Green’s matrices of second order elliptic systems with measurable coefficients in two dimensional domains

Dong, Hongjie; Kim, Seick

doi:10.1090/S0002-9947-09-04805-3

Green’s matrices of second order elliptic systems with measurable coefficients in two dimensional domains
HTML articles powered by AMS MathViewer

by Hongjie Dong and Seick Kim PDF

Trans. Amer. Math. Soc. 361 (2009), 3303-3323 Request permission

Abstract:

We study Green’s matrices for divergence form, second order strongly elliptic systems with bounded measurable coefficients in two dimensional domains. We establish existence, uniqueness, and pointwise estimates of Green’s matrices.

References

Pascal Auscher, Alan McIntosh, and Philippe Tchamitchian, Heat kernels of second order complex elliptic operators and applications, J. Funct. Anal. 152 (1998), no. 1, 22–73. MR 1600066, DOI 10.1006/jfan.1997.3156
Sagun Chanillo and Yan Yan Li, Continuity of solutions of uniformly elliptic equations in $\textbf {R}^2$, Manuscripta Math. 77 (1992), no. 4, 415–433. MR 1190215, DOI 10.1007/BF02567065
Cho, S.; Dong, H.; Kim, S. On the Green’s matrices of strongly parabolic systems of second order. arXiv:0705.1855; to appear in Indiana Univ. Math. J.
G. Dolzmann and S. Müller, Estimates for Green’s matrices of elliptic systems by $L^p$ theory, Manuscripta Math. 88 (1995), no. 2, 261–273. MR 1354111, DOI 10.1007/BF02567822
Joseph L. Doob, Classical potential theory and its probabilistic counterpart, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1984 edition. MR 1814344, DOI 10.1007/978-3-642-56573-1
Luis Escauriaza, Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form, Comm. Partial Differential Equations 25 (2000), no. 5-6, 821–845. MR 1759794, DOI 10.1080/03605300008821533
M. Fuchs, The Green matrix for strongly elliptic systems of second order with continuous coefficients, Z. Anal. Anwendungen 5 (1986), no. 6, 507–531 (English, with German and Russian summaries). MR 894243, DOI 10.4171/ZAA/219
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364, DOI 10.1007/978-3-642-61798-0
Michael Grüter and Kjell-Ove Widman, The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982), no. 3, 303–342. MR 657523, DOI 10.1007/BF01166225
Steve Hofmann and Seick Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math. 124 (2007), no. 2, 139–172. MR 2341783, DOI 10.1007/s00229-007-0107-1
Steve Hofmann and Seick Kim, Gaussian estimates for fundamental solutions to certain parabolic systems, Publ. Mat. 48 (2004), no. 2, 481–496. MR 2091016, DOI 10.5565/PUBLMAT_{4}8204_{1}0
Carlos E. Kenig and Wei-Ming Ni, On the elliptic equation $Lu-k+K\,\textrm {exp}[2u]=0$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 2, 191–224. MR 829052
Seick Kim, Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients, Trans. Amer. Math. Soc. 360 (2008), no. 11, 6031–6043. MR 2425701, DOI 10.1090/S0002-9947-08-04485-1
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43–77. MR 161019
Jan Malý and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, vol. 51, American Mathematical Society, Providence, RI, 1997. MR 1461542, DOI 10.1090/surv/051
Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511, DOI 10.1007/978-3-540-69952-1