## Advances in Differential Equations

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

### The two well problem for piecewise affine maps

Bernard Dacorogna, Paolo Marcellini, and Emanuele Paolini

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 17 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355702972

**Mathematical Reviews number (MathSciNet)**

MR2963800

**Zentralblatt MATH identifier**

1258.35061

**Subjects**

Primary: 35F50: Nonlinear first-order systems

#### Citation

Dacorogna, Bernard; Marcellini, Paolo; Paolini, Emanuele. The two well problem for piecewise affine maps. Adv. Differential Equations 17 (2012), no. 7/8, 673--696. https://projecteuclid.org/euclid.ade/1355702972.