Capacitary estimates of solutions of a class of nonlinear elliptic equations

Moshe Marcus; Laurent Véron

doi:10.1016/S1631-073X(03)00217-6

Partial Differential Equations

Capacitary estimates of solutions of a class of nonlinear elliptic equations
[Estimations capacitaires des solutions d'une classe d'équations elliptiques non linéaires]

Présenté par : Haïm Brezis

Moshe Marcus ¹ ; Laurent Véron ²

¹ Department of Mathematics, Israel Institute of Technology-Technion, 32000 Haifa, Israel
² Département de mathématiques, Faculté des sciences et techniques, Université de Tours, 37200 Tours, France

Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 913-918.

Résumé
Abstract

Soit $Ω$ un domaine borné régulier de $ℝ^{N}$ and K un sous-ensemble compact de $\partial Ω$ . Supposons q⩾(N+1)/(N−1) et soit U_K la solution maximale de $(ℰ) - Δ u + u^{q} = 0$ dans $Ω$ qui s'annulle sur $\partial Ω ⧹ K$ . Nous obtenons des majorations et minorations précises de U_K au moyen de la capacité de Bessel C_2/q,q′ et montrons que U_K est σ-modérée. En outre nous corrélons les points d'explosion forte de U_K et les points épais de K pour la topologie fine associée à C_2/q,q′ et caractérisons ces points par une condition d'intégrale de chemin portant sur U_K.

Let $Ω$ be a smooth bounded domain in $ℝ^{N}$ and K a compact subset of $\partial Ω$ . Assume that q⩾(N+1)/(N−1) and denote by U_K the maximal solution of −Δu+u^q=0 in $Ω$ which vanishes on $\partial Ω ⧹ K$ . We obtain sharp upper and lower estimates for U_K in terms of the Bessel capacity C_2/q,q′ and prove that U_K is σ-moderate. In addition we relate the strong ‘blow-up’ points of U_K on $\partial Ω$ to the ‘thick’ points of K in the fine topology associated with C_2/q,q′ and characterize these points by a path integral condition on U_K.

Reçu le : 2003-04-02
Accepté le : 2003-04-02
Publié le : 2003-06-01

DOI : 10.1016/S1631-073X(03)00217-6

Affiliations des auteurs :

Moshe Marcus ¹ ; Laurent Véron ²

¹ Department of Mathematics, Israel Institute of Technology-Technion, 32000 Haifa, Israel
² Département de mathématiques, Faculté des sciences et techniques, Université de Tours, 37200 Tours, France

@article{CRMATH_2003__336_11_913_0,
     author = {Moshe Marcus and Laurent V\'eron},
     title = {Capacitary estimates of solutions of a class of nonlinear elliptic equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {913--918},
     publisher = {Elsevier},
     volume = {336},
     number = {11},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00217-6},
     language = {en},
}

TY  - JOUR
AU  - Moshe Marcus
AU  - Laurent Véron
TI  - Capacitary estimates of solutions of a class of nonlinear elliptic equations
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 913
EP  - 918
VL  - 336
IS  - 11
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00217-6
LA  - en
ID  - CRMATH_2003__336_11_913_0
ER  -

%0 Journal Article
%A Moshe Marcus
%A Laurent Véron
%T Capacitary estimates of solutions of a class of nonlinear elliptic equations
%J Comptes Rendus. Mathématique
%D 2003
%P 913-918
%V 336
%N 11
%I Elsevier
%R 10.1016/S1631-073X(03)00217-6
%G en
%F CRMATH_2003__336_11_913_0

Moshe Marcus; Laurent Véron. Capacitary estimates of solutions of a class of nonlinear elliptic equations. Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 913-918. doi : 10.1016/S1631-073X(03)00217-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00217-6/

Bibliographie
Cité par

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

[2] 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

[3] E.B. Dynkin; S.E. Kuznetsov Solutions of Lu=u^α dominated by harmonic functions, J. Analyse Math., Volume 68 (1996), pp. 15-37

[4] D.A. Labutin, Wiener regularity for large solutions of nonlinear equations, Arch. Math., à paraı̂tre

[5] J.F. Legall The Brownian snake and solutions of Δu=u² in a domain, Probab. Theory Related Fields, Volume 102 (1995), pp. 393-432

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

[7] 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

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

[9] B. Mselati, Classification et représentation probabiliste des solutions positives de Δu=u² dans un domaine, Thèse de Doctorat, Université Paris 6, 2002

Cité par Sources :

^☆ This research was supported by RTN contract No. HPRN-CT-2002-00274.

Commentaires - Politique