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Studying Nonlinear pde by Geometry in Matrix Space

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Geometric Analysis and Nonlinear Partial Differential Equations

Summary

We outline an approach to study the properties of nonlinear partial differential equations through the geometric properties of a set in the space ofm xn matrices which is naturally associated to the equation. In particular, different notions of convex hulls play a crucial role. This work draws heavily on Tartar’s work on oscillations in nonlinear pde and compensated compactness and on Gromov’s work on partial differential relations and convex integration. We point out some recent successes of this approach and outline a number of open problems, most of which seem to require a better geometric understanding of the different convexity notions.

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Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103, Leipzig, Germany

    Bernd Kirchheim & Stefan Müller

  2. Department of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Vladimír Šverák

Authors
  1. Bernd Kirchheim
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  2. Stefan Müller
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  3. Vladimír Šverák
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Editors and Affiliations

  1. Department of Mathematics, University of Bonn, Beringstraße 1, 53115, Bonn, Germany

    Stefan Hildebrandt  & Hermann Karcher  & 

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Kirchheim, B., Müller, S., Šverák, V. (2003). Studying Nonlinear pde by Geometry in Matrix Space. In: Hildebrandt, S., Karcher, H. (eds) Geometric Analysis and Nonlinear Partial Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55627-2_19

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  • DOI: https://doi.org/10.1007/978-3-642-55627-2_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44051-2

  • Online ISBN: 978-3-642-55627-2

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