Studying Nonlinear pde by Geometry in Matrix Space

Bernd Kirchheim (), Stefan Müller () and Vladimír Šverák ()
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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

A chapter in Geometric Analysis and Nonlinear Partial Differential Equations, 2003, pp 347-395 from Springer

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.

Keywords: Elliptic System; Young Measure; Matrix Space; Nematic Elastomer; Gradient Young Measure (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-55627-2_19

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DOI: 10.1007/978-3-642-55627-2_19

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