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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), 68-77
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 | References | Similar Articles | Additional Information

Abstract: Our motivation is the following problem: to describe all positive solutions of a semilinear elliptic equation $L u=u^\alpha$ with $\alpha>1$ in a bounded smooth domain $E\subset \mathbb{R} ^d$. In 1998 Dynkin and Kuznetsov solved this problem for a class of solutions which they called $\sigma$-moderate. The question if all solutions belong to this class remained open. In 2002 Mselati proved that this is true for the equation $\Delta u=u^2$ in a domain of class $C^4$. His principal tool--the Brownian snake--is not applicable to the case $\alpha\neq 2$. 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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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/S1079-6762-04-00131-3
PII: S 1079-6762(04)00131-3
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 DMS-0204237
Communicated by: Mark Freidlin
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.