The asymptotics of the solution to the Neumann spectral problem in a domain of the “dense-comb” type

Mel’nik, T. A.; Назаров, С. А.

doi:10.1007/bf02355841

Cited by 57 publications

(53 citation statements)

References 8 publications

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“…If l = 0 or if g has mean value zero with respect to x N , u 0 coincides with the trace on Σ of the limit solution u in Ω + , while, if l ∈]0, +∞[ and g does not have mean value zero with respect to x N , it coincides with the trace of u on Σ only in Ω × ω. In the complementary set, it solves a system depending on l, which is an algebraic equation (see the third equation in (19)) coupled with a diffusion equation with respect to the y variable, involving, as boundary condition on ∂ ω, the trace of u on Σ (see the first two equations in (19)). Also, as for the dependance of the limit problem with respect to a, it may be surprising to notice that the limit problem does not depend on the restriction of a to Ω − × ω × R N .…”

Section: Resultsmentioning

confidence: 99%

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Limit behavior of thin heterogeneous domain with rapidly oscillating boundary

Mossino

Sili

2007

Ricerche mat.

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We give a generalization of the work presented in [6] where the asymptotic behaviour, as ε → 0, of a monotone nonlinear problem in a bounded multidomain of R N depending on ε was addressed. We extend the previous results to the case where the nonlinear operator depends both on the slow and rapid variable and we prove that, due to the presence of the rapid variable, the algebraic equation contained in the limit problem obtained in [6] must be replaced by a partial differential equation with respect to the microscopic variable y .

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Section: Resultsmentioning

confidence: 99%

“…For the study of highly oscillating boundaries we refer to [2], [3], [4], [7], [9], [10], [11], [14], [16], [18], [19] and [21]. For the junction of a plate with a beam we refer to [12], [13] and [15] and for the study of heterogeneous fibers we refer to [22].…”

mentioning

confidence: 99%

Limit behavior of thin heterogeneous domain with rapidly oscillating boundary

Mossino

Sili

2007

Ricerche mat.

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“…For these problem we do not need an additional condition like the condition (4) (see References [1][2][3]6]). …”

Section: Resultsmentioning

confidence: 99%

“…In References [1][2][3][4][5][6][7], convergence theorems were proved, the leading terms of asymptotics were constructed, and asymptotic estimates (as → 0) were proved for boundary value problems in thick junctions of di erent types. Also the in uence of boundary conditions (Neumann, Fourier, Steklov, Dirichlet) and the in uence of the geometric conÿguration of thick junctions (two bodies, di erent lengths of the joined thin domains) on the asymptotic behaviour of solutions were investigated.…”

Section: Reviewmentioning

confidence: 99%

Homogenization of the Neumann problem in thick multi‐structures of type 3 : 2 : 2

Maio

Mel'nyk

2004

Math Methods in App Sciences

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“…To obtain (11) we have to integrate by parts the last integral in (11) and take into account the boundary condition for Y and coordinates of the outward normal to the lateral surfaces of each of the cylinders G (i, j), i, j = 1,. . .…”

Section: Auxiliary Uniform Estimatesmentioning

confidence: 99%

Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem

Mel'nyk¹,

Nakvasiuk²,

Wendland³

2010

Math. Meth. Appl. Sci.

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We consider a mixed boundary-value problem for the Poisson equation in a thick junction X e which is the union of a domain X 0 and a large number of e-periodically situated thin cylinders. The non-uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as e → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non-uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as e → 0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non-standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in X 0 and an appropriate postprocessing. The equations in X 0 finally are also treated with boundary integral equations.

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The asymptotics of the solution to the Neumann spectral problem in a domain of the “dense-comb” type

Cited by 57 publications

References 8 publications

Limit behavior of thin heterogeneous domain with rapidly oscillating boundary

Limit behavior of thin heterogeneous domain with rapidly oscillating boundary

Homogenization of the Neumann problem in thick multi‐structures of type 3 : 2 : 2

Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem

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