The elastic flow with obstacles: small obstacle results

TL;DR

This paper studies the elastic flow of graphs with obstacles, proving existence and convergence results under small obstacle conditions, and analyzing qualitative properties like energy dissipation and regularity.

Contribution

It establishes global existence and subconvergence for elastic flow with obstacles, with improved convergence for symmetric cone obstacles, and examines qualitative behaviors.

Findings

Global existence and well-posedness under small obstacle conditions

Subconvergence of solutions, with convergence for symmetric cone obstacles

Analysis of energy dissipation and regularity properties

Abstract

We consider a parabolic obstacle problem for Euler's elastic energy of graphs with fixed ends. We show global existence, well-posedness and subconvergence provided that the obstacle and the initial datum are suitably 'small'. For symmetric cone obstacles we can improve the subconvergence to convergence. Qualitative aspects such as energy dissipation, coincidence with the obstacle and time regularity are also examined.