Partial Differential Equations
Hausdorff dimension of rupture sets and removable singularities
Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 27-32.

Given α>0 and a domain ΩRN, we show that for every finite energy solution u0 of the equation Δu+uα=f(x) in Ω, the set [u=0] has Hausdorff dimension at most N2+2α+1. The proof is based on a removable singularity property of the Laplacian Δ.

Étant donnés α>0 et un domaine borné ΩRN, nous prouvons que pour toute solution d'énergie finie u0 de l'équation Δu+uα=f(x) in Ω, l'ensemble [u=0] a une dimension de Hausdorff inférieure ou égale à N2+2α+1. La démonstration de ce résultat repose sur une propriété de singularité éliminable du laplacien Δ.

Accepted:
Published online:
DOI: 10.1016/j.crma.2007.11.007
Dávila, Juan 1; Ponce, Augusto C. 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
@article{CRMATH_2008__346_1-2_27_0,
     author = {D\'avila, Juan and Ponce, Augusto C.},
     title = {Hausdorff dimension of rupture sets and removable singularities},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {27--32},
     publisher = {Elsevier},
     volume = {346},
     number = {1-2},
     year = {2008},
     doi = {10.1016/j.crma.2007.11.007},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2007.11.007/}
}
TY  - JOUR
AU  - Dávila, Juan
AU  - Ponce, Augusto C.
TI  - Hausdorff dimension of rupture sets and removable singularities
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 27
EP  - 32
VL  - 346
IS  - 1-2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2007.11.007/
DO  - 10.1016/j.crma.2007.11.007
LA  - en
ID  - CRMATH_2008__346_1-2_27_0
ER  - 
%0 Journal Article
%A Dávila, Juan
%A Ponce, Augusto C.
%T Hausdorff dimension of rupture sets and removable singularities
%J Comptes Rendus. Mathématique
%D 2008
%P 27-32
%V 346
%N 1-2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2007.11.007/
%R 10.1016/j.crma.2007.11.007
%G en
%F CRMATH_2008__346_1-2_27_0
Dávila, Juan; Ponce, Augusto C. 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. http://www.numdam.org/articles/10.1016/j.crma.2007.11.007/

[1] Brezis, H.; Ponce, A.C. Remarks on the strong maximum principle, Differential Integral Equations, Volume 16 (2003), pp. 1-12

[2] Brezis, H.; Ponce, A.C. Kato's inequality when Δu is a measure, C.R. Math. Acad. Sci. Paris, Ser. I, Volume 338 (2004), pp. 599-604

[3] Dávila, J.; Montenegro, M. Positive versus free boundary solutions to a singular elliptic equation, J. Anal. Math., Volume 90 (2003), pp. 303-335

[4] Dávila, J.; Ponce, A.C. Variants of Kato's inequality and removable singularities, J. Anal. Math., Volume 91 (2003), pp. 143-178

[5] Dupaigne, L.; Ponce, A.C. Singularities of positive supersolutions in elliptic PDEs, Selecta Math. (N.S.), Volume 10 (2004), pp. 341-358

[6] Dupaigne, L.; Ponce, A.C.; Porretta, A. Elliptic equations with vertical asymptotes in the nonlinear term, J. Anal. Math., Volume 98 (2006), pp. 349-396

[7] Evans, L.C.; Gariepy, R.F. Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992

[8] Guo, Z.; Wei, J. Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscripta Math., Volume 120 (2006), pp. 193-209

[9] Jiang, H.; Lin, F.-H. Zero set of Sobolev functions with negative power of integrability, Chinese Ann. Math. Ser. B, Volume 25 (2004), pp. 65-72

[10] J. Van Schaftingen, M. Willem, Symmetry of solutions of semilinear elliptic problems, J. Eur. Math. Soc. (JEMS), in press

[11] Witelski, T.P.; Bernoff, A.J. Dynamics of three-dimensional thin film rupture, Physica D, Volume 147 (2000), pp. 155-176

Cited by Sources: