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July/August 2012 The two well problem for piecewise affine maps
Bernard Dacorogna, Paolo Marcellini, Emanuele Paolini
Adv. Differential Equations 17(7/8): 673-696 (July/August 2012).

Abstract

In the \textit{two-well problem} we look for a map $u$ which satisfies Dirichlet boundary conditions and whose gradient $Du$ assumes values in $SO\left( 2\right) A\cup SO\left( 2\right) B=\mathbb{S}_{A}\cup \mathbb{S}_{B},$ for two given invertible matrices $A,B$ (an element of $SO\left( 2\right) A$ is of the form $RA$ where $R$ is a rotation). In the original approach by Ball and James [1], [2] $A$, $B$ are two matrices such that $\det B>\det A>0$ and $\operatorname*{rank}\left\{ A-B\right\} =1.$ It was proved in the 1990's (see [4], [5], [6], [7], [17]) that a map $u$ satisfying given boundary conditions and such that $Du\in\mathbb{S} _{A}\cup\mathbb{S}_{B}$ exists in the Sobolev class $W^{1,\infty} (\Omega;\mathbb{R}^{2})$ of Lipschitz continuous maps. However, for orthogonal matrices it was also proved (see [3], [8], [9], [10], [11], [12], [16]) that solutions exist in the class of piecewise-$C^{1}$ maps, in particular in the class of piecewise-affine maps. We prove here that this possibility does not exist for other nonsingular matrices $A$, $B$: precisely, the two-well problem can be solved by means of piecewise-affine maps only for orthogonal matrices.

Citation

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Bernard Dacorogna. Paolo Marcellini. Emanuele Paolini. "The two well problem for piecewise affine maps." Adv. Differential Equations 17 (7/8) 673 - 696, July/August 2012.

Information

Published: July/August 2012
First available in Project Euclid: 17 December 2012

zbMATH: 1258.35061
MathSciNet: MR2963800

Subjects:
Primary: 35F50

Rights: Copyright © 2012 Khayyam Publishing, Inc.

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Vol.17 • No. 7/8 • July/August 2012
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