Gradient flows of $(K,N)$-convex functions with negative $N$

TL;DR

This paper explores the theory of gradient flows for $(K,N)$-convex functions in metric spaces when $N$ is negative, extending the framework to unbounded functionals and establishing key properties like contractivity and uniqueness.

Contribution

It introduces a novel analysis of $(K,N)$-convexity and gradient flows for negative $N$, including unbounded functionals, with proofs of fundamental properties.

Findings

Proved contractivity of gradient flows

Established regularity and uniqueness results

Extended theory to unbounded functionals with negative $N$

Abstract

We discuss $(K,N)$-convexity and gradient flows for $(K,N)$-convex functionals on metric spaces, in the case of real $K$ and negative $N$. In this generality, it is necessary to consider functionals unbounded from below and/or above, possibly attaining as values both the positive and the negative infinity. We prove several properties of gradient flows of $(K,N)$-convex functionals characterized by Evolution Variational Inequalities, including contractivity, regularity, and uniqueness.