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October 3, 2025Open Access

A Characterization of Solvability of the Parabolic Lᵖ Dirichlet Problem on Lipschitz Graph Domains Via Carleson Measure Estimates of Bounded Solutions

Key Points

Solvability of the parabolic Dirichlet problem improves when bounded solutions meet Carleson measures, ensuring well-defined parabolic measures.
A key finding shows that the parabolic measure on boundaries belongs to class A∞ due to meeting specific Carleson measure conditions.
This work enhances existing findings by relaxing assumptions related to uniform rectifiability for Lipschitz domains with parabolic problems.
Implications suggest a broader understanding of solvability in various domains while advancing the theoretical framework used in boundary measure studies.

Abstract

In this paper, we show that if the bounded solutions to the parabolic Dirichlet problem on a Lipshitz-1, 12 domain obey a Carleson measure estimate, then the corresponding parabolic measure on the boundary will belong to class A^, which is equivalent to Lᵖ solvability for some p<. This improves the existing literature which places additional assumptions on the parabolic uniform rectifiability or, equivalently, on the half-order time derivative of the function whose graph defines the boundary of the domain.

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