# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## A probabilistic approach to positive harmonic functions in a slab with alternating Dirichlet and Neumann boundary conditionsHTML articles powered by AMS MathViewer

by Ross G. Pinsky
Trans. Amer. Math. Soc. 352 (2000), 2445-2477 Request permission

## Abstract:

Let $\Omega = R^{d}\times (-1,1)$, $d\ge 2$, be a $d+1$ dimensional slab. Denote points $z\in R^{d+1}$ by $z=(r,\theta ,y)$, where $(r,\theta )\in [0,\infty )\times S^{d-1}$ and $y\in R$. Denoting the boundary of the slab by $\Gamma =\partial \Omega$, let $\Gamma _{D}=\{z=(r,\theta ,y)\in \Gamma : r\in \bigcup _{n=1}^{\infty }(a_{n},b_{n})\},$ where $\{(a_{n},b_{n})\}_{n=1}^{\infty }$ is an ordered sequence of intervals on the right half line (that is, $a_{n+1}>b_{n}$). Assume that the lengths of the intervals are bounded and that the spaces between consecutive intervals are bounded and bounded away from zero. Let $\Gamma _{N}=\Gamma -\bar \Gamma _{D}$. Let $C_{B}(\Omega ;\Gamma _{D}, \Gamma _{N})$ and $C_{P}(\Omega ; \Gamma _{D}, \Gamma _{N})$ denote respectively the cone of bounded, positive harmonic functions in $\Omega$ and the cone of positive harmonic functions in $\Omega$ which satisfy the Dirichlet boundary condition on $\Gamma _{D}$ and the Neumann boundary condition on $\Gamma _{N}$. Letting $\rho _{n}\equiv b_{n}-a_{n}$, the main result of this paper, under a modest assumption on the sequence $\{\rho _{n}\}$, may be summarized as follows when $d\ge 3$:

1. If $\sum _{n=1}^{\infty }\frac {n}{|\log \rho _{n}|} <\infty$, then $\mathcal C_B(\Omega ,\Gamma _D,\Gamma _N)$ and $\mathcal C_P(\Omega ,\Gamma _D,\Gamma _N)$ are both one-dimensional (as in the case of the Neumann boundary condition on the entire boundary). In particular, this occurs if $\rho _{n}=\exp (-n^{l})$ with $l>2$.

2. If $\sum _{n=1}^{\infty }\frac {n}{|\log \rho _{n}|} =\infty$ and $\sum _{n=1}^{\infty }\frac {|\log \rho _{n}|^{\frac {1}{2}}}{n^{2}}=\infty$, then $\mathcal C_B(\Omega ,\Gamma _D,\Gamma _N) =\varnothing$ and $\mathcal C_P(\Omega ,\Gamma _D,\Gamma _N)$ is one-dimensional. In particular, this occurs if $\rho _{n}=\exp (-n^{2})$.

3. If $\sum _{n=1}^{\infty }\frac {|\log \rho _{n}|^{\frac {1}{2}}}{n^{2}}<\infty$, then $\mathcal C_B(\Omega ,\Gamma _D,\Gamma _N)=\varnothing$ and the set of minimal elements generating $\mathcal C_P(\Omega ,\Gamma _D,\Gamma _N)$ is isomorphic to $S^{d-1}$ (as in the case of the Dirichlet boundary condition on the entire boundary). In particular, this occurs if $\rho _{n}=\exp (-n^{l})$ with $0\le l<2$. When $d=2$, $\mathcal C_B(\Omega ,\Gamma _D,\Gamma _N)=\varnothing$ as soon as there is at least one interval of Dirichlet boundary condition. The dichotomy for $\mathcal C_P(\Omega ,\Gamma _D,\Gamma _N)$ is as above.

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