Related papers: Large $m$ asymptotics for minimal partitions of th…

In this paper we study existence and regularity of solutions to Dirichlet problems as $$ \begin{cases} - {\rm div}\left(|u|^m\frac{D u}{|D u|}\right) = f & \text{in}\;\Omega,\\ \newline u=0 & \text{on}\;\partial\Omega, \end{cases} $$ where…

Given an open bounded subset $\Omega$ of $\mathbb{R}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-\Delta_{\infty} u = 1$ in $\Omega$, subject to the homogeneous boundary condition $u = 0$ on…

A plasmon of a bounded domain $\Omega\subset\mathbb{R}^n$ is a non-trivial bounded harmonic function on $\mathbb{R}^n\setminus\partial\Omega$ which is continuous at $\partial\Omega$ and whose exterior and interior normal derivatives at…

Given a closed symplectic manifold (M,\omega) of dimension greater than 2, we consider all Riemannian metrics on M, which are compatible with the symplectic structure \omega. For each such metric, we look at the first eigenvalue \lambda_1…

In this paper, we propose a new assumption (1.2) that involves a small oscillation and $C^2$ norms for maps from smooth bounded domains into Euclidean spaces. Furthermore, by assuming that the domain has non-negative Ricci curvature, we…

We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with non constant coefficients, over a fixed domain $\Omega$. Dirichlet conditions are imposed along $\partial \Omega$ and, in…

Let $\Omega\subset \mathbb{R}^d$ be an open set of finite measure and let $\Theta$ be a disjoint union of two balls of half measure. We study the stability of the full Dirichlet spectrum of $\Omega$ when its second eigenvalue is close to…

In this paper, we give pinching Theorems for the first nonzero eigenvalue $\lambda$ of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of $M$ is 1 then, for any $\epsilon>0$, there…

We consider a possibly multiply connected bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{\max\{1,m\},\alpha}$ for some $m\in {\mathbb{N}}$, $\alpha\in]0,1[$ and we plan to solve both the Dirichlet and the Neumann problem for…

We study the optimization of the positive principal eigenvalue of an indefinite weighted problem, associated with the Neumann Laplacian in a box $\Omega\subset\mathbb{R}^N$, which arises in the investigation of the survival threshold in…

Given an elliptic operator~$L$ on a bounded domain~$\Omega \subseteq {\bf R}^n$, and a positive Radon measure~$\mu$ on~$\Omega$, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of…

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…

Suppose that $M$ is a compact Riemannian manifold with boundary and $u$ is an $L^2$-normalized Dirichlet eigenfunction with eigenvalue $\lambda$. Let $\psi$ be its normal derivative at the boundary. Scaling considerations lead one to expect…

This work considers the Neumann eigenvalue problem for the weighted Laplacian on a Riemannian manifold $(M,g,\partial M)$ under the singular perturbation. This perturbation involves the imposition of vanishing Dirichlet boundary conditions…

We prove the existence of a smooth family of non-compact domains $Omega_s \subset R^{n+1}$ bifurcating from the straight cylinder $B^n \times R$ for which the first eigenfunction of the Laplacian with 0 Dirichlet boundary condition also has…

This paper investigates sloshing problems defined by $-\Delta u=0$ in $\Omega$, with mixed boundary conditions: $\partial_{\nu}u=\lambda u$ on $S$, and either $\partial_{\nu}u=0$ or $u=0$ on $W$. Here, $\Omega$ represents a smooth bounded…

We show optimal existence, nonexistence and regularity results for nonnegative solutions to Dirichlet problems as $$ \begin{cases} \displaystyle -\Delta_1 u = g(u)|D u|+h(u)f & \text{in}\;\Omega,\\ u=0 & \text{on}\;\partial\Omega,…

We consider spectral optimization problems of the form $$\min\Big\{\lambda_1(\Omega;D):\ \Omega\subset D,\ |\Omega|=1\Big\},$$ where $D$ is a given subset of the Euclidean space $\mathbb{R}^d$. Here $\lambda_1(\Omega;D)$ is the first…

We prove existence of partitions of an open set $\Omega$ with a given number of phases, which minimize the sum of the fractional perimeters of all the phases, with Dirichlet boundary conditions. In two dimensions we show that, if the…

We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $\Omega\subset {\bf R}^n.$ Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric…