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

Stefan Müller and Vladimír Šverák

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Submission date: 02. May. 1999
Pages: 33
published in: Journal of the European Mathematical Society, 1 (1999) 4, p. 393-422 
DOI number (of the published article): 10.1007/s100970050012
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Abstract:
We study solutions first order partial differential relations , where is a Lipschitz map and K is a bounded set in matrices, and extend Gromov's theory of convex integration in two ways. First, we allow for additional constraints on the minors of Du 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 motivated by questions in the analysis of crystal microstructure and we establish the existence of a wide class of solutions to the two-well problem in the theory of martensite.