Geometric and energetic criteria for the free boundary regularity in an obstacle-type problem

We consider an obstacle-type problem Δu = f(x)χΩ     in D, u = |∇u| = 0     on D \ Ω, where D is a given open set in ℝn and Ω is an unknown open subset of D. The problem originates in potential theory, in connection with harmonic continuation of potentials. The qualitative difference between this problem and the classical obstacle problem is that the solutions here are allowed to change sign. Using geometric and energetic criteria in delicate combination we show the C1,1 regularity of the solutions, and the regularity of the free boundary, below the Lipschitz threshold for the right-hand side.