Boundary regularity in the Dirichlet problem for the invariant Laplacians $\Delta _\gamma$ on the unit real ball
HTML articles powered by AMS MathViewer
- by Congwen Liu and Lizhong Peng PDF
- Proc. Amer. Math. Soc. 132 (2004), 3259-3268 Request permission
Abstract:
We study the boundary regularity in the Dirichlet problem of the differential operators \begin{equation*} \Delta _{\gamma }= (1-|x|^2)\bigg \{ \frac {1-|x|^2}4 \sum _j \frac {\partial ^2} {\partial x_j^2} + \gamma \sum _j x_j \frac {\partial }{\partial x_j} + \gamma \Big (\frac n2 -1 -\gamma \Big )\bigg \}. \end{equation*} Our main result is: if $\gamma >-1/2$ is neither an integer nor a half-integer not less than $n/2-1$, one cannot expect global smooth solutions of $\Delta _\gamma u=0$; if $u\in C^{\infty }(\overline {B}_n)$ satisfies $\Delta _\gamma u=0$, then $u$ must be either a polynomial of degree at most $2\gamma +2-n$ or a polyharmonic function of degree $\gamma +1$.References
- Patrick Ahern, Joaquim Bruna, and Carme Cascante, $H^p$-theory for generalized $M$-harmonic functions in the unit ball, Indiana Univ. Math. J. 45 (1996), no. 1, 103–135. MR 1406686, DOI 10.1512/iumj.1996.45.1961
- Lars V. Ahlfors, Möbius transformations in several dimensions, Ordway Professorship Lectures in Mathematics, University of Minnesota, School of Mathematics, Minneapolis, Minn., 1981. MR 725161
- Ömer Akın and Heinz Leutwiler, On the invariance of the solutions of the Weinstein equation under Möbius transformations, Classical and modern potential theory and applications (Chateau de Bonas, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 430, Kluwer Acad. Publ., Dordrecht, 1994, pp. 19–29. MR 1321603
- Nachman Aronszajn, Thomas M. Creese, and Leonard J. Lipkin, Polyharmonic functions, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1983. Notes taken by Eberhard Gerlach; Oxford Science Publications. MR 745128
- Adbelhamid Boussejra and Ahmed Intissar, $L^2$-concrete spectral analysis of the invariant Laplacian $\Delta _{\alpha \beta }$ in the unit complex ball $B^n$, J. Funct. Anal. 160 (1998), no. 1, 115–140. MR 1658708, DOI 10.1006/jfan.1998.3318
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- G. B. Folland, Spherical harmonic expansion of the Poisson-Szegő kernel for the ball, Proc. Amer. Math. Soc. 47 (1975), 401–408. MR 370044, DOI 10.1090/S0002-9939-1975-0370044-2
- Daryl Geller, Some results in $H^{p}$ theory for the Heisenberg group, Duke Math. J. 47 (1980), no. 2, 365–390. MR 575902
- C. Robin Graham, The Dirichlet problem for the Bergman Laplacian. I, Comm. Partial Differential Equations 8 (1983), no. 5, 433–476. MR 695400, DOI 10.1080/03605308308820275
- C. Robin Graham and John M. Lee, Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains, Duke Math. J. 57 (1988), no. 3, 697–720. MR 975118, DOI 10.1215/S0012-7094-88-05731-6
- Steven G. Krantz, Partial differential equations and complex analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. Lecture notes prepared by Estela A. Gavosto and Marco M. Peloso. MR 1207812
- Heinz Leutwiler, Best constants in the Harnack inequality for the Weinstein equation, Aequationes Math. 34 (1987), no. 2-3, 304–315. MR 921108, DOI 10.1007/BF01830680
- Song-Ying Li and Ezequias Simon, Boundary behavior of harmonic functions in metrics of Bergman type on the polydisc, Amer. J. Math. 124 (2002), no. 5, 1045–1057. MR 1925342
- C. Liu and L. Peng, Berezin-type transforms associated with the Weinstein equation, in preparation.
- Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Additional Information
- Congwen Liu
- Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
- Address at time of publication: School of Mathematical Sciences, Nankai University, Tianjin 300071, People’s Republic of China
- Email: cwliu@math.pku.edu.cn
- Lizhong Peng
- Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
- Email: lzpeng@pku.edu.cn
- Received by editor(s): July 4, 2003
- Published electronically: June 17, 2004
- Additional Notes: This research was supported by 973 project of China grant G1999075105
- Communicated by: Mei-Chi Shaw
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3259-3268
- MSC (2000): Primary 35J25, 32W50; Secondary 35C10, 35C15
- DOI: https://doi.org/10.1090/S0002-9939-04-07582-3
- MathSciNet review: 2073300