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Studying nonlinear pde by geometry in matrix space
Bernd Kirchheim, Stefan Müller and Vladimír Šverák
We outline an approach to study the properties of nonlinear partial differential equations through the geometric properties of a set in the space of $m \times n$ 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.