An isoperimetric inequality for twisted eigenvalues with one orthogonality constraint

TL;DR

This paper establishes an isoperimetric inequality for twisted eigenvalues with orthogonality constraints, identifying optimal shapes and providing bounds that extend classical eigenvalue inequalities without relying on Bessel functions.

Contribution

It introduces a novel isoperimetric inequality for twisted eigenvalues considering orthogonality constraints, revealing unique optimal shapes and interpolating between classical eigenvalue optimizers.

Findings

Optimal sets are unions of two disjoint balls with specific radii.

Lower bounds are attained when the orthogonality function is bang-bang type.

The inequality offers new insights into classical eigenvalue inequalities.

Abstract

We consider twisted eigenvalues $\lambda_{1}^{g}(\Omega)$, defined as the minimum of the Rayleigh quotient of functions in $H^1_{0}(\Omega)$ that are orthogonal to a given function $g\in L^2_\text{loc}(\mathbb R^d)$. We prove an isoperimetric inequality for $\lambda_1^g(\Omega)$, which provides a uniform bound on twisted eigenvalues -- not only with respect to the domain $\Omega$ (an open bounded set of $\mathbb R^d$) -- but also in relation to the orthogonality function $g$. Remarkably, the lower bound is uniquely attained when $\Omega$ is the union of two disjoint balls of specific radii, and when the function $g$ in the orthogonality constraint is of bang-bang type, i.e., constant on each ball. As a consequence, we obtain a continuous 1-parameter family of optimal sets -- each being the union of two disjoint balls -- that interpolates between the optimal shapes of the first two Dirichlet eigenvalues of the Laplacian. This new isoperimetric inequality offers fresh perspectives on well established results, such as the Hong-Krahn-Szeg\H{o} and the Freitas-Henrot inequalities. Notably, in these particular cases our proof avoids reliance on Bessel functions, suggesting potential extensions to nonlinear settings.