We prove that a nonnegative solution of Δu=u2 in a bounded and smooth domain in 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.
Nous montrons que les solutions positives de Δu=u2 dans un domaine lisse et borné de 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.
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Benoit Mselati 1
@article{CRMATH_2002__335_9_733_0, author = {Benoit Mselati}, title = {Classification et repr\'esentation probabiliste des solutions positives d'une \'equation elliptique semi-lin\'eaire}, journal = {Comptes Rendus. Math\'ematique}, pages = {733--738}, publisher = {Elsevier}, volume = {335}, number = {9}, year = {2002}, doi = {10.1016/S1631-073X(02)02557-8}, language = {fr}, }
TY - JOUR AU - Benoit Mselati TI - Classification et représentation probabiliste des solutions positives d'une équation elliptique semi-linéaire JO - Comptes Rendus. Mathématique PY - 2002 SP - 733 EP - 738 VL - 335 IS - 9 PB - Elsevier DO - 10.1016/S1631-073X(02)02557-8 LA - fr ID - CRMATH_2002__335_9_733_0 ER -
Benoit Mselati. Classification et représentation probabiliste des solutions positives d'une équation elliptique semi-linéaire. Comptes Rendus. Mathématique, Volume 335 (2002) no. 9, pp. 733-738. doi : 10.1016/S1631-073X(02)02557-8. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02557-8/
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