Optimal regularity of lower dimensional obstacle problems

In this paper we prove that solutions to the “boundary obstacle problem” have the optimal regularity, $C^{1,1/2}$, in any space dimension. This bound depends only on the local $L^2$ norm of the solution. Main ingredients in the proof are the quasiconvexity of the solution and a monotonicity formula for an appropriate weighted average of the local energy of the normal derivative of the solution.