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Studying Nonlinear pde by Geometry in Matrix Space

In: Geometric Analysis and Nonlinear Partial Differential Equations

Author

Listed:
  • Bernd Kirchheim

    (Max Planck Institute for Mathematics in the Sciences)

  • Stefan Müller

    (Max Planck Institute for Mathematics in the Sciences)

  • Vladimír Šverák

    (University of Minnesota, Department of Mathematics)

Abstract

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.

Suggested Citation

  • Bernd Kirchheim & Stefan Müller & Vladimír Šverák, 2003. "Studying Nonlinear pde by Geometry in Matrix Space," Springer Books, in: Stefan Hildebrandt & Hermann Karcher (ed.), Geometric Analysis and Nonlinear Partial Differential Equations, pages 347-395, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-55627-2_19
    DOI: 10.1007/978-3-642-55627-2_19
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