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


Author: Ross G. Pinsky
Journal: Trans. Amer. Math. Soc. 352 (2000), 2445-2477
MSC (1991): Primary 35J05, 31C35, 31B05, 60J50
DOI: https://doi.org/10.1090/S0002-9947-00-02594-0
Published electronically: February 24, 2000
MathSciNet review: 1709778
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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

\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 $l>2$.

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 $d=2$, $\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.


References [Enhancements On Off] (What's this?)

  • 1. Ioffe, D. and Pinsky, R., Positive harmonic functions vanishing on the boundary for the Laplacian in unbounded horn-shaped domains, Trans. Amer. Math. Soc. 342 (1994), 773-791. MR 94h:60114
  • 2. Pinsky, R., A new approach to the Martin boundary via diffusions conditioned to hit a compact set, Annals of Probab. 21 (1993), 453-481. MR 94f:60098
  • 3. Pinsky, R., Positive Harmonic Functions and Diffusion, Cambridge Univ. Press, 1995. MR 96m:60179

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Additional Information

Ross G. Pinsky
Affiliation: Department of Mathematics, Technion-Israel Institute of Mathematics, Haifa 32000, Israel
Email: pinsky@techunix.technion.ac.il

DOI: https://doi.org/10.1090/S0002-9947-00-02594-0
Keywords: Positive harmonic functions, Martin boundary, Dirichlet boundary condition, Neumann boundary condition, harmonic measure
Received by editor(s): January 4, 1999
Published electronically: February 24, 2000
Additional Notes: This research was done while the author was on sabbatical at the Courant Institute of Mathematical Sciences
Article copyright: © Copyright 2000 American Mathematical Society

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