Lecture note 16/2003
Rigidity and Geometry of Microstructures
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Submission date: 17. Jan. 2003
MSC-Numbers: 74N15, 35L70, 26B25, 35R70, 74N05, 35G30, 35F20
Keywords and phrases: hyperbolic monge-ampere equation, partial differential inclusions, rank-one connection and rigidity, geometry and stability of rank-one convexity
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Existence and regularity of solutions to partial differential inclusion is investigated. A central question is whether pairwise incompatibility of the admissible gradients enforces triviality of all solutions. We obtain first finite counterexamples and exactly describe the complexity which they must have - a question partially motivated by the study of the non-convex energy landscapes of shape memory alloys.
For this purpose we develop a flexible and unifying method how to solve partial differential inclusion, a new kind of regularity results for the hyperbolic Monge-Ampere equation and a geometric approach to rank-one convexity establishing its stability properties.