The $L^1$ gradient flow of a generalized scale invariant Willmore energy for radially non increasing functions

TL;DR

This paper investigates the gradient flow of a relaxed Willmore energy functional for radially non-increasing functions, revealing an erosion process governed by a first-order ODE, using minimizing movement theory and coarea formula.

Contribution

It introduces a novel gradient flow analysis for a relaxed Willmore energy on L1 functions with radial symmetry, including a coarea formula and erosion dynamics.

Findings

Flow results in erosion of initial data

Erosion speed described by a first-order ODE

Coarea formula established for relaxed energy

Abstract

We use the minimizing movement theory to study the gradient flow associated with a non-regular relaxation of a geometric functional derived from the Willmore energy. Thanks to the coarea formula, one can define a Willmore energy on regular functions of L 1 (R d). This functional is extended to every L 1 function by taking its lower semi-continuous envelope. We study the flow generated by this relaxed energy for radially non-increasing functions, i.e. functions with balls as level sets. In the first part of the paper, we prove a coarea formula for the relaxed energy of such functions. Then we show that the flow consists on an erosion of the initial data. The erosion speed is given by a first order ordinary equation.