On the $p$-Laplacian evolution equation in metric measure spaces

Abstract

The $p$-Laplacian evolution equation in metric measure spaces has been studied as the gradient flow in $L^2$ of the $p$-Cheeger energy (for $1 < p < \infty$). In this paper, using the first-order differential structure on a metric measure space introduced by Gigli, we characterize the subdifferential in $L^2$ of the $p$-Cheeger energy. This gives rise to a new definition of the $p$-Laplacian operator in metric measure spaces, which allows us to work with this operator in more detail. In this way, we introduce a new notion of solutions to the $p$-Laplacian evolution equation in metric measure spaces. For $p = 1$, we obtain a Green-Gauss formula similar to the one by Anzellotti for Euclidean spaces, and use it to characterise the $1$-Laplacian operator and study the total variation flow. We also study the asymptotic behaviour of the solutions of the $p$-Laplacian evolution equation, showing that for $1 \leq p < 2$ we have finite extinction time.