Composition Operators on Sobolev Spaces and Neumann Eigenvalues

TL;DR

This paper explores how composition operators on Sobolev spaces influence the spectral properties of non-linear elliptic operators, providing lower bounds for Neumann eigenvalues in cusp-shaped domains.

Contribution

It applies geometric theory of composition operators to estimate Neumann eigenvalues of the p-Laplace operator in cusp domains, advancing spectral analysis techniques.

Findings

Lower bounds for first non-trivial Neumann eigenvalues in cusp domains

Application of geometric theory to spectral estimates

Insights into non-linear elliptic operator spectra

Abstract

In this paper we discuss applications of the geometric theory of composition operators on Sobolev spaces to the spectral theory of non-linear elliptic operators. The lower estimates of the first non-trivial Neumann eigenvalues of the $p$-Laplace operator in cusp domains $\Omega\subset\mathbb R^n$, $n\geq 2$, are given.