Trace theory for parabolic boundary value problems with rough boundary conditions

TL;DR

This work develops a weighted, anisotropic trace theory for parabolic boundary value problems, characterising higher-order traces of intersections of weighted Sobolev spaces and transferring results from half-spaces to rough domains via the Dahlberg–Kenig–Stein pullback. It establishes continuity, surjectivity, and right inverses for trace operators and applies the theory to obtain well-posedness and maximal regularity for the heat equation with rough boundary data on domains with limited smoothness. A key advance is the independence of trace spaces from the spatial regularity parameter and the accommodation of higher-order traces and vector-valued data with temporal weights, enabling robust analysis of parabolic problems on rough domains. The framework unifies and extends prior results, offering a versatile tool for studying boundary data with boundary-layer behavior in weighted, anisotropic settings.

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 $C^{1,κ}$, with $κ\in [0,1)$.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3
• Theorem 4.1
• Remark 4.2