Hausdorff dimension of rupture sets and removable singularities
Juan Dávila, Augusto C. Ponce · Comptes Rendus Mathématique · 2008
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.