Weak solutions to gradient flows in metric measure spaces

Abstract

Due to the fact that in a metric space there is (in general) no notion of directional derivatives, the definition of solutions to gradient flows in metric measure spaces necessarily avoid their direct use. The most classical example is the heat flow, or more generally the p‐Laplacian evolution equation, which has been studied as the gradient flow in L2 of the p‐Cheeger energy. Typically, solutions are defined using the semigroup theory through a subdifferential of the energy. Other popular approaches include variational solutions and the weighted energy‐dissipation formalism. In this paper, using the first‐order differential structure on a metric measure space introduced by Gigli, we characterize the subdifferential in L2 of the p‐Cheeger energy. This gives rise to a new notion of solutions to the p‐Laplacian evolution equation in metric measure spaces. In the case p = 1, we introduce a metric analogue of the Anzellotti pairing and obtain a Green‐Gauss formula, which is then used in place of Gigli's structure to characterise the 1‐Laplacian operator and study the total variation flow.