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Dimension reduction for functionals on solenoidal vector fields

Stefan Krömer — 2012

ESAIM: Control, Optimisation and Calculus of Variations

We study integral functionals constrained to divergence-free vector fields in on a thin domain, under standard -growth and coercivity assumptions, 1    ∞. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in is always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject to the limiting...

Dimension reduction for functionals on solenoidal vector fields

Stefan Krömer — 2012

ESAIM: Control, Optimisation and Calculus of Variations

We study integral functionals constrained to divergence-free vector fields in on a thin domain, under standard -growth and coercivity assumptions, 1    ∞. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in is always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject to the limiting...

Injective weak solutions in second-gradient nonlinear elasticity

Timothy J. HealeyStefan Krömer — 2009

ESAIM: Control, Optimisation and Calculus of Variations

We consider a class of second-gradient elasticity models for which the internal potential energy is taken as the sum of a convex function of the second gradient of the deformation and a general function of the gradient. However, in consonance with classical nonlinear elasticity, the latter is assumed to grow unboundedly as the determinant of the gradient approaches zero. While the existence of a minimizer is routine, the existence of weak solutions is not, and we focus our efforts on that question...

Injective weak solutions in second-gradient nonlinear elasticity

Timothy J. HealeyStefan Krömer — 2008

ESAIM: Control, Optimisation and Calculus of Variations

We consider a class of second-gradient elasticity models for which the internal potential energy is taken as the sum of a convex function of the second gradient of the deformation and a general function of the gradient. However, in consonance with classical nonlinear elasticity, the latter is assumed to grow unboundedly as the determinant of the gradient approaches zero. While the existence of a minimizer is routine, the existence of weak solutions is not, and we focus our efforts on that question...

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