Some estimates and maximum principles for weakly coupled systems of elliptic PDE

A study was conducted to discuss maximum principles and Harnack type estimates for systems of linear elliptic PDEs of second order given in a bounded domain. The study obtained a generalized maximum principle and a Harnack inequality for these PDEs. The results and their extensions provided the lower bounds for the eigenvalue in terms of the coefficients of Li and the domain, which can further be used to verify the condition of positivity of the eigenvalue. It was observed that the maximum principle hold for domains with sufficiently small measure. It was also found that the maximum principle was verified if either the matrix was semi-negative definite or the operators coincided. The study concluded that the maximum principle hold provided the operators coincided and can be written in divergence form.