Optimal boundary regularity of harmonic maps from $RCD(K,N)$-spaces to $CAT(0)$-spaces

TL;DR

This work addresses boundary regularity for harmonic maps from $RCD(K,N)$ spaces to $CAT(0)$ spaces, a setting with non-smooth domains and targets. The authors develop a sharp Gauss-Green framework in $RCD$ spaces, derive optimal Dirichlet heat-kernel and Green-function bounds near boundaries with exterior-ball geometry, and obtain a quantitative boundary gradient estimate for energy-minimizing maps with Lipschitz boundary data. The main contribution is a full boundary Lipschitz control, $${\rm Lip}\,u(x)\le c_{1} L\cdot \ln\left(\frac{2e\cdot {\rm diam}(\Omega)}{d(x,\partial\Omega)}\right),$$ valid under perimetrical regularity and exterior-ball conditions, with the method relying on a novel Gauss-Green formula for sets of finite perimeter in $RCD(K,N)$ spaces. This extends classical boundary regularity results to a broad class of singular spaces and provides a blueprint for boundary estimates in geometric-analytic problems on non-smooth spaces, with potential implications for qualitative regularity in nonlinear harmonic-approximation problems on metric-measure spaces.

Abstract

In 1983, Schoen-Uhlenbeck \cite{SU83} established boundary regularity for energy-minimizing maps between smooth manifolds with the Dirichlet boundary condition under the assumption that both the boundary and the data are of $C^{2,α}$. A natural problem is to study the qualitative boundary behavior of harmonic maps with rough boundary and/or non-smooth boundary data. For the special case where $u$ is a harmonic function on a domain $Ω\subset \mathbb R^n$, this problem has been extensively studied (see, for instance, the monograph \cite{Kenig94}, the proceedings of ICM 2010 \cite{Tor10} and the recent work of Mourgoglou-Tolsa \cite{MT24}). The $W^{1,p}$-regularity ($1<p<\infty$) has been well-established when $\partialΩ$ is Lipschitz (or even more general) and the boundary data belongs to $W^{1,p}(\partialΩ)$. However, for the endpoint case where the boundary data is Lipschitz continous, as demonstrated by Hardy-Littlewood's classical examples \cite{HL32}, the gradient $|\nabla u|(x)$ may have logarithmic growth as $x$ approaches the boundary $\partial Ω$ even if the boundary is smooth. In this paper, we first establish a version of the Gauss-Green formula for bounded domains in $RCD(K, N)$ metric measure space. We then apply it to obtain the optimal boundary regularity of harmonic maps from $RCD(K, N)$ metric measure spaces into $CAT(0)$ metric spaces. Our result is new even for harmonic functions on Lipschitz domains of Euclidean spaces.

Theorems & Definitions (74)

• Theorem 1.1: Schoen-Uhlenbeck SU83
• Theorem 1.2
• Remark 1.3
• Definition 1.4
• Definition 1.5
• Remark 1.6
• Theorem 1.7: Zhang-Zhu ZZ24
• Theorem 1.8
• Example 1.9: Hardy-Littlewood
• Theorem 1.10