Related papers: Small-hole minimization of the first Dirichlet eig…

We consider the well-known shape optimization problem with spectral cost: minimizing the first eigenvalue of the Dirichlet Laplacian among all subdomains $\Omega$ having prescribed volume and contained in a fixed box $D$; equivalently, we…

A generic model for the shape optimization problems we consider in this paper is the optimization of the Dirichlet eigenvalues of the Laplace operator with a volume constraint. We deal with an obstacle placement problem which can be…

We study the behaviour of extremal eigenvalues of the Dirichlet biharmonic operator over rectangles with a given fixed area. We begin by proving that the principal eigenvalue is minimal for a rectangle for which the ratio between the…

Inside a fixed bounded domain $\Omega$ of the plane, we look for the best compact connected set $K$, of given perimeter, in order to maximize the first Dirichlet eigenvalue $\lambda_1(\Omega\setminus K)$. We discuss some of the qualitative…

Let $\Omega\subset\mathbb{R}^N$, $N\geq 2$, be an open bounded connected set. We consider the fractional weighted eigenvalue problem $(-\Delta)^s u =\lambda \rho u$ in $\Omega$ with homogeneous Dirichlet boundary condition, where…

We deal with the following eigenvalue optimization problem: Given a bounded domain $D\subset \R^2$, how to place an obstacle $B$ of fixed shape within $D$ so as to maximize or minimize the fundamental eigenvalue $\lambda_1$ of the Dirichlet…

We consider the problem of minimizing convex combinations of the first two eigenvalues of the Dirichlet-Laplacian among open sets of $R^N$ of fixed measure. We show that, by purely elementary arguments, based on the minimality condition, it…

In this paper, we consider the optimization problem for the first Dirichlet eigenvalue $\lambda_1(\Omega)$ of the $p$-Laplacian $\Delta_p$, $1< p< \infty$, over a family of doubly connected planar domains $\Omega= B \setminus \overline{P}$,…

We consider the following eigenvalue optimization problem: Given a bounded domain $\Omega\subset\R^n$ and numbers $\alpha\geq 0$, $A\in [0,|\Omega|]$, find a subset $D\subset\Omega$ of area $A$ for which the first Dirichlet eigenvalue of…

We compute the first-order correction in the quartic coupling $\alpha$ to the entanglement entropy of a massive, non-minimally coupled scalar across the horizon of a four-dimensional Schwarzschild black hole, treating the non-minimal…

In this article, we study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian $(-\Delta)^s$ in bounded open Lipschitz sets in the small order limit $s \to 0^+$. While it is easy to see that all…

Maximization and minimization problems of the principle eigenvalue for divergence form second order elliptic operators with the Dirichlet boundary condition are considered. The principal eigen map of such elliptic operators is introduced…

We consider $\mathbb{S}^2$-valued maps on a domain $\Omega\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy with vertical penalization in $\Omega$ and horizontal penalization on $\partial\Omega$. We first show the…

In this paper we prove an existence and uniqueness result for the double phase Dirichlet problem when the lowest exponent is equal to 1. Our solution is a function of bounded variation that simultaneously lies in a suitable weighted Sobolev…

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair $\boldsymbol\varepsilon = (\varepsilon_1, \varepsilon_2 )$ of positive parameters, we consider a perforated domain…

We study the minimizers of the sum of the principal Dirichlet eigenvalue of the negative Laplacian and the perimeter with respect to a general norm in the class of Jordan domains in the plane. This is equivalent (modulo scaling) to…

We prove an isoperimetric inequality of the Rayleigh-Faber-Krahn type for a nonlinear generalization of the first twisted Dirichlet eigenvalue. More precisely, we show that the minimizer among sets of given volume is the union of two equal…

In this note, we study the Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in $\CAT(1)$ space. Under the setting, we prove that the Korevaar-Schoen energy admits a unique minimizer.

The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of $\mathbb{S}^2$-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of…

We consider an energy functional combining the square of the local oscillation of a one--dimensional function with a double well potential. We establish the existence of minimal heteroclinic solutions connecting the two wells of the…