Hausdorff dimension of rupture sets and removable singularities

Juan Dávila; Augusto C. Ponce

doi:10.1016/j.crma.2007.11.007

Partial Differential Equations

Hausdorff dimension of rupture sets and removable singularities

Presented by: Haïm Brezis

Juan Dávila ¹; Augusto C. Ponce ²

¹ Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
² Laboratoire de mathématiques et physique théorique (UMR CNRS 6083), Université de Tours, 37200 Tours, France

Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 27-32.

Abstract
Résumé

Given $α > 0$ and a domain $Ω \subset R^{N}$ , 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 $N - 2 + \frac{2}{α + 1}$ . The proof is based on a removable singularity property of the Laplacian Δ.

Étant donnés $α > 0$ et un domaine borné $Ω \subset R^{N}$ , nous prouvons que pour toute solution d'énergie finie $u ⩾ 0$ de l'équation $- Δ u + u^{- α} = f (x) in Ω$ , l'ensemble $[u = 0]$ a une dimension de Hausdorff inférieure ou égale à $N - 2 + \frac{2}{α + 1}$ . La démonstration de ce résultat repose sur une propriété de singularité éliminable du laplacien Δ.

Accepted: 2007-11-05
Published online: 2008-01-01

DOI: 10.1016/j.crma.2007.11.007

Author's affiliations:

Juan Dávila ¹; Augusto C. Ponce ²

¹ Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
² Laboratoire de mathématiques et physique théorique (UMR CNRS 6083), Université de Tours, 37200 Tours, France

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     author = {Juan D\'avila and Augusto C. Ponce},
     title = {Hausdorff dimension of rupture sets and removable singularities},
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     year = {2008},
     doi = {10.1016/j.crma.2007.11.007},
     language = {en},
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Juan Dávila; Augusto C. Ponce. Hausdorff dimension of rupture sets and removable singularities. Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 27-32. doi : 10.1016/j.crma.2007.11.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.11.007/

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