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 DuKDu \in K, where u:Ω\bbfRn\bbfRmu: \Omega \subset \bbfR^n \to \bbfR^m and where KK is a given set of m×nm\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 2×22 \times 2 (variational) second order strongly elliptic system divσ(Du)=0{div}\sigma(Du) = 0 that admits a Lipschitz solution which is nowhere C1C^1.