Weak solutions to gradient flows of functionals with inhomogeneous growth in metric spaces

TL;DR

This paper introduces a new approach to define and analyze weak solutions for gradient flows in complex mathematical spaces.

Contribution

The paper introduces a novel framework for weak solutions in metric spaces with inhomogeneous growth functionals.

Findings

A new definition of weak solutions for gradient flows is established.

Existence and uniqueness of these solutions are proven.

The approach is applied to functionals with inhomogeneous growth.

Abstract

We use the framework of the first-order differential structure in metric measure spaces introduced by Gigli to define a notion of weak solution to gradient flows of convex, lower semicontinuous and coercive functionals. We prove their existence and uniqueness and show that they are also variational solutions; in particular, this is an existence result for variational solutions. Then, we apply this technique in the case of a gradient flow of a functional with inhomogeneous growth.