On the asymptotic formula for the solution of degenerate elliptic partial differential equations

Wei, Dong-Huang

doi:10.1090/S0002-9939-2011-10780-9

On the asymptotic formula for the solution of degenerate elliptic partial differential equations
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Proc. Amer. Math. Soc. 139 (2011), 2863-2875 Request permission

Abstract:

This paper gives an asymptotic expansion for the solution of the boundary problem with respect to the Laplace-Beltrami operator. We also consider some examples in the domain whose boundary is real ellipsoid, where the boundary problem does not have a $C^n$ solution up to the boundary.

References

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
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, DOI 10.1353/ajm.2002.0028
Song-Ying Li and DongHuan Wei, On the rigidity theorem for harmonic functions in Kähler metric of Bergman type, Sci. China Math. 53 (2010), no. 3, 779–790. MR 2608333, DOI 10.1007/s11425-010-0040-8
Manfred Stoll, Invariant potential theory in the unit ball of $\textbf {C}^n$, London Mathematical Society Lecture Note Series, vol. 199, Cambridge University Press, Cambridge, 1994. MR 1297545, DOI 10.1017/CBO9780511526183
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Der-Chen Chang and Song-Ying Li, A zeta function associated to the sub-Laplacian on the unit sphere in $\Bbb C^N$, J. Anal. Math. 86 (2002), 25–48. MR 1894476, DOI 10.1007/BF02786643