Sobolev $(p,q)$-extension operators and Neumann eigenvalues

TL;DR

This paper develops $(p,q)$-extension operators for Sobolev spaces in cuspidal domains and uses them to estimate nonlinear Neumann eigenvalues of the $p$-Laplace operator.

Contribution

It introduces new $(p,q)$-extension operators based on composition operators and applies them to bound Neumann eigenvalues in complex domains.

Findings

Constructed $(p,q)$-extension operators with norm estimates

Derived bounds for nonlinear Neumann eigenvalues in cuspidal domains

Extended Sobolev space techniques to non-standard domain geometries

Abstract

In this article, we consider $(p,q)$-extension operators, $1 < q \le p < \infty$, on Sobolev spaces. Based on composition operators on Sobolev spaces, we construct the extension operators in outward cuspidal domains with estimates of their norms. Using these $(p,q)$-extension operators, we prove estimates for the non-linear Neumann eigenvalues of the $p$-Laplace operator in outward cuspidal domains.