The positive Jacobian constraint in elasticity theory and orientation-preserving Young measures

WHERE:

MR14 - Centre for Mathematical Sciences
Cambridge, United Kingdom
WHEN:

Tuesday, 13 October 2015
from 14:00 to 18:00
SPEAKERS:

Dr Filip Rindler
University of Warwick
ABSTRACT
In elasticity theory, one naturally requires that the Jacobian determinant of the deformation is positive or even a-priori prescribed (for example in the case of incompressibility). However, such strongly non-linear and non-convex constraints are difficult to deal with in mathematical models. In this minicourse, I will present various recent results on how this constraint can be manipulated in subcritical Sobolev spaces, where the integrability exponent is less than the dimension. This setting is related to cavitation and fracture phenomena in materials. In particular, after introducing the appropriate notions, I will present a characterization of such constraint on the Jacobian determinant formulated in the language of Young measures. These objects, which I will briefly introduce, are widely used in the Calculus of Variations to model limits of nonlinear functions of weakly converging “generating” sequences. I will also discuss relations to convex integration and “geometry” in matrix space. Finally, I will show some applications to the minimization of integral functionals, the theory of semiconvex hulls and incompressible extensions.
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