Unexpected solutions of first and second order partial differential equations. (English) Zbl 0896.35029

Müller, Stefan; Šverák, Vladimir

Doc. Math., Extra Vol. ICM Berlin 1998, vol. II, 691-702 (1998).

Summary: 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 \( \text{div}\sigma(Du) = 0 \) that admits a Lipschitz solution which is nowhere \(C^1\).

MSC:

35F30	Boundary value problems for nonlinear first-order PDEs
35J55	Systems of elliptic equations, boundary value problems (MSC2000)
74B20	Nonlinear elasticity
49J24	Optimal control problems with differential inclusions (existence) (MSC2000)

Keywords:

phase transitions; regularity; singular solutions; microstructure; convex integration