New conditions for averaging of nonlinear dirichlet problems in perforated domains

1996 ◽  
Vol 48 (5) ◽  
pp. 753-774 ◽  
Author(s):  
I. V. Skrypnik
Keyword(s):  
Dirichlet Problems ◽  
Perforated Domains

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.

Homogenization of singularly perturbed Dirichlet problems in perforated domains

2000 ◽  
Vol 130 (1) ◽  
pp. 35-51
Author(s):  
Viêt Há Hoáng
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behaviour ◽  
Error Estimates ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Perforated Domain ◽  
Dirichlet Problems ◽  
Perforated Domains

We study the singularly perturbed problem —εαΔuε + uε = f (α > 0) with the Dirichlet boundary condition in a perforated domain, in which the holes are distributed periodically with period 2ε throughout a fixed domain Ω. The asymptotic behaviour of uε when ε → 0, together with corrector results and error estimates in L2(Ω), are deduced for all sizes of holes. The behaviour of uε in is obtained for the cases where the size of holes is of order ε or is of a sufficiently smaller order. When the holes' size is of a sufficiently small order, as expected, uε has similar behaviour to that in the case of a non-varying domain.

Two-scale convergence for nonlinear Dirichlet problems in perforated domains

2000 ◽  
Vol 130 (2) ◽  
pp. 249-276 ◽  
Author(s):  
J. Casado-Díaz
Keyword(s):  
Dirichlet Problem ◽  
Periodic Structure ◽  
Limit Problem ◽  
Dirichlet Problems ◽  
Perforated Domains

The aim of the present paper is to adapt the method of two-scale convergence to the homogenization of a pseudomonotone Dirichlet problem in perforated domains with periodic structure. The limit problem and a corrector result are obtained.

scholarly journals Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains

1995 ◽  
Vol 125 (1) ◽  
pp. 99-114 ◽  
Author(s):  
Gianni Dal Maso ◽  
Annalisa Malusa
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problems ◽  
Elliptic Operator ◽  
Bounded Domain ◽  
Radon Measure ◽  
Boundary Value ◽  
Approximation Procedure ◽  
Dirichlet Problems ◽  
Perforated Domains ◽  
Positive Radon Measure

Given an elliptic operator L on a bounded domain Ω ⊆ Rn, and a positive Radon measure μ on Ω, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains Ωh ⊇ Ω with the following property: for every f ∈ H−1(Ω) the sequence uh of the solutions of the Dirichlet problems Luh = f in Ωh, uh = 0 on ∂Ωh, extended to 0 in Ω\Ωh, converges to the solution of the “relaxed Dirichlet problem” Lu + μu = f in Ω, u = 0 on ∂Ω.

Homogenization of nonlinear Dirichlet problems in random perforated domains

2016 ◽  
Vol 133 ◽  
pp. 250-274 ◽  
Author(s):  
Carmen Calvo-Jurado ◽  
Juan Casado-Díaz ◽  
Manuel Luna-Laynez
Keyword(s):  
Dirichlet Problems ◽  
Perforated Domains
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