A probabilistic approach to positive harmonic functions in a slab with alternating Dirichlet and Neumann boundary conditions期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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A probabilistic approach to positive harmonic functions in a slab with alternating Dirichlet and Neumann boundary conditions

Authors:	Ross G Pinsky

Institution:	Department of Mathematics, Technion-Israel Institute of Mathematics, Haifa 32000, Israel

Abstract:	Let $\Omega = R^{d}\times (-1,1)$ , $d\ge 2$ , be a 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 $\begin{displaymath}\Gamma _{D}=\{z=(r,\theta ,y)\in \Gamma : r\in \bigcup _{n=1}^{\infty }(a_{n},b_{n})\},\end{displaymath}$ 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 $\sum _{n=1}^{\infty }\frac{n}{\vert\log \rho _{n}\vert} <\infty$ , 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 . 2. If $\sum _{n=1}^{\infty }\frac{n}{\vert\log \rho _{n}\vert} =\infty$ and $\sum _{n=1}^{\infty }\frac{\vert\log \rho _{n}\vert^{\frac{1}{2}}}{n^{2}}=\infty$ , 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 $\sum _{n=1}^{\infty }\frac{\vert\log \rho _{n}\vert^{\frac{1}{2}}}{n^{2}}<\infty$ , 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 , $\mathcal C_B(\Omega ,\Gamma _D,\Gamma _N)=\varnothing$ 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.

Keywords:	Positive harmonic functions Martin boundary Dirichlet boundary condition Neumann boundary condition harmonic measure

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