Relations between principal eigenvalue and torsional rigidity with Robin boundary conditions

2025/12/16 by Buttazzo, Giuseppe, Cito, Simone, Solombrino, Francesco
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.2512.14927

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.

Citations

• Asymptotics of nonlinear Robin energies
• Minimization of the k -th eigenvalue of the Robin-Laplacian with perimeter constraint
• Sharp Quantitative Stability of the Dirichlet spectrum near the ball
• On a reverse Kohler-Jobin inequality
• On the relations between principal eigenvalue and torsional rigidity
• Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability and the Mumford-Shah conjecture
• Optimization problems involving the first Dirichlet eigenvalue and the\n torsional rigidity

Related

• Inequalities between torsional rigidity and principal eigenvalue of the p-Laplacian
• A Rigidity Result for the Robin Torsion Problem
• On the relations between principal eigenvalue and torsional rigidity
• Sharp estimates for the Gaussian torsional rigidity with Robin boundary conditions
• On asymptotics of Robin eigenvalues in the Dirichlet limit