A counter example to weak maximum principles for locally vanishing elliptic operators

For the validity of the weak maximum principle for classical solutions of elliptic partial differential equations it is sufficient that the coefficient matrix a(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 . We demonstrate that the assumption that the coefficient matrix a(x) is positive almost everywhere is essential and cannot be weakened. To this end we give a counter example originating from geometrically linear elasticity. Note After submission of this paper we learned that there are much simpler examples which demonstrate that the positivity of the coefficient matrix a(x) is essential and cannot be weakened. In two space dimensions, let the coefficient matrix be given by a(x1, x2) = ( x2 −x1x2 −x1x2 x 2 1 ) . Then any u that is some C-function of x1 + x 2 2 is a solution of Equation (1.3) below with f ≡ 0. Hence u does not need to satisfy supΩ u = sup∂Ω u. We remark that this example works already for classical solutions of equations in divergence form. Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, D-04103 Leipzig, Germany, email: [email protected] University of Erlangen-Nuremberg, Department of Mathematics, Applied Mathematics 1, Martensstr. 3, D-91058 Erlangen, Germany, email: [email protected]