Harnack Principle for Weakly Coupled Elliptic Systems

Abstract We prove the Harnack inequality for the weakly coupled elliptic system S , where Su=\big (\begin{align} L_1 & & \\ & \cdots & \\ & & L_N\end{align}\big )u+Qu and u=\big (\begin{align} u_1\\ \cdot \\ \cdot \\ \cdot \\ u_N\end{align}\big ). { L k , k =1, …,  N } are second order elliptic operators with Holder continuous coefficients and Q is a matrix-valued function with singular entries. In the case that Q is irreducible, a full Harnack principle is proved.

[1]  Mark Freidlin,et al.  On the dirichlet problem for a class of second order pde systems with small parameter , 1990 .

[2]  Zhongxin Zhao.  Green Functions and Conditioned Gauge Theorem for a 2-Dimensional Domain , 1988 .

[3]  Y. Kifer.  Principal eigenvalues and equilibrium states corresponding to weakly coupled parabolic systems of PDE , 1992 .

[4]  A. M. Hinz,et al.  Subsoluton estimates and Harnack's inequality for Schrödinger operators. , 1990 .

[5]  Zhen-Qing Chen,et al.  Switched diffusion processes and systems of elliptic equations: a Dirichlet space approach , 1994, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[6]  Mark Freidlin,et al.  Averaging principle for perturbed random evolution equations and corresponding Dirichlet problems , 1993 .

[7]  A. Ancona.  Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien , 1978 .

[8]  B. Simon,et al.  Brownian motion and harnack inequality for Schrödinger operators , 1982 .

[9]  J. Serrin.  On the Harnack inequality for linear elliptic equations , 1954 .

[10]  Guido Sweers,et al.  Strong positivity in $$C(\bar \Omega )$$ for elliptic systemsfor elliptic systems , 1992 .

[11]  Zhen-Qing Chen,et al.  Potential theory for elliptic systems , 1996 .

[12]  M. Sieveking,et al.  Uniform bounds for quotients of Green functions on $C^{1,1}$-domains , 1982 .

[13]  Zhong-Hua Zhao.  Conditional gauge with unbounded potential , 1983 .

[14]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

[15]  C. Simader.  An elementary proof of Harnack’s inequality for Schrödinger operators and related topics , 1990 .

[16]  E. Fabes,et al.  Harnack’s inequality for Schrödinger operators and the continuity of solutions , 1986 .