Trace theory for parabolic boundary value problems with rough boundary conditions

TL;DR

This paper develops a trace theory for weighted Sobolev spaces to handle boundary value problems with rough boundary conditions, and applies it to prove well-posedness of the heat equation with irregular boundary data.

Contribution

It introduces a new characterization of trace spaces for weighted Sobolev spaces and applies this to establish well-posedness for heat equations with rough boundary data.

Findings

Characterization of trace spaces for weighted Sobolev spaces.

Well-posedness of heat equation with rough boundary data.

Applicability to domains with $C^{1,ppa}$ regularity.

Abstract

We characterise the trace spaces arising from intersections of weighted, vector-valued Sobolev spaces, where the weights are powers of the distance to the boundary. These weighted function spaces are particularly suitable for treating boundary value problems where derivatives of the solution blow up at the boundary. As an application of our trace theory, we prove well-posedness for the heat equation with rough inhomogeneous boundary data in Sobolev spaces of higher regularity in domains of fixed regularity $C^{1,\kappa}$, with $\kappa \in [0,1)$.