envelope of the boundary data.…

We prove the (optimal) $W^{1,\infty}$-regularity of weak solutions to the equation $-\Delta u = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ in a domain $\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions, where $\Gamma \subset…

In this paper, we investigate the first eigenvalue $\Lambda_1$ of the area Jacobi operator for complex curves in K\"ahler surfaces, establishing an extrinsic counterpart to the classical Lichnerowicz theorem for the Laplace-Beltrami…

In this paper we consider minimizers of the functional \begin{equation*} \min \big\{ \lambda_1(\Omega)+\cdots+\lambda_k(\Omega) + \Lambda|\Omega|, \ : \ \Omega \subset D \text{ open} \big\} \end{equation*} where $D\subset\mathbb{R}^d$ is a…

Comparing Neumann and Dirichlet eigenvalues of the Laplacian on a bounded domain $\Omega\subseteq\Rbb^n$ is a topic that goes back at least to the work of P\'olya \cite{polya}. We study the effect of the isoperimetric ratio of $\Omega$ on…

Given a frequency sequence $\omega=(\omega_n)$ and a finite subset $J \subset \mathbb{N}$, we study the space $\mathcal{H}_{\infty}^{J}(\omega)$ of all Dirichlet polynomials $D(s) := \sum_{n \in J} a_n e^{-\omega_n s}, \, s \in \mathbb{C}$.…

Let $\Sigma$ be a closed embedded minimal hypersurface in the unit sphere $\mathbb{S}^{m+1}$ and let $\Lambda=\max\limits_{\Sigma}|A|$ be the norm of its second fundamental form. In this work we prove that the first eigenvalue of the…

Given $(M,g)$ a smooth compact Riemannian manifold without boundary of dimension $n\geq 3$, we consider the first conformal eigenvalue which is by definition the supremum of the first eigenvalue of the Laplacian among all metrics conformal…

We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is…

We compute the whole spectrum of the Dirichlet-to-Neumann operator acting on differential p-forms on the unit Euclidean ball. Then, we prove a new upper bound for its first eigenvalue on a domain $\Omega$ in Euclidean space in terms of the…

In this paper, we present a nonlocal model for Poisson equation and corresponding eigenproblem with Dirichlet boundary condition. In the direct derivation of the nonlocal model, normal derivative is required which is not known for Dirichlet…

Let $A$ be a compact $d$-dimensional $C^2$ Riemannian manifold with boundary, embedded in ${\bf R}^m$ where $m \geq d \geq 2$, and let $B$ be a nice subset of $A$ (possibly $B=A$). Let $X_1,X_2, \ldots $ be independent random uniform points…

In this paper, we study the relationship between the type problem and the asymptotic behavior of the first eigenvalues $\lambda_1(B_r)$ of ``balls'' $B_r:=\{\rho<r\}$ on a complete Riemannian manfold $M$ as $r\rightarrow +\infty$, where…