On the Precise Asymptotics of the Constant in Friedrich's Inequality for Functions Vanishing on the Part of the Boundary with Microinhomogeneous Structure

We study the asymptotic behavior of solutions to a boundary value problem for the Poisson equation with a singular right-hand side, singular potential and with alternating type of the boundary condition. Assuming that the boundary microstructure is periodic, we construct the limit problem and prove the homogenization theorem by means of the unfolding method. The proof requires that the dimension be larger than two. RésuméLe but de cet article est d'étudier le comportement asymptotique des solutions d'une équation de Poisson avec un potentiel et un membre de droite singuliers et des conditions aux limites oscillantes. Le problème aux limites est posé dans un domaine de R n , n 3. Sous l'hypothèse que la microstructure de la frontière est périodique, on démontre un théorème d'homogénéisation en utilisant la méthode d'éclatement périodique.

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“…On the other hand, by [8], there is a Poincaré-Friedrichs inequality in the set B for functions vanishing on D ∩ B. By scaling, it follows that…”

Section: Proposition 31 There Is a Constant C Independent Of ε And mentioning

confidence: 98%

On boundary value problem with singular inhomogeneity concentrated on the boundary

Chechkin

Cioranescu

Damlamian

et al. 2012

Journal de Mathématiques Pures et Appliquées

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“…The uniform boundedness of the family of operators in the corresponding operator norm sup ε A ε L (H ε ) < M easily follows from the integral identity (2.4), the Cauchy-Bunyakowsky-Schwarz inequality, and a Friedrichs type inequality (cf. [40]). We have…”

Section: )mentioning

confidence: 99%

Plane Scalar Analog of Linear Degenerate Hydrodynamic Problem with Nonperiodic Microstructure of Free Surface

Чечкин

Chechkina

2015

J Math Sci

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We consider the degenerate problem on low-frequency oscillations of a heavy viscous incompressible fluid in a vessel with free surface of inhomogeneous nonperiodic microstructure. For the obtained quadratic operator pencil we construct the limit (homogenized) pencil which turns out to be degenerate. We prove a homogenization theorem. Bibliography: 40 titles. Illustration: 2 figures. Dedicated to Nina Nikolaevna UraltsevaA scalar analog of linear hydrodynamics was first described in [1] in detail. Such problems with surfaces of microinhomogeneous periodic structure were considered in [2]-[4], where homogenization of an operator pencil was studied. This study was based on homogenization methods (cf., for example, [5]-[13] and the references therein), asymptotic analysis, and matching asymptotic expansions (cf., for example, [14]). were used. Owing to this approach, it becomes possible to construct effective models of inhomogeneous media or study singular perturbations of problems without microinhomogeneities; moreover, these methods provide rigorous justifications of constructions and accurate proofs. In this direction, various problems with singular perturbation were considered, in particular, perturbation of geometry (cf., for example, [7,12], [15]-[17]), coefficients (cf. [18, 19]), and the type of boundary condition (cf., for example, [20]-[27] and [13]). Similar methods were used in hydrodynamics for studying strongly inhomogeneous fluids with different rheology (cf., for example, [28]-[39]).In this paper, we study a plane analog of the problem on low-frequency oscillations of a heavy viscous incompressible fluid in a vessel with free surface covered by a perforated cap with nonperiodic structure (cf. Figure 1).

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“…What has not been treated systematically is the case where only a part D of the boundary of the underlying domain Ω is involved, reflecting the Dirichlet condition of the equation on this part -while on ∂Ω \ D other boundary conditions may be imposed, compare [11,26,2,24,8] including references therein. The aim of this paper is to set up a geometric framework for the domain Ω and the Dirichlet boundary part D that allow to deduce the corresponding Hardy inequality…”

Section: Introductionmentioning

confidence: 99%

Hardy’s Inequality for Functions Vanishing on a Part of the Boundary

2015

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On the Precise Asymptotics of the Constant in Friedrich's Inequality for Functions Vanishing on the Part of the Boundary with Microinhomogeneous Structure

Cited by 20 publications

References 9 publications

On boundary value problem with singular inhomogeneity concentrated on the boundary

On boundary value problem with singular inhomogeneity concentrated on the boundary

Plane Scalar Analog of Linear Degenerate Hydrodynamic Problem with Nonperiodic Microstructure of Free Surface

Hardy’s Inequality for Functions Vanishing on a Part of the Boundary

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