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Equi-integrability results for 3D-2D dimension reduction problems

Marian BoceaIrene Fonseca — 2002

ESAIM: Control, Optimisation and Calculus of Variations

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients α u ε | 1 ε 3 u ε bounded in L p ( Ω ; 9 ) , 1 < p < + . Here it is shown that, up to a subsequence, u ε may be decomposed as w ε + z ε , where z ε carries all the concentration effects, i.e. α w ε | 1 ε 3 w ε p is equi-integrable, and w ε captures the oscillatory behavior, i.e. z ε 0 in measure. In addition, if { u ε } is a recovering sequence then z ε = z ε ( x α ) nearby Ω .

Equi-integrability results for 3D-2D dimension reduction problems

Marian BoceaIrene Fonseca — 2010

ESAIM: Control, Optimisation and Calculus of Variations

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients α u ε | 1 ε 3 u ε bounded in L p ( Ω ; 9 ) , 1 < p < + . Here it is shown that, up to a subsequence, u ε may be decomposed as w ε + z ε , where z ε carries all the concentration effects, α w ε | 1 ε 3 w ε p is equi-integrable, and w ε captures the oscillatory behavior, z ε 0 in measure. In addition, if { u ε } is a recovering sequence then z ε = z ε ( x α ) nearby Ω .

Oscillations and concentrations generated by 𝒜 -free mappings and weak lower semicontinuity of integral functionals

Irene FonsecaMartin Kružík — 2010

ESAIM: Control, Optimisation and Calculus of Variations

DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps { u k } k L p ( Ω ; m ) satisfying a linear differential constraint 𝒜 u k = 0 . Applications to sequential weak lower semicontinuity of integral functionals on 𝒜 -free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det ϕ k * det ϕ in measures on the closure of Ω n if ϕ k ϕ in W 1 , n ( Ω ; n ) . This convergence holds, for example, under...

An existence result for a nonconvex variational problem via regularity

Irene FonsecaNicola FuscoPaolo Marcellini — 2002

ESAIM: Control, Optimisation and Calculus of Variations

Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The x -dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.

Topological degree, Jacobian determinants and relaxation

Irene FonsecaNicola FuscoPaolo Marcellini — 2005

Bollettino dell'Unione Matematica Italiana

A characterization of the total variation T V u , Ω of the Jacobian determinant det D u is obtained for some classes of functions u : Ω R n outside the traditional regularity space W 1 , n Ω ; R n . In particular, explicit formulas are deduced for functions that are locally Lipschitz continuous away from a given one point singularity x 0 Ω . Relations between T V u , Ω and the distributional determinant Det D u are established, and an integral representation is obtained for the relaxed energy of certain polyconvex functionals at maps u W 1 , p Ω ; R n W 1 , Ω x 0 ; R n .

A-Quasiconvexity: Relaxation and Homogenization

Andrea BraidesIrene FonsecaGiovanni Leoni — 2010

ESAIM: Control, Optimisation and Calculus of Variations

Integral representation of relaxed energies and of -limits of functionals ( u , v ) Ω f ( x , u ( x ) , v ( x ) ) d x are obtained when sequences of fields may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell's equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in , are recovered.

An existence result for a nonconvex variational problem via regularity

Irene FonsecaNicola FuscoPaolo Marcellini — 2010

ESAIM: Control, Optimisation and Calculus of Variations

Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are . In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands with respect to the gradient variable. The -dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.

Regularity results for an optimal design problem with a volume constraint

Menita CarozzaIrene FonsecaAntonia Passarelli di Napoli — 2014

ESAIM: Control, Optimisation and Calculus of Variations

Regularity results for minimal configurations of variational problems involving both bulk and surface energies and subject to a volume constraint are established. The bulk energies are convex functions with -power growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (), Hölder continuity of the function is proved as well as partial regularity of the boundary of the minimal set . Moreover, full regularity of the boundary of the minimal set is obtained...

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