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
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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.
@article{JEMS_1999_1_4_a1,
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/}
}
TY - JOUR AU - Stefan Müller AU - Vladimír Šverák TI - Convex integration with constraints and applications to phase transitions and partial differential equations JO - Journal of the European Mathematical Society PY - 1999 SP - 393 EP - 422 VL - 1 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.1007/s100970050012/ DO - 10.1007/s100970050012 ID - JEMS_1999_1_4_a1 ER -
%0 Journal Article %A Stefan Müller %A Vladimír Šverák %T Convex integration with constraints and applications to phase transitions and partial differential equations %J Journal of the European Mathematical Society %D 1999 %P 393-422 %V 1 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.1007/s100970050012/ %R 10.1007/s100970050012 %F JEMS_1999_1_4_a1
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
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