Trace maps under weak regularity assumptions

TL;DR

This paper develops a new approach to bounded trace maps on hypersurfaces under minimal regularity assumptions, enabling solutions to boundary value problems with very weak geometric conditions.

Contribution

It introduces a novel method for constructing bounded trace maps for Sobolev spaces on hypersurfaces with minimal regularity assumptions, such as mere continuity or Lebesgue measurability.

Findings

Constructed bounded trace maps under weak regularity conditions.

Proved a coarea formula for Lebesgue measurable hypersurfaces.

Applied results to Dirichlet problems with minimal boundary regularity.

Abstract

We study bounded trace maps on hypersurfaces for Sobolev spaces from a point of view that is fundamentally different from the one in the classical theory. This allows us to construct bounded trace maps under weak regularity assumptions on the hypersurfaces. In the case of bounded domains in $\mathbf R^n$ we only require the continuity of the boundary. For hypersurfaces in the whole space $\mathbf R^n$ we only assume that the hypersurfaces are Lebesgue measurable. As an application of our trace maps we consider the Dirichlet problem and we prove a coarea formula where the level sets are only assumed to be Lebesgue measurable hypersurfaces.