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

Thomas Blesgen and Anja Schlömerkemper

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Submission date: 15. Apr. 2010 (revised version: July 2010)
Pages: 12
Bibtex
MSC-Numbers: 35B50, 35J15, 74B20
Keywords and phrases: Maximum principles, General theory of second order of second-order ell, nonlinear elasticity
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Abstract:
For the validity of the weak maximum principle for classical solutions of elliptic partial differential equations it is sufficient that the coefficient matrix is non-negative. In this note we consider maximum principles for weak solutions of elliptic partial differential equations in divergence form with bounded coefficients . We demonstrate that the assumption that the coefficient matrix is positive almost everywhere is essential and cannot be weakened. To this end we give a counter example originating from geometrically linear elasticity.