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 C2,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⊂ 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 W1,p-regularity (1<p<∞) has been well-established when ∂Ω is Lipschitz (or even more general) and the boundary data belongs to W1,p(∂Ω). 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 ∂ 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.