A new inequality for superdiffusions and its applications to nonlinear differential equations
Author:
E. B. Dynkin
Journal:
Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 6877
MSC (2000):
Primary 60H30; Secondary 35J60, 60J60
Published electronically:
August 2, 2004
Comment(s):
Additional information about this paper
MathSciNet review:
2075898
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Abstract: Our motivation is the following problem: to describe all positive solutions of a semilinear elliptic equation with in a bounded smooth domain . In 1998 Dynkin and Kuznetsov solved this problem for a class of solutions which they called moderate. The question if all solutions belong to this class remained open. In 2002 Mselati proved that this is true for the equation in a domain of class . His principal toolthe Brownian snakeis not applicable to the case . In 2003 Dynkin and Kuznetsov modified most of Mselati's arguments by using superdiffusions instead of the snake. However a critical gap remained. A new inequality established in the present paper allows us to close this gap.
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 , On upper bounds for positive solutions of semilinear equations, J. Functional Analysis 210 (2004), 73100. MR 2051633
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 S. E. Kuznetsov, An upper bound for positive solutions of the equation , Amer. Math. Soc., Electronic Research Announcements, to appear.
 [MV04]
 M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion, J. European Math. Soc., to appear.
 [Ms02]
 B. Mselati, Classification et représentation probabiliste des solutions positives de dans un domaine, Thése de Doctorat de l'Université Paris 6, 2002.
 [Ms04]
 B. Mselati, Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation, Memoirs of the American Mathematical Society 168 (2004), no. 798, to appear.
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Additional Information
E. B. Dynkin
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853
Email:
ebd1@cornell.edu
DOI:
http://dx.doi.org/10.1090/S1079676204001313
PII:
S 10796762(04)001313
Keywords:
Positive solutions of semilinear elliptic PDEs,
superdiffusions,
conditional diffusions,
$\mathbb{N}$measures
Received by editor(s):
April 23, 2004
Published electronically:
August 2, 2004
Additional Notes:
Partially supported by the National Science Foundation Grant DMS0204237
Communicated by:
Mark Freidlin
Article copyright:
© Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
