Stability of eigenvalues for variable exponent problems

TL;DR

This paper proves that variational eigenvalues in variable exponent Sobolev spaces remain stable when the exponents converge uniformly, ensuring robustness of spectral properties in these variable settings.

Contribution

It establishes the stability of eigenvalues under uniform convergence of exponents in variable exponent Sobolev spaces, a novel result in spectral theory.

Findings

Eigenvalues are stable under uniform convergence of exponents.

Variational eigenvalues defined by Rayleigh ratios are robust.

Results apply to Luxemburg norm-based eigenvalues.

Abstract

In the framework of variable exponent Sobolev spaces, we prove that the variational eigenvalues defined by inf sup procedures of Rayleigh ratios for the Luxemburg norms are all stable under uniform convergence of the exponents.