The spectral drop problem

TL;DR

This paper investigates spectral optimization problems involving the first eigenvalue of the Laplace operator with mixed boundary conditions, establishing existence results and qualitative properties of optimal domains within a given set.

Contribution

It introduces a new spectral drop problem framework, providing existence theorems and analyzing qualitative features of optimal shapes under general shape cost functionals.

Findings

Existence of optimal domains is proven for general shape cost functionals.

Optimal domains exhibit specific qualitative properties.

The problem generalizes classical drop problems by involving spectral quantities.

Abstract

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 eigenvalue of the Laplace operator $-\Delta$ with Dirichlet conditions on $\partial\Omega\cap D$ and Neumann or Robin conditions on $\partial\Omega\cap\partial D$. The equivalent variational formulation $$\lambda_1(\Omega;D)=\min\left\{\int_\Omega|\nabla u|^2\,dx+k\int_{\partial D}u^2\,d\mathcal{H}^{d-1}\ :\ u\in H^1(D),\ u=0\hbox{ on }\partial\Omega\cap D,\ \|u\|_{L^2(\Omega)}=1\right\}$$ reminds the classical drop problems, where the first eigenvalue replaces the total variation functional. We prove an existence result for general shape cost functionals and we show some qualitative properties of the optimal domains.