Interpolation theory for Sobolev functions with partially vanishing trace on irregular open sets

TL;DR

This paper develops a comprehensive interpolation theory for Sobolev functions with fractional smoothness that vanish on parts of irregular open sets, extending previous results beyond Lipschitz domains using measure-theoretic geometric assumptions.

Contribution

It introduces a new interpolation framework for Sobolev spaces with partially vanishing trace on irregular sets, incorporating porous boundary conditions and measure-theoretic geometry.

Findings

Established interpolation results for Sobolev functions with fractional smoothness.

Extended the theory to irregular domains beyond Lipschitz regularity.

Analyzed the role of porous boundaries and characteristic multipliers.

Abstract

A full interpolation theory for Sobolev functions with smoothness between 0 and 1 and vanishing trace on a part of the boundary of an open set is established. Geometric assumptions are of mostly measure theoretic nature and reach beyond Lipschitz regular domains. Previous results were limited to regular geometric configurations or Hilbertian Sobolev spaces. Sets with porous boundary and their characteristic multipliers on smoothness spaces play a major role in the arguments.