Compactness of weighted Sobolev trace operators and non-linear Steklov problems

TL;DR

This paper establishes the compactness of weighted Sobolev trace operators in cuspidal domains, enabling the proper formulation and solution of non-linear Steklov problems in such geometries.

Contribution

It introduces a novel approach using composition operators to prove compactness, facilitating the analysis of non-linear Steklov problems in complex domains.

Findings

Proved compactness of weighted Sobolev trace operators in cuspidal domains.

Formulated the non-linear Steklov problem in these domains.

Established existence of non-trivial solutions.

Abstract

We prove the compactness of weighted Sobolev trace operators in outward cuspidal domains by using composition operators on Sobolev spaces. This result allows us to formulate the non-linear Steklov problem in outward cuspidal domains in a correct functional setting and to establish the existence of its non-trivial solution.