An isoperimetric inequality for twisted eigenvalues with one orthogonality constraint

TL;DR

This work introduces and analyzes twisted Dirichlet eigenvalues $\lambda_1^g(\Omega)$, defined via an orthogonality constraint to a function $g$, and proves a sharp isoperimetric inequality that bounds $|\Omega|^{2/d}\lambda_1^g(\Omega)$ uniformly in $\Omega$ and $g$. The minimizers are shown to be unions of two disjoint balls with bang-bang orthogonality $g=\chi_\alpha$, yielding a continuous 1-parameter family $\{(m(\alpha),\lambda(\alpha))\}_{\alpha\in(0,1]}$ that interpolate between the optimal shapes for the first two Dirichlet eigenvalues. The analysis combines variational methods, rearrangement techniques, and shape derivatives, and, in the case $\alpha=1$, recovers the Freitas–Henrot and Hong–Krahn–Szegő inequalities without resorting to Bessel functions in key regimes. A companion implicit-function theorem argument shows the precise regularity of the dependence on $\alpha$, establishing monotonicity and differentiability of the optimal radii ratio and eigenvalue, with explicit asymptotics as $\alpha\to0^+$ and $\alpha\to1^-$. Overall, the paper advances the theory of nonlocal (twisted) eigenvalue problems and provides a robust framework for extending isoperimetric-type results to nonlinear or more general settings.

Abstract

We consider twisted eigenvalues $λ_{1}^{g}(Ω)$, defined as the minimum of the Rayleigh quotient of functions in $H^1_{0}(Ω)$ that are orthogonal to a given function $g\in L^2_\text{loc}(\mathbb R^d)$. We prove an isoperimetric inequality for $λ_1^g(Ω)$, which provides a uniform bound on twisted eigenvalues -- not only with respect to the domain $Ω$ (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 $Ω$ 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-Szegő and the Freitas-Henrot inequalities. Notably, in these particular cases our proof avoids reliance on Bessel functions, suggesting potential extensions to nonlinear settings.

Figures (3)

• Figure 1: The optimal shape (up to scaling) $B_+^\alpha\cup B_-^\alpha$ corresponding to the values $\alpha=1$, $\alpha=\frac{1}{6}$ and $\alpha=\frac{1}{20}$, in the case $d=2$.
• Figure 2: Plots of the functions $m(\alpha)$ and $\lambda(\alpha)$ (continuous lines) with their lower bounds in \ref{['dis.radius']} (dashed lines), in the case $d=2$.
• Figure 3: Plots (for $d=2$) of $\lambda(m,\alpha)=\lambda_1^{\chi_\alpha}(B_+\cup B_-)$ in terms of the variable $m=|B_-|/|B_+|$, at fixed $\alpha=1,1/6,1/20$ (continuous lines) with their upper bounds in Proposition \ref{['p.upper']} (dashed lines).

Theorems & Definitions (56)

• Theorem 1.1: Isoperimetric inequality for twisted eigenvalues
• Corollary 1.2: Freitas-Henrot and Hong-Krahn-Szegő inequalities
• Theorem 1.3: Continuous maps connecting the optimal shapes of Dirichlet eigenvalues
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Remark 2.4