Weak Maximum Principle for Strongly Coupled Elliptic Differential Systems

A classical counterexample due to E. De Giorgi, shows that the weak maximum principle does not remain true for general linear elliptic differential systems. After that, there are some efforts to establish the weak maximum principle for special elliptic differential systems, but the existing works are addressing only the cases of weakly coupled systems, or almost-diagonal systems, or even some systems coupling in various lower order terms. In this paper, by contrast, we present maximum modulus estimates for weak solutions to two classes of coupled linear elliptic differential systems with different principal parts, under considerably mild and physically reasonable assumptions. The systems under consideration are strongly coupled in the second order terms and other lower order terms, without restrictions on the size of ratios of the different principal part coefficients, or on the number of equations and space variables.