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Transactions of the American Mathematical Society

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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
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\;$positive$ \;$harmonic$ \;$in$ \; 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

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