Relations between principal eigenvalue and torsional rigidity with Robin boundary conditions

TL;DR

This work analyzes how Robin boundary conditions change shape-optimization bounds for the product $F_{\beta,q}(\Omega)=\lambda_\beta(\Omega) T_\beta(\Omega)^q$ at fixed measure. It establishes a Robin-specific threshold $m_q=0$ iff $q>1/(d+1)$ (independent of $\beta$) and shows $M_1=1$ with finite $M_q$ for $q>1$, using homogenization of perforated lattices and $\gamma$-convergence to characterize extremal behavior. The approach blends variational characterizations, sharp inequalities, and perforated-domain constructions to contrast Robin and Dirichlet scenarios, highlighting a richer threshold structure. It also outlines open questions about Kohler–Jobin-type inequalities in the Robin setting and potential reverse inequalities for large $q$.

Abstract

We consider the torsional rigidity and the principal eigenvalue related to the Laplace operator with Dirichlet and Robin boundary conditions. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in the class of Lipschitz domains. The threshold exponent for the Robin case is explicitly recovered and shown to be strictly smaller than in the Dirichlet one.

Figures (1)

• Figure 1: Approximation of a smooth boundary by means of rough boundaries.

Theorems & Definitions (16)

• Proposition 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1
• proof