Geodesic
Convex integration with constraints and applications to phase transitions and partial differential equations
Journal of the European Mathematical Society, Tome 1 (1999) no. 4, pp. 393-422.

Voir la notice de l'article provenant de la source EMS Press

Abstract. We study solutions of first order partial differential relations Du]K, where u2ÂnMÂm 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.
DOI : 10.1007/s100970050012
Classification : 35-XX, 00-XX
Keywords: 
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     author = {Stefan M\"uller and Vladim{\'\i}r \v{S}ver\'ak},
     title = {Convex integration with constraints and applications to phase transitions and partial differential equations},
     journal = {Journal of the European Mathematical Society},
     pages = {393--422},
     publisher = {mathdoc},
     volume = {1},
     number = {4},
     year = {1999},
     doi = {10.1007/s100970050012},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s100970050012/}
}
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Stefan Müller; Vladimír Šverák. Convex integration with constraints and applications to phase transitions and partial differential equations. Journal of the European Mathematical Society, Tome 1 (1999) no. 4, pp. 393-422. doi : 10.1007/s100970050012. http://geodesic.mathdoc.fr/articles/10.1007/s100970050012/

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