scholarly journals Γ-convergence of nonconvex integrals in Cheeger--Sobolev spaces and homogenization

2017 ◽  
Vol 10 (4) ◽  
pp. 381-405 ◽  
Author(s):  
Omar Anza Hafsa ◽  
Jean-Philippe Mandallena
Keyword(s):  
Calculus Of Variations ◽  
Sobolev Spaces ◽  
Variational Integrals ◽  
Γ Convergence

AbstractWe study Γ-convergence of nonconvex variational integrals of the calculus of variations in the setting of Cheeger–Sobolev spaces. Applications to relaxation and homogenization are given.

scholarly journals On the relaxation of variational integrals in metric Sobolev spaces

2015 ◽  
Vol 8 (1) ◽  
Author(s):  
Omar Anza Hafsa ◽  
Jean-Philippe Mandallena
Keyword(s):  
Integral Representation ◽  
Calculus Of Variations ◽  
Sobolev Spaces ◽  
General Framework ◽  
Metric Measure Spaces ◽  
Representation Theorems ◽  
Differentiable Structure ◽  
Convex Case ◽  
Measure Spaces ◽  
Variational Integrals

AbstractWe give an extension of the theory of relaxation of variational integrals in classical Sobolev spaces to the setting of metric Sobolev spaces. More precisely, we establish a general framework to deal with the problem of finding an integral representation for “relaxed” variational functionals of variational integrals of the calculus of variations in the setting of metric measure spaces. We prove integral representation theorems, both in the convex and non-convex case, which extend and complete previous results in the setting of euclidean measure spaces to the setting of metric measure spaces. We also show that these integral representation theorems can be applied in the setting of Cheeger–Keith's differentiable structure.

scholarly journals Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-16
Author(s):  
Brahim Amaziane ◽  
Leonid Pankratov
Keyword(s):  
Sobolev Spaces ◽  
Variational Problems ◽  
Double Porosity ◽  
Energy Characteristics ◽  
Perforated Domains ◽  
Growth Functions ◽  
Homogenization Approach ◽  
Convergence Method ◽  
Oscillating Coefficients ◽  
Γ Convergence

We review recent results on the homogenization in Sobolev spaces with variable exponents. In particular, we are dealing with the Γ-convergence of variational functionals with rapidly oscillating coefficients, the homogenization of the Dirichlet and Neumann variational problems in strongly perforated domains, as well as double porosity type problems. The growth functions also depend on the small parameter characterizing the scale of the microstructure. The homogenization results are obtained by the method of local energy characteristics. We also consider a parabolic double porosity type problem, which is studied by combining the variational homogenization approach and the two-scale convergence method. Results are illustrated with periodic examples, and the problem of stability in homogenization is discussed.

ON MINIMIZATION OF A NON-CONVEX FUNCTIONAL IN VARIABLE EXPONENT SPACES

2014 ◽  
Vol 25 (01) ◽  
pp. 1450011 ◽  
Author(s):  
GERARDO R. CHACÓN ◽  
RENATO COLUCCI ◽  
HUMBERTO RAFEIRO ◽  
ANDRÉS VARGAS
Keyword(s):  
Calculus Of Variations ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Direct Method ◽  
Convex Functional ◽  
Lebesgue Spaces ◽  
Variable Exponent Spaces ◽  
Existence Of Minimizers ◽  
Variable Exponent Sobolev Spaces

We study the existence of minimizers of a regularized non-convex functional in the context of variable exponent Sobolev spaces by application of the direct method in the calculus of variations. The results are new even in the framework of classical Lebesgue spaces.

On a necessary condition in the calculus of variations in Sobolev spaces with variable exponent

2014 ◽  
Vol 41 (2) ◽  
pp. 165-174
Author(s):  
Elhoussine Azroul ◽  
Meryem El Lekhlifi ◽  
Badr Lahmi ◽  
Abdelfattah Touzani
Keyword(s):  
Calculus Of Variations ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Necessary Condition

The gap phenomenon for variational integrals in Sobolev spaces

1992 ◽  
Vol 120 (1-2) ◽  
pp. 93-98 ◽  
Author(s):  
M. Giaquinta ◽  
G. Modica ◽  
J. Souček
Keyword(s):  
Riemannian Manifold ◽  
Sobolev Spaces ◽  
Variational Integrals

SynopsisWe show that a gap phenomenon occurs for general variational integrals for mappings from a domain Rn into a Riemannian manifold if has a non-trivial topology.

Γ-Convergence of Inhomogeneous Functionals in Orlicz–Sobolev Spaces

2015 ◽  
Vol 58 (2) ◽  
pp. 287-303 ◽  
Author(s):  
Marian Bocea ◽  
Mihai Mihăilescu
Keyword(s):  
Asymptotic Behaviour ◽  
Recent Work ◽  
Power Law ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Γ Convergence

AbstractThe asymptotic behaviour of inhomogeneous power-law type functionals is undertaken via De Giorgi’s Γ-convergence. Our results generalize recent work dealing with the asymptotic behaviour of power-law functionals acting on fields belonging to variable exponent Lebesgue and Sobolev spaces to the Orlicz–Sobolev setting.

Rank-one convexity does not imply quasiconvexity

1992 ◽  
Vol 120 (1-2) ◽  
pp. 185-189 ◽  
Author(s):  
Vladimír Šverák
Keyword(s):  
Continuous Function ◽  
Calculus Of Variations ◽  
Open Subset ◽  
Lower Semicontinuous ◽  
Regular Functions ◽  
Rank One ◽  
Variational Integrals ◽  
Bounded Open Subset ◽  
Image Position

We consider variational integralsdefined for (sufficiently regular) functionsu: Ω→Rm. Here Ω is a bounded open subset ofRn,Du(x) denotes the gradient matrix ofuatxandfis a continuous function on the space of all realm×nmatrices Mm×n. One of the important problems in the calculus of variations is to characterise the functionsffor which the integralIis lower semicontinuous. In this connection, the following notions were introduced (see [3], [9], [10]).

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