A Semigroup Approach to Harmonic Maps

We present a semigroup approach to harmonic maps between metric spaces. Our basic assumption on the target space (N,d) is that it admits a barycenter contraction, i.e. a contracting map which assigns to each probability measure q on N a point b(q) in N. This includes all metric spaces with globally nonpositive curvature in the sense of Alexandrov as well as all metric spaces with globally nonpositive curvature in the sense of Busemann. It also includes all Banach spaces.The analytic input comes from the domain space (M,) where we assume that we are given a Markov semigroup (p_t)_t>0. Typical examples come from elliptic or parabolic second-order operators on Rⁿ, from Lévy type operators, from Laplacians on manifolds or on metric measure spaces and from convolution operators on groups. In contrast to the work of Korevaar and Schoen (1993, 1997), Jost (1994, 1997), Eells and Fuglede (2001) our semigroups are not required to be symmetric.The linear semigroup acting, e.g., on the space of bounded measurable functions u:MR gives rise to a nonlinear semigroup (P_t^*)_t acting on certain classes of measurable maps f:MN. We will show that contraction and smoothing properties of the linear semigroup (p_t)_t can be extended to the nonlinear semigroup (P_t^*)_t, for instance, L_p–L_q smoothing, hypercontractivity, and exponentially fast convergence to equilibrium. Among others, we state existence and uniqueness of the solution to the Dirichlet problem for harmonic maps between metric spaces. Moreover, for this solution we prove Lipschitz continuity in the interior and Hölder continuity at the boundary.Our approach also yields a new interpretation of curvature assumptions which are usually required to deduce regularity results for the harmonic map flow: lower Ricci curvature bounds on the domain space are equivalent to estimates of the L₁-Wasserstein distance between the distribution of two Brownian motions in terms of the distance of their starting points; nonpositive sectional curvature on the target space is equivalent to the fact that the L₁-Wasserstein distance of two distributions always dominates the distance of their barycenters.Dedicated to the memory of Professor Dr. Heinz Bauer