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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Trace on the boundary for solutions of nonlinear differential equations


Authors: E. B. Dynkin and S. E. Kuznetsov
Journal: Trans. Amer. Math. Soc. 350 (1998), 4499-4519
MSC (1991): Primary 60J60, 35J60; Secondary 60J80, 60J45, 35J65
MathSciNet review: 1422602
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Abstract: Let $L$ be a second order elliptic differential operator in $\mathbb{R}^{d}$ with no zero order terms and let $E$ be a bounded domain in $\mathbb{R}^{d}$ with smooth boundary $\partial E$. We say that a function $h$ is $L$-harmonic if $Lh=0$ in $E$. Every positive $L$-harmonic function has a unique representation

\begin{equation*}h(x)=\int _{\partial E} k(x,y) \nu (dy), \end{equation*}

where $k$ is the Poisson kernel for $L$ and $\nu $ is a finite measure on $\partial E$. We call $\nu $ the trace of $h$ on $\partial E$. Our objective is to investigate positive solutions of a nonlinear equation

\begin{equation*}L u=u^{\alpha }\quad \text{in } E \end{equation*}

for $1<\alpha \le 2$ [the restriction $\alpha \le 2$ is imposed because our main tool is the $\alpha $-superdiffusion which is not defined for $\alpha >2$]. We associate with every solution $u$ a pair $(\Gamma ,\nu )$, where $\Gamma $ is a closed subset of $\partial E$ and $\nu $ is a Radon measure on $O=\partial E\setminus \Gamma $. We call $(\Gamma ,\nu )$ the trace of $u$ on $\partial E$. $\Gamma $ is empty if and only if $u$ is dominated by an $L$-harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. We establish necessary and sufficient conditions for a pair $(\Gamma ,\nu )$ to be a trace, and we give a probabilistic formula for the maximal solution with a given trace.


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Additional Information

E. B. Dynkin
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-7901; Department of Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309-0395
Email: ebd1@cornell.edu

S. E. Kuznetsov
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-7901
Email: sk47@cornell.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01952-7
PII: S 0002-9947(98)01952-7
Additional Notes: Partially supported by National Science Foundation Grant DMS-9301315.
Article copyright: © Copyright 1998 American Mathematical Society