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

Stefan Müller; Vladimír Šverák

Journal of the European Mathematical Society (1999)

  • Volume: 001, Issue: 4, page 393-422
  • ISSN: 1435-9855

Abstract

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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 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.

How to cite

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Müller, Stefan, and Šverák, Vladimír. "Convex integration with constraints and applications to phase transitions and partial differential equations." Journal of the European Mathematical Society 001.4 (1999): 393-422. <http://eudml.org/doc/277630>.

@article{Müller1999,
abstract = {We study solutions of first order partial differential relations $Du\in K$, where $u:\Omega \subset \mathbb \{R\}^n\rightarrow \mathbb \{R\}^m$ is a Lipschitz map and $K$ is a bounded set in $m\times n$ 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.},
author = {Müller, Stefan, Šverák, Vladimír},
journal = {Journal of the European Mathematical Society},
keywords = {first order partial differential relations; convex integration; Gromov’s theory of convex integration; $P$−convex hull; rank-one convex hull; theory of martensite; differential relation; convex integration},
language = {eng},
number = {4},
pages = {393-422},
publisher = {European Mathematical Society Publishing House},
title = {Convex integration with constraints and applications to phase transitions and partial differential equations},
url = {http://eudml.org/doc/277630},
volume = {001},
year = {1999},
}

TY - JOUR
AU - Müller, Stefan
AU - Šverák, Vladimír
TI - Convex integration with constraints and applications to phase transitions and partial differential equations
JO - Journal of the European Mathematical Society
PY - 1999
PB - European Mathematical Society Publishing House
VL - 001
IS - 4
SP - 393
EP - 422
AB - We study solutions of first order partial differential relations $Du\in K$, where $u:\Omega \subset \mathbb {R}^n\rightarrow \mathbb {R}^m$ is a Lipschitz map and $K$ is a bounded set in $m\times n$ 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.
LA - eng
KW - first order partial differential relations; convex integration; Gromov’s theory of convex integration; $P$−convex hull; rank-one convex hull; theory of martensite; differential relation; convex integration
UR - http://eudml.org/doc/277630
ER -

Citations in EuDML Documents

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  1. Andrew Lorent, A two well Liouville theorem
  2. Andrew Lorent, A Two Well Liouville Theorem
  3. Sergio Conti, Georg Dolzmann, Bernd Kirchheim, Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions
  4. Andrew Lorent, The regularisation of the N -well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions
  5. Andrew Lorent, The regularisation of the -well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions
  6. Krzysztof Chełmiński, Agnieszka Kałamajska, New convexity conditions in the calculus of variations and compensated compactness theory
  7. Krzysztof Chełmiński, Agnieszka Kałamajska, New convexity conditions in the calculus of variations and compensated compactness theory

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