Preprint 49/2002

Studying nonlinear pde by geometry in matrix space

Bernd Kirchheim, Stefan Müller, and Vladimír Šverák

Contact the author: Please use for correspondence this email.
Submission date: 21. Jun. 2002
Pages: 63
published in: Geometric analysis and nonlinear partial differential equations / S. Hildebrandt (ed.)
Berlin [u.a.] : Springer, 2003. - P. 347 - 395 
MSC-Numbers: 35J45, 35F30, 35D05, 49J40
Keywords and phrases: nonlinear partial differential equations, convex integration, microstructure, convex hulls
Download full preprint: PDF (525 kB), PS ziped (342 kB)

We outline an approach to study the properties of nonlinear partial differential equations through the geometric properties of a set in the space of formula3 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.

04.01.2023, 02:12