New formulas for the Laplacian of distance functions and applications

By Fabio Cavalletti, Andrea Mondino. Analysis & PDE, 2018

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The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially non-branching MCP(K,N)-spaces). Such a representation formula makes apparent the classical upper bounds and also some new lower bounds, together with a precise description of the singular part. The exact representation formula for the Laplacian of 1-Lipschitz functions (in particular for distance functions) holds also (and seems new) in a general complete Riemannian manifold. We apply these results to prove the equivalence of CD(K,N) and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic Splitting Theorem for infinitesimally Hilbertian essentially non-branching spaces verifying MCP(0,N).

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/7e23071fd5176d218a2235f99bad3b916a891d5c