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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