Hausdorff dimension of rupture sets and removable singularities

Abstract

Given α > 0 and a domain Ω ⊂ R N , we show that for every finite energy solution u ⩾ 0 of the equation − Δ u + u − α = f ( x ) in Ω, the set [ u = 0 ] has Hausdorff dimension at most N − 2 + 2 α + 1 . The proof is based on a removable singularity property of the Laplacian Δ. To cite this article: J. Dávila, A.C. Ponce, C. R. Acad. Sci. Paris, Ser. I 346 (2008).