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

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.

[1]  C. B. Morrey Multiple Integrals in the Calculus of Variations , 1966 .

[2]  S. Müller,et al.  Estimates for Green's matrices of elliptic systems byLp theory , 1995 .

[3]  M. Fuchs The Green Matrix for Strongly Elliptic Systems of Second Order with Continuous Coefficients , 1986 .

[4]  G. Stampacchia,et al.  Regular points for elliptic equations with discontinuous coefficients , 1963 .

[5]  J. Doob Classical potential theory and its probabilistic counterpart , 1984 .

[6]  Yanyan Li,et al.  Continuity of solutions of uniformly elliptic equations in R2 , 1992 .

[7]  Jan Malý,et al.  Fine Regularity of Solutions of Elliptic Partial Differential Equations , 1997 .

[8]  L. Escauriaza Bounds for the fundamental solutions of elliptic and parabolic equations , 2000 .

[9]  Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients , 2007, 0704.1372.

[10]  Hongjie Dong,et al.  On the Green's matrices of strongly parabolic systems of second order , 2007, 0705.1855.

[11]  S. Hofmann,et al.  The Green function estimates for strongly elliptic systems of second order , 2007, 0704.1352.

[12]  L. Escauriaza BOUNDS FOR THE FUNDAMENTAL SOLUTION OF ELLIPTIC AND PARABOLIC EQUATIONS IN NONDIVERGENCE FORM , 2000 .

[13]  O. Ladyženskaja Linear and Quasilinear Equations of Parabolic Type , 1968 .

[14]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[15]  S. Hofmann,et al.  Gaussian estimates for fundamental solutions to certain parabolic systems. , 2004 .

[16]  P. Tchamitchian,et al.  Heat Kernels of Second Order Complex Elliptic Operators and Applications , 1998 .

[17]  G. M. Lieberman SECOND ORDER PARABOLIC DIFFERENTIAL EQUATIONS , 1996 .

[18]  Michael Grüter,et al.  The Green function for uniformly elliptic equations , 1982 .

[19]  C. Kenig,et al.  On the elliptic equation $Lu - k + K \exp [2u] = 0$ , 1985 .