  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.
