Traces and embeddings of anisotropic function spaces

TL;DR

This paper characterizes the trace spaces of weighted intersection-type function spaces with mixed regularities and applies these results to establish maximal regularity for a two-phase Stefan problem with Gibbs-Thomson correction.

Contribution

It introduces a novel approach to handle mixed regularity scales in weighted function spaces and applies it to a complex free boundary problem.

Findings

Characterization of trace spaces for weighted intersection function spaces.

Overcoming difficulties in mixed scales via microscopic improvements.

Proving maximal $L^p$-$L^q$-regularity for the Stefan problem.

Abstract

In this paper we characterize the trace spaces of a class of weighted function spaces of intersection type with mixed regularities. To a large extent we can overcome the difficulty of mixed scales by employing a microscopic improvement in Sobolev and mixed derivative embeddings with fixed integrability. We apply the general results to prove maximal $L^p$-$L^q$-regularity for the linearized, fully inhomogeneous two-phase Stefan problem with Gibbs-Thomson correction.