A necessary and sufficient condition for $C^1$-regularity of solutions of one-dimensional variational obstacle problems

Abstract

In this paper we study the $C^1$-regularity of solutions of one-dimensional variational obstacle problems in $W^{1,1}$ when the obstacles are $C^{1,\sigma }$ and the Lagrangian is locally Hölder continuous and globally elliptic. In this framework, we prove that the solutions of one-dimensional variational obstacle problems are $C^1$ for all boundary data if and only if the value function is Lipschitz continuous at all boundary data.