The degenerate two well problem for piecewise affine maps

The two well problem consists in finding maps u which satisfy some boundary conditions and whose gradient Du assumes values in the two wells . Here (similarly ) is the well generated by a 2 x 2 matrix A, i.e., 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 . 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 [15]. We also show that the two well problem can be solved in some cases involving singular matrices, in strict contrast to the nonsingular (and not orthogonal) case.


Published in:
Nodea-Nonlinear Differential Equations And Applications, 20, 2, 345-359
Year:
2013
Publisher:
Basel, Springer Basel Ag
ISSN:
1021-9722
Keywords:
Laboratories:




 Record created 2013-10-01, last modified 2018-03-17


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