*Published Paper*

**Inserted:** 1 may 2012

**Last Updated:** 14 jun 2014

**Journal:** Nonlinear Differ. Equ. Appl. (NoDEA)

**Volume:** 20

**Number:** 2

**Pages:** 345--359

**Year:** 2013

**Doi:** 10.1007/s00030-012-0169-y

**Abstract:**

The *two wells problem* consists in finding maps $u$ which satisfy some
boundary conditions and whose gradient $Du$ assumes values in the two wells
$\mathbb{S}_{A}$, $\mathbb{S}_{B}$. Here $\mathbb{S}_{A}$ (similarly
$\mathbb{S}_{B}$) is the *well* generated by a square matrix $A$, i.e.,
$\mathbb{S}_{A}$ is the set of matrices of the form $RA$, where $R$ is a
rotation. We study specifically the case when at least one of the two matrices
$A$, $B$ is singular and we characterize piecewise affine maps $u$ satisfying
almost everywhere the differential inclusion $Du\left( x\right)
\in\mathbb{S}_{A}\cup\mathbb{S}_{B}$. In particular we describe the
*lamination* and *angle* properties, which turn out to be different
from those of the nonsingular case described in detail in
DMP-11. We also show that the two wells problem can be
solved in some cases involving singular matrices, in strict contrast to the
nonsingular (and not *orthogonal*) case.

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