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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Martin kernel and infima of positive harmonic functions


Author: Zoran Vondraček
Journal: Trans. Amer. Math. Soc. 335 (1993), 547-557
MSC: Primary 31C35; Secondary 31B05, 35B05, 35J15
DOI: https://doi.org/10.1090/S0002-9947-1993-1104202-1
MathSciNet review: 1104202
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Abstract: Let $D$ be a bounded Lipschitz domain in ${{\mathbf {R}}^n}$ and let $K(x,z)$, $x \in D$, $z \in \partial D$, be the Martin kernel based at ${x_0} \in D$. For $x,y \in D$, let $k(x,y) = \inf \{ h(x):h\;\text {positive}\;\text {harmonic}\;\text {in}\; D, h(y) = 1\}$. We show that the function $k$ completely determines the family of positive harmonic functions on $D$. Precisely, for every $z \in \partial D$, ${\lim _{y \to z}}k(x,y)/k({x_0},y) = K(x,z)$. The same result is true for second-order uniformly elliptic operators and Schrödinger operators.


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Keywords: Positive harmonic functions, Martin kernel
Article copyright: © Copyright 1993 American Mathematical Society