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On the regularity of solutions of one dimensional variational obstacle problems

We study the regularity of solutions of one dimensional variational obstacle problems in $W^{1,1}$ when the Lagrangian is locally Hölder 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 $C^{1,σ}$ then every Sobolev solution has Tonelli's partial regularity.