The structure of the free boundary for lower dimensional obstacle problems

Abstract. We study the regularity of the “free surface ” in boundary obstacle problems. We show that near a non-degenerate point the free boundary is a C1,α (n − 2)-dimensional surface in Rn−1. 1. Introduction. The purpose of this paper is to study the structure and reg-ularity of the free boundary in “boundary obstacle ” problems. Boundary obstacle problems concern the following question: We are given a smooth Ω in Rn, n ≥ 3, and seek a function u that: (a) In the interior of Ω, u satisfies a nice, elliptic equation, say ∆u = f