The Martin kernel and infima of positive harmonic functions

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.