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MiS Preprint
18/2010
A counter example to weak maximum principles for locally vanishing elliptic operators
Thomas Blesgen and Anja Schlömerkemper
Abstract
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.
