An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equations

Abstract

This work addresses the uniqueness and regularity of solutions to integral equations associated with elliptic boundary value problems in irregular domains. Traditional results often assume smooth (Lipschitz) boundaries, but this study extends these results to more general domains with irregular boundaries. By leveraging Sobolev spaces, particularly fractional Sobolev spaces , and the properties of the Slobodetskii norm, we develop a robust theoretical framework. Our main theorem demonstrates that, under suitable conditions, has a unique solution in , and this solution inherits the regularity properties from the function . The results provide significant advancements in the mathematical understanding of boundary value problems in non-smooth domains, with potential applications in various fields of physics and engineering.