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On Uniqueness of a Solution of $Lu=u^\alpha$ with Given Trace

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1.	Title	Title of document	On Uniqueness of a Solution of $Lu=u^\alpha$ with Given Trace
2.	Creator	Author's name, affiliation, country	Sergei E. Kuznetsov; University of Colorado at Boulder
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	superdiffusion, moderate solutions, sigma-moderate solutions, stochastic boundary values, trace of a solution, explosion points.
3.	Subject	Subject classification	35J67, 35J75, 60J50, 60J60, 60J85, 60H30
4.	Description	Abstract	A boundary trace $(\Gamma, \nu)$ of a solution of $\Delta u = u^\alpha$ in a bounded smooth domain in $\mathbb{R}^d$ was first constructed by Le Gall \cite{LGOne} who described all possible traces for $\alpha = 2, d= 2$ in which case a solution is defined uniquely by its trace. In a number of publications, Marcus, V\'eron, Dynkin and Kuznetsov gave analytic and probabilistic generalization of the concept of trace to the case of arbitrary $\alpha > 1, d \ge 1$. However, it was shown by Le Gall  that the trace, in general, does not define a solution uniquely in case $d\ge (\alpha +1)/(\alpha -1)$. He offered a sufficient condition for the uniqueness and conjectured that a uniqueness should be valid if the singular part $\Gamma$ of the trace coincides with the set of all explosion points of the measure $\nu$. Here, we establish a necessary condition for the uniqueness which implies a negative answer to the above conjecture.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2000-05-07
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/1027
10.	Identifier	Digital Object Identifier	10.1214/ECP.v5-1027
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 5
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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