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

 

Solutions of nonlinear differential equations
on a Riemannian manifold
and their trace on the Martin boundary


Authors: E. B. Dynkin and S. E. Kuznetsov
Journal: Trans. Amer. Math. Soc. 350 (1998), 4521-4552
MSC (1991): Primary 60J60, 35J60; Secondary 60J80, 60J45, 35J65
MathSciNet review: 1443191
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Abstract: Let $L$ be a second order elliptic differential operator on a Riemannian manifold $E$ with no zero order terms. We say that a function $h$ is $L$-harmonic if $Lh=0$. Every positive $L$-harmonic function has a unique representation

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

where $k$ is the Martin kernel, $E'$ is the Martin boundary and $\nu $ is a finite measure on $E'$ concentrated on the minimal part $E^{*}$ of $E'$. We call $\nu $ the trace of $h$ on $E'$. Our objective is to investigate positive solutions of a nonlinear equation

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

for $1<\alpha \le 2$ [the restriction $\alpha \le 2$ is imposed because our main tool is the $(L,\alpha )$-superdiffusion, which is not defined for $\alpha >2$]. We associate with every solution $u$ of (*) a pair $(\Gamma ,\nu )$, where $\Gamma $ is a closed subset of $E'$ and $\nu $ is a Radon measure on $O=E'\setminus \Gamma $. We call $(\Gamma ,\nu )$ the trace of $u$ on $E'$. $\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. In an earlier paper, we investigated the case when $L$ is a second order elliptic differential operator in $\mathbb{R}^{d}$ and $E$ is a bounded smooth domain in $\mathbb{R}^{d}$. We obtained necessary and sufficient conditions for a pair $(\Gamma ,\nu )$ to be a trace, and we gave a probabilistic formula for the maximal solution with a given trace. The general theory developed in the present paper is applicable, in particular, to elliptic operators $L$ with bounded coefficients in an arbitrary bounded domain of $\mathbb{R}^{d}$, assuming only that the Martin boundary and the geometric boundary coincide.


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Additional Information

E. B. Dynkin
Affiliation: Department of Mathematics, White Hall, Cornell University, Ithaca, New York 14853-7901
Email: ebd1@cornell.edu

S. E. Kuznetsov
Affiliation: Central Econ.-Math. Institute, Russian Academy of Sciences, 32 Krasikowa, Moscow 117418, Russia
Address at time of publication: Department of Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309-0395
Email: sk47@cornell.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02006-6
PII: S 0002-9947(98)02006-6
Received by editor(s): August 6, 1996
Additional Notes: Partially supported by National Science Foundation Grant DMS-9301315
Article copyright: © Copyright 1998 American Mathematical Society