Relations between principal eigenvalue and torsional rigidity with Robin boundary conditions

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.

💡 Research Summary

This paper investigates the interplay between the principal (first) eigenvalue and the torsional rigidity of a bounded Lipschitz domain Ω⊂ℝⁿ under Robin boundary conditions. For a fixed Robin parameter β>0, the eigenvalue λ_β(Ω) is defined variationally by the Rayleigh quotient (1.1) and the torsional rigidity T_β(Ω) by the dual variational formulation (1.2). The authors study the shape functional

  F_{β,q}(Ω)=λ_β(Ω)·