JournalsjemsVol. 1, No. 4pp. 393–422

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

  • Stefan Müller

    Universität Bonn, Germany
  • Vladimír Šverák

    University of Minnesota, Minneapolis, USA
Convex integration with constraints and applications to phase transitions and partial differential equations cover
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Abstract

Abstract. We study solutions of first order partial differential relations Du]K, where u

²ÂnMÂm is a Lipschitz map and K is a bounded set in m2n 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.

Cite this article

Stefan Müller, Vladimír Šverák, Convex integration with constraints and applications to phase transitions and partial differential equations. J. Eur. Math. Soc. 1 (1999), no. 4, pp. 393–422

DOI 10.1007/S100970050012