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

Published electronically:
February 24, 2000

MathSciNet review:
1709778

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Abstract | References | Similar Articles | Additional Information

Abstract: Let , , be a dimensional slab. Denote points by , where and . Denoting the boundary of the slab by , let

where is an ordered sequence of intervals on the right half line (that is, ). Assume that the lengths of the intervals are bounded and that the spaces between consecutive intervals are bounded and bounded away from zero. Let . Let and denote respectively the cone of bounded, positive harmonic functions in and the cone of positive harmonic functions in which satisfy the Dirichlet boundary condition on and the Neumann boundary condition on .

Letting , the main result of this paper, under a modest assumption on the sequence , may be summarized as follows when :

1. If , then and are both one-dimensional (as in the case of the Neumann boundary condition on the entire boundary). In particular, this occurs if with .

2. If and , then and is one-dimensional. In particular, this occurs if .

3. If , then and the set of minimal elements generating is isomorphic to (as in the case of the Dirichlet boundary condition on the entire boundary). In particular, this occurs if with .

When , as soon as there is at least one interval of Dirichlet boundary condition. The dichotomy for is as above.

**1.**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**, 10.1090/S0002-9947-1994-1211410-9**2.**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****3.**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

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