On the regularity of solutions of one dimensional variational obstacle problems

TL;DR

This paper investigates the regularity of solutions to one-dimensional variational obstacle problems with locally Hölder continuous Lagrangians, introducing a direct method to establish partial regularity based on obstacle smoothness.

Contribution

It presents a new direct method for analyzing regularity in obstacle problems, linking obstacle smoothness to solution regularity within a specific function class.

Findings

Solutions with $C^{1,\sigma}$ obstacles have Tonelli's partial regularity.

Introduces a subclass of $W^{1,1}$ functions with guaranteed regularity properties.

Provides a framework connecting obstacle regularity to solution regularity.

Abstract

We study the regularity of solutions of one dimensional variational obstacle problems in $W^{1,1}$ when the Lagrangian is locally H\"older continuous and globally elliptic. In the spirit of the work of Sychev ([Syc89, Syc91, Syc92]), a direct method is presented for investigating such regularity problems with obstacles. This consists of introducing a general subclass $\mathcal{L}$ of $W^{1,1}$, related in a certain way to one dimensional variational obstacle problems, such that every function of $\mathcal{L}$ has Tonelli's partial regularity, and then to prove that, depending of the regularity of the obstacles, solutions of corresponding variational problems belong to $\mathcal{L}$. As an application of this direct method, we prove that if the obstacles are $C^{1,\sigma}$ then every Sobolev solution has Tonelli's partial regularity.