Unexpected solutions of first and second order partial differential equations

  • Stefan Müller

  • Vladimir Šverák


This note discusses a general approach to construct Lipschitz solutions of , where \( u: \Omega \subset \bbfR^n \to \bbfR^m \) and where is a given set of 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 (variational) second order strongly elliptic system that admits a Lipschitz solution which is nowhere .