scholarly journals Γ-convergence approach to variational problems in perforated domains with Fourier boundary conditions

2008 ◽  
Vol 16 (1) ◽  
pp. 148-175 ◽  
Author(s):  
Valeria Chiadò Piat ◽  
Andrey Piatnitski
Keyword(s):  
Boundary Conditions ◽  
Variational Problems ◽  
Perforated Domains ◽  
Γ Convergence

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.

FRACTURE MECHANICS IN PERFORATED DOMAINS: A VARIATIONAL MODEL FOR BRITTLE POROUS MEDIA

2009 ◽  
Vol 19 (11) ◽  
pp. 2065-2100 ◽  
Author(s):  
MATTEO FOCARDI ◽  
M. S. GELLI ◽  
M. PONSIGLIONE
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Fracture Mechanics ◽  
Special Functions ◽  
Variational Model ◽  
Perforated Domain ◽  
Functions Of Bounded Variation ◽  
Perforated Domains ◽  
Γ Convergence ◽  
Internal Scale

This paper deals with fracture mechanics in periodically perforated domains. Our aim is to provide a variational model for brittle porous media in the case of anti-planar elasticity. Given the perforated domain Ωε ⊂ ℝN (ε being an internal scale representing the size of the periodically distributed perforations), we will consider a total energy of the type [Formula: see text] Here u is in SBV(Ωε) (the space of special functions of bounded variation), Su is the set of discontinuities of u, which is identified with a macroscopic crack in the porous medium Ωε, and [Formula: see text] stands for the (N - 1)-Hausdorff measure of the crack Su. We study the asymptotic behavior of the functionals [Formula: see text] in terms of Γ-convergence as ε → 0. As a first (nontrivial) step we show that the domain of any limit functional is SBV(Ω) despite the degeneracies introduced by the perforations. Then we provide explicit formula for the bulk and surface energy densities of the Γ-limit, representing in our model the effective elastic and brittle properties of the porous medium, respectively.

The equivalence of some variational problems for surfaces of prescribed mean curvature

1979 ◽  
Vol 20 (1) ◽  
pp. 87-104 ◽  
Author(s):  
Graham H. Williams
Keyword(s):  
Boundary Conditions ◽  
Minimization Problem ◽  
Mean Curvature ◽  
Variational Problems ◽  
Generalized Solution ◽  
Smooth Functions ◽  
Prescribed Mean Curvature ◽  
Class Of Functions ◽  
General Conditions ◽  
Non Parametric

One method of finding non-parametric hypersurfaces of prescribed mean curvature which span a given curve in Rn is to find a function which minimizes a particular integral amongst all smooth functions satisfying certain boundary conditions. A new problem can be considered by changing the integral slightly and then minimizing over a larger class of functions. It is possible to show that a solution to this new problem exists under very general conditions and it is usually known as the generalized solution. In this paper we show that the two problems are equivalent in the sense that the least value for the original minimization problem and the generalized problem are the same even though no solution may exist. The case where the surfaces are constrained to lie above an obstacle is also considered.

NEW RESULTS ON THE ASYMPTOTIC BEHAVIOR OF DIRICHLET PROBLEMS IN PERFORATED DOMAINSDIRICHLET PROBLEMS IN PERFORATED DOMAINS

1994 ◽  
Vol 04 (03) ◽  
pp. 373-407 ◽  
Author(s):  
GIANNI DAL MASO ◽  
ADRIANA GARRONI
Keyword(s):  
Adjoint Operator ◽  
Arbitrary Sequence ◽  
Symmetric Part ◽  
Dirichlet Problems ◽  
Limit Measure ◽  
Compactness Result ◽  
Perforated Domains ◽  
Measurable Coefficients ◽  
Open Set ◽  
Γ Convergence

Let A be a linear elliptic operator of the second order with bounded measurable coefficients on a bounded open set Ω of Rn and let (Ωh) be an arbitrary sequence of open subsets of Ω. We prove the following compactness result: there exist a subsequence, still denoted by (Ωh), and a positive Borel measure μ on Ω, not charging polar sets, such that, for every f∈H−1(Ω) the solutions [Formula: see text] of the equations Auh=f in Ωh, extended to 0 on Ω\Ωh, converge weakly in [Formula: see text] to the unique solution [Formula: see text] of the problem [Formula: see text] When A is symmetric, this compactness result is already known and was obtained by Γ-convergence techniques. Our new proof, based on the method of oscillating test functions, extends the result to the non-symmetric case. The new technique, which is completely independent of Γ-convergence, relies on the study of the behavior of the solutions [Formula: see text] of the equations [Formula: see text] where A* is the adjoint operator. We prove also that the limit measure μ does not change if A is replaced by A*. Moreover, we prove that µ depends only on the symmetric part of the operator A, if the coefficients of the skew-symmetric part are continuous, while an explicit example shows that μ may depend also on the skew-symmetric part of A, when the coefficients are discontinuous.

scholarly journals Natural Boundary Conditions in Geometric Calculus of Variations

2015 ◽  
Vol 65 (6) ◽  
Author(s):  
Giovanni Moreno ◽  
Monika Ewa Stypa
Keyword(s):  
Boundary Conditions ◽  
Variational Problems ◽  
Boundary Values ◽  
Natural Boundary ◽  
Natural Behavior ◽  
Geometric Calculus ◽  
Jet Bundles ◽  
Independent Variables ◽  
Natural Boundary Conditions ◽  
Dimensional Domain

AbstractIn this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of Y - the manifold of dependent and independent variables underlying a given problem - as well as the order of its Lagrangian. Our result follows from the natural behavior, under boundary-friendly transformations, of an operator, similar to the Euler map, constructed in the context of relative horizontal forms on jet bundles (or Grassmann fibrations) over Y . Explicit examples of natural boundary conditions are obtained when Y is an (n + 1)-dimensional domain in ℝ

Homogenization of Dirichlet parabolic systems with variable monotone operators in general perforated domains

2003 ◽  
Vol 133 (6) ◽  
pp. 1231-1248 ◽  
Author(s):  
Carmen Calvo-Jurado ◽  
Juan Casado-Díaz
Keyword(s):  
Boundary Conditions ◽  
Elliptic Problem ◽  
Dirichlet Boundary ◽  
Monotone Operators ◽  
Parabolic Systems ◽  
Limit Problem ◽  
Dirichlet Boundary Conditions ◽  
Perforated Domains ◽  
Parabolic Case

We consider the homogenization of parabolic systems with Dirichlet boundary conditions when the operators and the domains in which the problems are posed vary simultaneously. We assume the operators do not depend on t. Then we show that the corrector obtained in a previous paper for the elliptic problem still gives a corrector for the parabolic one. From this result, we easily obtain the limit problem in the parabolic case.

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