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MiS Preprint
42/1998
Unexpected solutions of first and second order partial differential equations
Stefan Müller and Vladimír Šverák
Abstract
We discuss 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 $div \sigma (Du)=0$ that admits a Lipschitz solution which is nowhere $C^1$.
