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Comptes Rendus Mathématique
Volume 335, n° 9
pages 733-738 (novembre 2002)
Doi : S1631-073X(02)02557-8
Received : 11 September 2002 ;  accepted : 13 September 2002
Classification et représentation probabiliste des solutions positives d'une équation elliptique semi-linéaire
Classification and probabilistic representation for the positive solutions of a semilinear elliptic equation

Benoit Mselati
Département de mathématiques et applications, École normale supérieure, 45, rue d'Ulm, 75230 Paris cedex 05, France 

Note présentée par Haïm Brézis


Nous montrons que les solutions positives de Δu =u 2 dans un domaine lisse et borné de Rd sont uniquement caractérisées par leur trace fine au bord définie dans [6], répondant ainsi à une question ouverte importante de [2]. Une formule probabiliste faisant intervenir le serpent brownien et reliant une solution à sa trace fine est également obtenue. Nous prouvons en outre que toute solution est limite croissante de solutions majorées par des fonctions harmoniques dans D. Pour citer cet article : B. Mselati, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 733-738.

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We prove that a nonnegative solution of Δu =u 2 in a bounded and smooth domain in Rd is uniquely determined by its fine trace on the boundary as defined in [6], thus answering a major open question of [2]. A probabilistic formula for a solution in terms of its fine trace and of the Brownian snake is also provided. Moreover, we show that every solution is the increasing limit of solutions which are dominated by a harmonic function in D. To cite this article: B. Mselati, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 733-738.

The full text of this article is available in PDF format.

© 2002  Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. All Rights Reserved.
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