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

TL;DR

This paper develops a framework for weak solutions to gradient flows of convex functionals with inhomogeneous growth in metric spaces, proving existence, uniqueness, and their variational nature, with applications to specific growth conditions.

Contribution

It introduces a new approach to defining and analyzing weak solutions for gradient flows of inhomogeneous growth functionals in metric measure spaces, extending previous theories.

Findings

Existence and uniqueness of weak solutions are established.

Weak solutions are shown to be variational solutions.

Application to functionals with inhomogeneous growth demonstrates the framework's effectiveness.

Abstract

We use the framework of the first-order differential structure in metric measure spaces introduced by Gigli to define a notion of weak solutions 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.