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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Boundary regularity in the Dirichlet problem for the invariant Laplacians $\Delta_\gamma$ on the unit real ball


Authors: Congwen Liu and Lizhong Peng
Journal: Proc. Amer. Math. Soc. 132 (2004), 3259-3268
MSC (2000): Primary 35J25, 32W50; Secondary 35C10, 35C15
Published electronically: June 17, 2004
MathSciNet review: 2073300
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Abstract: We study the boundary regularity in the Dirichlet problem of the differential operators

\begin{displaymath}\Delta_{\gamma}= (1-\vert x\vert^2)\bigg\{ \frac {1-\vert x\v... ...}{\partial x_j} + \gamma\Big(\frac n2 -1 -\gamma \Big)\bigg\}. \end{displaymath}

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 $\gamma+1$.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07582-3
PII: S 0002-9939(04)07582-3
Keywords: Invariant Laplacians, Laplace-Beltrami operator, Weinstein equation, boundary regularity, polyharmonicity
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
Article copyright: © Copyright 2004 American Mathematical Society