On Uniqueness of a Solution of Lu=u^a with Given Trace
Sergei E. Kuznetsov, University of Colorado at Boulder
Abstract
A boundary trace (G, m) of a solution of Delta u = u^a in a bounded smooth domain in R^d was first constructed by Le Gall who described all possible traces for a = 2, d= 2 in which case a solution is defined uniquely by its trace. In a number of publications, Marcus, Veron, Dynkin and Kuznetsov gave analytic and probabilistic generalization of the concept of trace to the case of arbitrary a > 1, d > 1. However, it was shown by Le Gall that the trace, in general, does not define a solution uniquely in case d >= (a +1)/(a -1). He offered a sufficient condition for the uniqueness and conjectured that a uniqueness should be valid if the singular part G of the trace coincides with the set of all explosion points of the measure m. Here, we establish a necessary condition for the uniqueness which implies a negative answer to the above
conjecture.
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