Boundary regularity in the Dirichlet problem for the invariant Laplacians Δᵧ on the unit real ball

Liu, Congwen; Peng, Lizhong

doi:10.1090/S0002-9939-04-07582-3

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
DOI: https://doi.org/10.1090/S0002-9939-04-07582-3
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

, 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

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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