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

In 1983, Schoen-Uhlenbeck 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,. 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 Rⁿ, this problem has been extensively studied (see, for instance, the monograph Kenig94, the proceedings of ICM 2010 Tor10 and the recent work of Mourgoglou-Tolsa MT24). The W^1, p-regularity (1<p<) has been well-established when is Lipschitz (or even more general) and the boundary data belongs to W^1, p (). However, for the endpoint case where the boundary data is Lipschitz continous, as demonstrated by Hardy-Littlewood's classical examples HL32, the gradient | u| (x) may have logarithmic growth as x approaches the boundary 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.