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

For an harmonic map \(u\) from a domain \(U\subset{\rm X}\) in an \({\sf RCD}(K,N)\) space \({\rm X}\) to a \({\sf CAT}(0)\) space \({\rm Y}\) we prove the Lipschitz estimate \[ {\rm Lip}(u|_B)\leq \frac {C(K^-R^2,N)}r\inf_{{\sf o}\in {\rm Y}}\,\sqrt{\frac1{{\mathfrak m}(2B)}\int_{2B}{\sf d}_{\rm Y}^2(u(\cdot),{\sf o})\, {\rm d}{\mathfrak m}}, \qquad \forall 2B\subset U \] where \(r\in(0,R)\) 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 Poincar’e 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 \({\sf RCD}\) spaces is particularly relevant. It can be roughly stated as: if \(({\rm X},{\sf d},{\mathfrak m})\) is \({\sf RCD}(K,\infty)\) and \(f\in C_b({\rm X})\) is so that \(\Delta f\leq C\) for some constant \(C>0\), then for every \(t>0\) and \({\mathfrak m}\)-a.e. \(x\in{\rm X}\) there is a unique minimizer \(F_t(x)\) for \(y\ \mapsto\ f(y)+\frac{{\sf d}^2(x,y)}{2t}\) and the map \(F_t\) satisfies \[ (F_t)_*{\mathfrak m}\leq e^{t(C+2K^-{\sf Osc}(f))}{\mathfrak m},\qquad\text{where}\qquad {\sf Osc}(f):=\sup f-\inf f. \]
Here 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.