Advances in Differential Equations

The two well problem for piecewise affine maps

Bernard Dacorogna, Paolo Marcellini, and Emanuele Paolini

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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.

Article information

Adv. Differential Equations Volume 17, Number 7/8 (2012), 673-696.

First available in Project Euclid: 17 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35F50: Nonlinear first-order systems


Dacorogna, Bernard; Marcellini, Paolo; Paolini, Emanuele. The two well problem for piecewise affine maps. Adv. Differential Equations 17 (2012), no. 7/8, 673--696.

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