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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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.

Received:
Apr 15, 2010
Published:
Apr 23, 2010
MSC Codes:
35B50, 35J15, 74B20
Keywords:
Maximum principles, General theory of second order of second-order ell, nonlinear elasticity

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Preprint
2010 Repository Open Access
Thomas Blesgen and Anja Schlömerkemper

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