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.
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