The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary -Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially nonbranching -spaces). Such a representation formula makes apparent the classical upper bounds together with lower bounds and a precise description of the singular part. The exact representation formula for the Laplacian of a general 1-Lipschitz function holds also (and seems new) in a general complete Riemannian manifold.
We apply these results to prove the equivalence of and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic splitting theorem for infinitesimally Hilbertian, essentially nonbranching spaces satisfying .
Fabio Cavalletti. Andrea Mondino. "New formulas for the Laplacian of distance functions and applications." Anal. PDE 13 (7) 2091 - 2147, 2020. https://doi.org/10.2140/apde.2020.13.2091