The precise boundary trace of solutions of a class of supercritical nonlinear equations

Moshe Marcus; Laurent Véron

doi:10.1016/j.crma.2006.11.028

Partial Differential Equations

The precise boundary trace of solutions of a class of supercritical nonlinear equations
[La trace au bord précise des solutions d'une classe d'équations non linéaires sur-critiques]

Présenté par : Haïm Brezis

Moshe Marcus ¹ ; Laurent Véron ²

¹ Department of Mathematics, Technion, Haifa 32000, Israel
² Laboratoire de mathématiques et physique théorique, faculté des sciences, parc de Grandmont, 37200 Tours, France

Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 181-186.

Résumé
Abstract

Nous construisons et étudions les propriétés de la trace au bord précise des solutions positives de $- Δ u + u^{q} = 0$ dans un domaine régulier de $R^{N}$ , dans le cas sur-critique $q ⩾ q_{c} = (N + 1) / (N - 1)$ .

We construct and study the properties of the precise boundary trace of positive solutions of $- Δ u + u^{q} = 0$ in a smooth bounded domain of $R^{N}$ , in the supercritical case $q ⩾ q_{c} = (N + 1) / (N - 1)$ .

Accepté le : 2006-11-14
Publié le : 2006-12-21

DOI : 10.1016/j.crma.2006.11.028

Affiliations des auteurs :

Moshe Marcus ¹ ; Laurent Véron ²

¹ Department of Mathematics, Technion, Haifa 32000, Israel
² Laboratoire de mathématiques et physique théorique, faculté des sciences, parc de Grandmont, 37200 Tours, France

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     title = {The precise boundary trace of solutions of a class of supercritical nonlinear equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {181--186},
     publisher = {Elsevier},
     volume = {344},
     number = {3},
     year = {2007},
     doi = {10.1016/j.crma.2006.11.028},
     language = {en},
}

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Moshe Marcus; Laurent Véron. The precise boundary trace of solutions of a class of supercritical nonlinear equations. Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 181-186. doi : 10.1016/j.crma.2006.11.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.11.028/

Bibliographie
Cité par

[1] D.R. Adams; L.I. Hedberg Function Spaces and Potential Theory, Grundlehren Math. Wiss., vol. 314, Springer, 1996

[2] E.B. Dynkin Diffusions, Superdiffusions and Partial Differential Equations, Colloquium Publications, vol. 50, Amer. Math. Soc., Providence, RI, 2002

[3] E.B. Dynkin Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations, Colloquium Publications, vol. 34, Amer. Math. Soc., Providence, RI, 2004

[4] E.B. Dynkin; S.E. Kuznetsov Superdiffusions and removable singularities for quasilinear partial differential equations, Comm. Pure Appl. Math., Volume 49 (1996), pp. 125-176

[5] E.B. Dynkin; S.E. Kuznetsov Fine topology and fine trace on the boundary associated with a class of quasilinear differential equations, Comm. Pure Appl. Math., Volume 51 (1998), pp. 897-936

[6] S.E. Kuznetsov σ-moderate solutions of $L u = u^{α}$ and fine trace on the boundary, C. R. Acad. Sci. Paris, Ser. I, Volume 326 (1998), pp. 1189-1194

[7] J.F. Legall The Brownian snake and solutions of $Δ u = u^{2}$ in a domain, Probab. Theory Related Fields, Volume 102 (1995), pp. 393-432

[8] J.F. Legall Spatial Branching Processes, Random Snakes and Partial Differential Equations, Birkhäuser, Basel, 1999

[9] M. Marcus; L. Véron The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Arch. Rat. Mech. Anal., Volume 144 (1998), pp. 201-231

[10] M. Marcus; L. Véron The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case, J. Math. Pures Appl., Volume 77 (1998), pp. 481-524

[11] M. Marcus; L. Véron Removable singularities and boundary trace, J. Math. Pures Appl., Volume 80 (2000), pp. 879-900

[12] M. Marcus; L. Véron Capacitary estimates of positive solutions of semilinear elliptic equations with absorption, J. Eur. Math. Soc., Volume 6 (2004), pp. 483-527

[13] B. Mselati Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation, Mem. Amer. Math. Soc., Volume 168 (2004)

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