Positive harmonic functions vanishing on the boundary for the Laplacian in unbounded horn-shaped domains

Authors:
Dimitry Ioffe and Ross Pinsky

Journal:
Trans. Amer. Math. Soc. **342** (1994), 773-791

MSC:
Primary 60J50; Secondary 31C35

DOI:
https://doi.org/10.1090/S0002-9947-1994-1211410-9

MathSciNet review:
1211410

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Abstract: Denote points , , by , where , , and . Let be a nondecreasing -function and define the "horn-shaped" domain and its unit "cylinder" . Under appropriate regularity conditions on *a*, we prove the following theorem: (i) If , then the Martin boundary at infinity for in is a single point, (ii) If , then the Martin boundary at infinity for in is homeomorphic to . More specifically, a sequence satisfying is a Martin sequence if and only if exists on . From (i), it follows that the cone of positive harmonic functions in vanishing continuously on is one-dimensional. From (ii), it follows easily that the cone of positive harmonic functions on vanishing continuously on is generated by a collection of minimal elements which is homeomorphic to .

In particular, the above result solves a problem stated by Kesten, who asked what the Martin boundary is for in in the case , . Our method of proof involves an analysis as of the exit distribution on for Brownian motion starting from and conditioned to hit *D* before exiting .

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DOI:
https://doi.org/10.1090/S0002-9947-1994-1211410-9

Article copyright:
© Copyright 1994
American Mathematical Society