On the regularity of harmonic maps from ${\sf RCD}(K,N)$ to ${\sf CAT}(0)$ spaces and related results

Abstract

For an harmonic map u from a domain U ⊂ X in an RCD(K, N ) space X to a CAT(0) space Y we prove the Lipschitz estimatewhere r ∈ (0, R) is the radius of B. This is obtained by combining classical Moser's iteration, a Bochner-type inequality that we derive (guided by recent works of Zhang-Zhu) together with a reverse Poincaré inequality that is also established here. A direct consequence of our estimate is a Lioville-Yau type theorem in the case K = 0. Among the ingredients we develop for the proof, a variational principle valid in general RCD spaces is particularly relevant. It can be roughly stated as: if (X, d, m) is RCD(K, ∞) and f ∈ C b (X) is so that ∆f ≤ C for some constant C > 0, then for every t > 0 and m-a.e. x ∈ X there is a unique minimizer Ft(x) for y → f (y) +and the map Ft satisfiesHere existence is in place without any sort of compactness assumption and uniqueness should be intended in a sense analogue to that in place for Regular Lagrangian Flows and Optimal Maps (and is related to both these concepts). Finally, we also obtain a Rademacher-type result for Lipschitz maps between spaces as above.

Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in $\CAT(1)$ space

“…One of the research directions is to generalize the theory for non-smooth source spaces. Such attempts have been done by Gregori [17], Eells-Fuglede [11], Kuwae-Shioya [32], and so on. Gregori [17] and Eells-Fuglede [11] have dealt with a domain of a Lipschitz manifold and that of a Riemannian polyhedron, respectively.…”

“…Such attempts have been done by Gregori [17], Eells-Fuglede [11], Kuwae-Shioya [32], and so on. Gregori [17] and Eells-Fuglede [11] have dealt with a domain of a Lipschitz manifold and that of a Riemannian polyhedron, respectively. Kuwae-Shioya [32] have examined a metric measure space satisfying the so-called strongly measure contraction property of Bishop-Gromov type, called SMCPBG space.…”

“…Thanks to Theorem 1.1, one can introduce the notion of harmonic map from an RCD(K, N) space to a CAT(0) space. Very recently, Gigli [17] has shown a quantitative Lipschitz estimate for such harmonic maps, and produced a Cheng type Liouville theorem ( [7]) based on [10], [18], [22] (see also [41], [42]). Mondino-Semola [36] have also obtained a similar result, independently.…”