Remarks on the maximal regularity for parabolic boundary value problems with inhomogeneous data

TL;DR

The paper addresses maximal parabolic regularity for the heat equation in the upper half-space with inhomogeneous boundary data by developing a unified framework based on isotropic and anisotropic function spaces and interpolation. Using Fourier analysis of the heat kernel and anisotropic Littlewood-Paley theory, it derives sharp derivative estimates that bound interior derivatives $\nabla_x^{\alpha}\partial_t^{\beta} v$ in $L^p$ by boundary data norms in homogeneous anisotropic Besov spaces, $\|g\|_{\dot B^{m-\frac{1}{p},\frac{m}{2}-\frac{1}{2p}}_{p,p}}$. The main contributions are the maximal-regularity estimates for all derivative orders (including fractional orders), trace inequalities, and the corresponding equivalences in $W^{2m,m}_p$, $H^{s,s/2}_p$, and $B^{s,s/2}_{p,p}$ via interpolation; the results also extend to Neumann boundary data and hold under $p=1$ in some cases. These findings unify and extend prior work of Ogawa-Shimizu and Chen-Liang-Tsai, offering a versatile tool for parabolic boundary problems and potential applications to Stokes/Navier-Stokes boundary phenomena and free-boundary problems.

Abstract

Inspired by Ogawa-Shimizu [JEE 2022] and Chen-Liang-Tsai [IMRN 2025] on the second and first order derivative estimates of solution of heat equation in the upper half space with boundary data in homogeneous Besov spaces, we extend the estimates to any order of derivatives, including fractional derivatives.

Theorems & Definitions (15)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Definition 2.1: Isotropic function spaces
• Lemma 2.2: Basic properties
• proof
• Remark 2.3
• Lemma 2.4
• proof
• Definition 2.5: Anisotropic function spaces