Martin boundary in potential theory

The ideal boundary of a Green space $\Omega$( see also Boundary (in the theory of uniform algebras)), which allows one to construct the characteristic representation of positive harmonic functions in $\Omega$. Let $\Omega$ be a locally compact, non-compact, topological space, and let $\Phi$ be a family of continuous functions $f : \Omega \rightarrow [ - \infty , + \infty ]$. The Constantinescu–Cornea theorem [2] asserts that, up to a homeomorphism, there is a unique compact space $\widehat \Omega$ with the following properties: 1) $\Omega$ is an everywhere-dense subspace of $\widehat \Omega$; 2) each $f \in \Phi$ extends continuously to a function $\widehat{f}$ on $\widehat \Omega$, separating points on the ideal boundary $\Delta = \widehat \Omega \setminus \Omega$ of $\Omega$ relative to $\Phi$; and 3) $\Omega$ is an open set in $\widehat \Omega$.

Now, let $\Omega$ be a bounded domain in a Euclidean space $\mathbf R ^ {n}$, $n \geq 2$, or, more generally, a Green space; let $G = G ( x , y )$ be the Green function on $\Omega$ with pole $y \in \Omega$ and let $y _ {0} \in \Omega$ be fixed. The Martin space or Martin compactification $\widehat \Omega$ of $\Omega$ is obtained via the Constantinescu–Cornea theorem by taking for $\Phi$ the family

$$\Phi = \ \left \{ { x \in \Omega \rightarrow K ( x , y ) = \frac{G ( x , y ) }{G ( x , y _ {0} ) } } : { y \in \Omega } \right \} ,$$

where, by definition, $K ( x _ {0} , y _ {0} ) = 1$. The Martin boundary is the corresponding ideal boundary $\Delta = \widehat \Omega \setminus \Omega$. The Martin topology $T$ is the topology on the Martin space $\widehat \Omega$. Two Martin spaces $\widehat \Omega {} ^ \prime$, $\widehat \Omega {} ^ {\prime\prime}$ corresponding to different points $y _ {0} ^ \prime , y _ {0} ^ {\prime\prime} \in \Omega$ are homeomorphic. The function $\widehat{K} ( \xi , y) : \Delta \times \Omega \rightarrow [ 0 , + \infty ]$, the extension of $K ( x , y )$, is harmonic in $y$ and jointly continuous in the variables $( \xi , y)$; $\widehat \Omega$ is a metrizable space. Martin's fundamental theorem [1] asserts: The class of all positive harmonic functions $u ( y) \geq 0$ on $\Omega$ is characterized by the Martin representation:

$$\tag{* } u ( y) = \int\limits K ( \xi , y ) d \mu ( \xi ) ,$$

where $\mu$ is a positive Radon measure on $\Delta$. The measure $\mu$ in (*) is not uniquely determined by the function $u$. A harmonic function $v \geq 0$ is called minimal in $\Omega$ if each harmonic function $w$ such that $0 \leq w \leq v$ in $\Omega$ is proportional to $v$. Minimal harmonic functions $v \neq 0$ are proportional to $\widehat{K} ( \xi , y)$, the corresponding points $\xi \in \Delta$ are called minimal, and the set of minimal points $\Delta _ {1} \subset \Delta$ is called the minimal Martin boundary. If one poses the additional condition that $\mu$ in (*) be concentrated on $\Delta _ {1}$, one obtains the canonical Martin representation:

$$u ( y) = \int\limits \widehat{K} ( \xi , y ) d \mu _ {1} ( \xi ) ,$$

in which the measure $\mu _ {1} \geq 0$ is uniquely determined by $u$.

Examples. a) If $\Omega = \{ {x \in \mathbf R ^ {n} } : {| x | < R } \}$ is a ball of radius $R$ in $\mathbf R ^ {n}$, $n \geq 2$, then

$$\widehat{K} ( \xi , y ) = \ \frac{R ^ {n-2} ( R ^ {2} - | y | ^ {2} ) }{| \xi - y | ^ {n} }$$

is the Poisson kernel, $\widehat \Omega$ is the Euclidean closure $\widehat \Omega = \overline \Omega \;$, the Martin boundary $\Delta$ is the sphere $\{ {\xi \in \mathbf R ^ {n} } : {| \xi | = R } \}$, all points of which are minimal. The representation (*) in this case reduces to the Poisson–Herglotz formula (see Integral representation of an analytic function; Poisson integral).

b) The Martin boundary $\Delta$ coincides with the Euclidean boundary $\Gamma = \overline \Omega \; \setminus \Omega$ whenever $\Gamma$ is a sufficiently smooth hypersurface in $\mathbf R ^ {n}$, $n \geq 2$.

c) If $\Omega$ is a simply-connected domain in the plane, then the Martin boundary $\Delta$ coincides with the set of limit elements, or Carathéodory prime ends. Thus, an element of the Martin boundary $\xi \in \Delta$ can be considered as a generalization of the notion of a prime end to dimension $n \geq 2$.

References

 [1] R.S. Martin, "Minimal positive harmonic functions" Trans. Amer. Math. Soc. , 49 (1941) pp. 137–172 [2] C. Constantinescu, A. Cornea, "Ideale Ränder Riemannscher Flächen" , Springer pp. 1963 [3] M. Brelot, "On topologies and boundaries in potential theory" , Springer (1971)