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

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 $C^{2,\alpha}$. 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 $\Omega\subset \mathbb R^n$, 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 $\partial\Omega$ is Lipschitz (or even more general) and the boundary data belongs to $W^{1,p}(\partial\Omega)$. 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 $\partial \Omega$ 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.