MiS Preprint Repository

For the validity of the weak maximum principle for classical solutions of elliptic partial differential equations it is sufficient that the coefficient matrix $a^{ij}(x)$ is non-negative. In this note we consider maximum principles for weak solutions of elliptic partial differential equations in divergence form with bounded coefficients $a^{ij}$. We demonstrate that the assumption that the coefficient matrix $a^{ij}(x)$ is positive almost everywhere is essential and cannot be weakened. To this end we give a counter example originating from geometrically linear elasticity.