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2000 On Uniqueness of a Solution of $Lu=u^\alpha$ with Given Trace
Sergei Kuznetsov
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Electron. Commun. Probab. 5: 137-147 (2000). DOI: 10.1214/ECP.v5-1027


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 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, Véron, Dynkin and Kuznetsov gave analytic and probabilistic generalization of the concept of trace to the case of arbitrary $\alpha \gt 1, d \ge 1$. 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.


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Sergei Kuznetsov. "On Uniqueness of a Solution of $Lu=u^\alpha$ with Given Trace." Electron. Commun. Probab. 5 137 - 147, 2000.


Accepted: 7 May 2000; Published: 2000
First available in Project Euclid: 2 March 2016

zbMATH: 0954.35072
MathSciNet: MR1781847
Digital Object Identifier: 10.1214/ECP.v5-1027

Primary: 35J67
Secondary: 35J75, 60H30, 60J50, 60J60, 60J85


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