On the regularity of solutions of one-dimensional variational obstacle problems

Abstract

Abstract We study the regularity of solutions of one-dimensional variational obstacle problems in W 1 , 1 {W^{1,1}} when the Lagrangian is locally Hölder continuous and globally elliptic. In the spirit of the work of Sychev [5, 6, 7], a direct method is presented for investigating such regularity problems with obstacles. This consists of introducing a general subclass ℒ {\mathcal{L}} of W 1 , 1 {W^{1,1}} , related in a certain way to one-dimensional variational obstacle problems, such that every function of ℒ {\mathcal{L}} has Tonelli’s partial regularity, and then to prove that, depending on the regularity of the obstacles, solutions of corresponding variational problems belong to ℒ {\mathcal{L}} . As an application of this direct method, we prove that if the obstacles are C 1 , σ {C^{1,\sigma}} , then every Sobolev solution has Tonelli’s partial regularity.