H convergence for quasi-linear elliptic equations with quadratic growth
Bensoussan, Alain; Boccardo, L.; Murat, F. (1992), H convergence for quasi-linear elliptic equations with quadratic growth, Applied Mathematics and Optimization, 26, 3, p. 253-272. http://dx.doi.org/10.1007/BF01371084
TypeArticle accepté pour publication ou publié
Journal nameApplied Mathematics and Optimization
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Abstract (EN)We consider in this paper the limit behavior of the solutionsu ɛ of the problem −div(aεDuε)+γuε=Hε(x,uε,Duε),uε∈H10(Ω)∩L∞(Ω), whereH ɛ has quadratic growth inDu ɛ anda ɛ (x) is a family of matrices satisfying the general assumptions of abstract homogenization. We also consider the problem −div(aεDuε)+Gε(x,uε,Duε)=f∈H−1(Ω),uε∈H10(Ω),Gε(x,uε,Duε)∈L1(Ω),uεGε(x,uε,Duε)∈L1(Ω) whereG ɛ has quadratic growth inDu ɛ and satisfiesG ɛ (x, s, ξ)s ≥ 0. Note that in this last modelu ɛ is in general unbounded, which gives extra difficulties for the homogenization process. In both cases we pass to the limit and obtain an homogenized equation having the same structure.
Subjects / KeywordsHomogenization; G-convergence; H-convergence; Quasi-linear PDE
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