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 L²-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. Bibliography: 8 titles. Dedicated to Nina Nikolaevna Uraltseva on the occasion of her 70th birthday __________ Published in Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 49–66.