Markov Semigroups and Harmonic Maps

Summary

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 Banach spaces.

The analytic input comes from the domain space (M, p) where we assume that we are given a Markov semigroup (pt)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 [25], [26], [21], [22], [8], our semigroups are not required to be symmetric.

The linear semigroup acting e.g. on the space of bounded measurable functions u : M → ℝ gives rise to a nonlinear semigroup acting on certain classes of measurable maps f :M →N. We will show that contraction and smoothing properties of the linear semigroup (pt)t can be extended to the nonlinear semigroup. Among others, we state existence, uniqueness and Lipschitz continuity of the solution to the Dirichlet problem for harmonic maps between metric spaces.