A Semigroup Approach to Harmonic Maps

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