# On Gradient Flows with Obstacles and Euler's Elastica

## TL;DR

This paper studies a gradient flow with obstacle constraints in higher order energy settings, focusing on elastic flow of graph curves with Navier boundary conditions, establishing existence and long-term behavior.

## Contribution

It introduces a new approach to construct obstacle-constrained gradient flows in complex energy frameworks using De Giorgi's scheme, applied to elastic graph curves.

## Key findings

- Established long-time existence of elastic flow with obstacles.
- Analyzed asymptotic behavior of the flow.
- Developed a general construction method for obstacle gradient flows.

## Abstract

We examine a steepest energy descent flow with obstacle constraint in higher order energy frameworks where the maximum principle is not available. We construct the flow under general assumptions using De Giorgi's minimizing movement scheme. Our main application will be the elastic flow of graph curves with Navier boundary conditions for which we study long-time existence and asymptotic behavior.