Positive harmonic functions vanishing on the boundary for the Laplacian in unbounded horn-shaped domains
HTML articles powered by AMS MathViewer
- by Dimitry Ioffe and Ross Pinsky PDF
- Trans. Amer. Math. Soc. 342 (1994), 773-791 Request permission
Abstract:
Denote points $\bar x \in {R^{d + 1}}$, $d \geq 2$, by $\bar x = (\rho ,\theta ,z)$, where $\rho > 0$, $\theta \in {S^{d - 1}}$, and $z \in R$. Let $a:[0,\infty ) \to (0,\infty )$ be a nondecreasing ${C^2}$-function and define the "horn-shaped" domain $\Omega = \{ \bar x = (\rho ,\theta ,z):|z| < a(\rho )\}$ and its unit "cylinder" $D = \{ \bar x = (\rho ,\theta ,z) \in \Omega :\rho < 1\}$. Under appropriate regularity conditions on a, we prove the following theorem: (i) If ${\smallint ^\infty }a(\rho )/{\rho ^2}d\rho = \infty$, then the Martin boundary at infinity for $\frac {1}{2}\Delta$ in $\Omega$ is a single point, (ii) If ${\smallint ^\infty }a(\rho )/{\rho ^2}d\rho < \infty$, then the Martin boundary at infinity for $\frac {1}{2}\Delta$ in $\Omega$ is homeomorphic to ${S^{d - 1}}$. More specifically, a sequence $\{ ({\rho _n},{\theta _n},{z_n})\} _{n = 1}^\infty \subset \Omega$ satisfying ${\lim _{n \to \infty }}{\rho _n} = \infty$ is a Martin sequence if and only if ${\lim _{n \to \infty }}{\theta _n}$ exists on ${S^{d - 1}}$. From (i), it follows that the cone of positive harmonic functions in $\Omega$ vanishing continuously on $\partial \Omega$ is one-dimensional. From (ii), it follows easily that the cone of positive harmonic functions on $\Omega$ vanishing continuously on $\partial \Omega$ is generated by a collection of minimal elements which is homeomorphic to ${S^{d - 1}}$. In particular, the above result solves a problem stated by Kesten, who asked what the Martin boundary is for $\frac {1}{2}\Delta$ in $\Omega$ in the case $a(\rho ) = 1 + {\rho ^\gamma }$, $0 < \gamma < 1$. Our method of proof involves an analysis as $\rho \to \infty$ of the exit distribution on $\partial D$ for Brownian motion starting from $(\rho ,\theta ,z) \in \Omega$ and conditioned to hit D before exiting $\Omega$.References
- Alano Ancona, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 4, 169–213, x (French, with English summary). MR 513885
- J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431–458. MR 109961
- Richard A. Hunt and Richard L. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc. 147 (1970), 507–527. MR 274787, DOI 10.1090/S0002-9947-1970-0274787-0
- Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252
- Harry Kesten, Positive harmonic functions with zero boundary values, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 349–352. MR 545274
- Ross Pinsky, A probabilistic approach to a theorem of Gilbarg and Serrin, Israel J. Math. 74 (1991), no. 1, 1–12. MR 1135225, DOI 10.1007/BF02777812
- 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, DOI 10.1017/CBO9780511526244
- Hiroaki Aikawa, On the Martin boundary of Lipschitz strips, J. Math. Soc. Japan 38 (1986), no. 3, 527–541. MR 845719, DOI 10.2969/jmsj/03830527
Additional Information
- © Copyright 1994 American Mathematical Society
- 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