A new minimizing-movements scheme for curves of maximal slope

TL;DR

This paper introduces an alternative minimizing-movements scheme for curves of maximal slope, improving regularity, providing convergence estimates, and simplifying proofs in the context of gradient flows in metric spaces.

Contribution

It proposes a new minimizing-movements approach that enhances regularity, offers a posteriori convergence estimates, and simplifies the convergence proof for curves of maximal slope.

Findings

Provides a more regular discretization scheme.

Enables a posteriori convergence estimation.

Simplifies the convergence proof for gradient flows.

Abstract

Curves of maximal slope are a reference gradient-evolution notion in metric spaces and arise as variational formulation of a vast class of nonlinear diffusion equations. Existence theories for curves of maximal slope are often based on minimizing-movements schemes, most notably on the Euler scheme. We present here an alternative minimizing-movements approach, yielding more regular discretizations, serving as a-posteriori convergence estimator, and allowing for a simple convergence proof.