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

Weak Maximum Principle for Strongly Coupled Elliptic Differential Systems