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

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- by Ross G. Pinsky PDF
- 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.

## References

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*Positive harmonic functions and diffusion*, Cambridge Studies in Advanced Mathematics, vol. 45, Cambridge University Press, Cambridge, 1995. MR**1326606**, DOI 10.1017/CBO9780511526244

## Additional Information

**Ross G. Pinsky**- Affiliation: Department of Mathematics, Technion-Israel Institute of Mathematics, Haifa 32000, Israel
- Email: pinsky@techunix.technion.ac.il
- 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
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1709778