On trace Theorems for weighted mixed norm Sobolev spaces and applications

TL;DR

This paper establishes new trace theorems for weighted mixed norm Sobolev spaces in the upper-half space, revealing how the differentiability of trace functions depends on weight and integrability parameters, with applications to elliptic and parabolic PDEs.

Contribution

It introduces novel trace theorems for weighted mixed norm Sobolev spaces, applicable even in unweighted cases, advancing the understanding of boundary regularity for anisotropic PDEs.

Findings

Trace differentiability depends on weight power and vertical integrability

Results are new even without weights and generalize classical un-mixed norm theorems

Provides tools for regularity analysis of fractional elliptic and parabolic equations

Abstract

We prove trace theorems for weighted mixed norm Sobolev spaces in the upper-half space where the weight is a power function of the vertical variable. The results show the differentiability order of the trace functions depends only on the power in the weight function and the integrability power for the integration with respect to the vertical variable but not on the integrability powers for the integration with respect to the horizontal ones. They are new even in the un-weighted case and they recover classical results in the case of un-mixed norm spaces. The work is motivated by the study of regularity theory for solutions of elliptic and parabolic equations with anisotropic features and with non-homogeneous boundary conditions. The results provide an essential ingredient to the study of fractional elliptic and parabolic equations in divergence form with measurable coefficients.