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Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional

Luca Mugnai (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We establish some new results about the Γ-limit, with respect to the L1-topology, of two different (but related) phase-field approximations { ℰ } , { ℰ ˜ } ℰ ε ε ,   x10ff65; ℰ ε ε of the so-called Euler’s Elastica Bending Energy for curves in the plane. In particular we characterize theΓ-limit as ε → 0 of ℰε, and show that in general the Γ-limits of ℰεand ℰ ˜ x10ff65; ℰ ε do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of theΓ-limit of ℰ ˜ x10ff65;...

Gehring's lemma for metrics and higher integrability of the gradient for minimizers of noncoercive variational functionals

Bruno Franchi, Francesco Serra Cassano (1996)

Studia Mathematica

We prove a higher integrability result - similar to Gehring's lemma - for the metric space associated with a family of Lipschitz continuous vector fields by means of sub-unit curves. Applications are given to show the higher integrability of the gradient of minimizers of some noncoercive variational functionals.

General method of regularization. I: Functionals defined on BD space

Jarosław L. Bojarski (2004)

Applicationes Mathematicae

The aim of this paper is to prove that the relaxation of the elastic-perfectly plastic energy (of a solid made of a Hencky material) is the lower semicontinuous regularization of the plastic energy. We find the integral representation of a non-locally coercive functional. In part II, we will show that the set of solutions of the relaxed problem is equal to the set of solutions of the relaxed problem proposed by Suquet. Moreover, we will prove the existence theorem for the limit analysis problem.

General method of regularization. II: Relaxation proposed by suquet

Jarosław L. Bojarski (2004)

Applicationes Mathematicae

The aim of this paper is to prove that the relaxation of the elastic-perfectly plastic energy (of a solid made of a Hencky material) is the lower semicontinuous regularization of the plastic energy. We find the integral representation of a non-locally coercive functional. We show that the set of solutions of the relaxed problem is equal to the set of solutions of the relaxed problem proposed by Suquet. Moreover, we prove an existence theorem for the limit analysis problem.

General method of regularization. III: The unilateral contact problem

Jarosław L. Bojarski (2004)

Applicationes Mathematicae

The aim of this paper is to prove that the relaxation of the elastic-perfectly plastic energy (of a solid made of a Hencky material with the Signorini constraints on the boundary) is the weak* lower semicontinuous regularization of the plastic energy. We consider an elastic-plastic solid endowed with the von Mises (or Tresca) yield condition. Moreover, we show that the set of solutions of the relaxed problem is equal to the set of solutions of the relaxed problem proposed by Suquet. We deduce that...

Generalised functions of bounded deformation

Gianni Dal Maso (2013)

Journal of the European Mathematical Society

We introduce the space G B D of generalized functions of bounded deformation and study the structure properties of these functions: the rectiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for G B D , which leads to a compactness result for the space G S B D of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation of some variational...

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