Hausdorff dimension of rupture sets and removable singularities

Davila, Juan;Ponce, Augusto
(2008) Comptes rendus - Mathématique — Vol. 346, n° 1-2, p. 27-32 (2008)

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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.
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Davila, J., & Ponce, A. (2008). Hausdorff dimension of rupture sets and removable singularities. Comptes rendus - Mathématique, 346(1-2), 27-32. https://doi.org/10.1016/j.crma.2007.11.007 (Original work published 2008)