Unexpected solutions of first and second order partial differential equations

Abstract

This note discusses a general approach to construct Lipschitz solutions of $Du \in K$, where $u: \Omega \subset \bbfR^n \to \bbfR^m$ 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 $C^1$.