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 | References | Similar Articles | Additional Information

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.

- Dimitry Ioffe and Ross Pinsky,
*Positive harmonic functions vanishing on the boundary for the Laplacian in unbounded horn-shaped domains*, Trans. Amer. Math. Soc.**342**(1994), no. 2, 773–791. MR**1211410**, DOI https://doi.org/10.1090/S0002-9947-1994-1211410-9 - Ross G. Pinsky,
*A new approach to the Martin boundary via diffusions conditioned to hit a compact set*, Ann. Probab.**21**(1993), no. 1, 453–481. MR**1207233** - Ross G. Pinsky,
*Positive harmonic functions and diffusion*, Cambridge Studies in Advanced Mathematics, vol. 45, Cambridge University Press, Cambridge, 1995. MR**1326606**

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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

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