DOCUMENTA MATHEMATICA, Extra Volume ICM II (1998), 691-702

Stefan Müller and Vladimir Sverák

Title: Unexpected Solutions of First and Second Order Partial Differential Equations

This note discusses a general approach to construct Lipschitz solutions of $Du \in K$, where $u: \Omega \subset {\mathbb R}^n \to {\mathbb R}^m$ and where $K$ is a given set of $m\times n$ matrices. The approach is an extension of Gromov's method of convex integration. One application concerns variational problems that arise in models of microstructure in solid-solid phase transitions. Another application is the systematic construction of singular solutions of elliptic systems. In particular, there exists a $2 \times 2$ (variational) second order strongly elliptic system $ \mbox{\rm div} \, \sigma(Du) = 0 $ that admits a Lipschitz solution which is nowhere $C^1$.

1991 Mathematics Subject Classification: 35F30, 35J55, 73G05

Keywords and Phrases: Partial differential equations, elliptic systems, regularity, variational problems, microstructure, convex integration

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