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Convex integration with constraints and applications to phase transitions and partial differential equations

Stefan MüllerVladimír Šverák — 1999

Journal of the European Mathematical Society

We study solutions of first order partial differential relations D u K , where u : Ω n m is a Lipschitz map and K is a bounded set in m × n matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of D u and second we replace Gromov’s P −convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our work was originally...

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