Hausdorff dimension of rupture sets and removable singularities

Abstract

Given alpha > 0 and a domain Omega C R-N, we show that for every finite energy solution it u >= 0 of the equation -Delta u + u(-alpha) = f (x) 2 in Omega, the set [u = 0] has Hausdorff dimension at most N - 2 + 2/alpha+1. The proof is based on a removable singularity property of the Laplacian Delta.