Spectral problems in singularly perturbed domains and selfadjoint extensions of differential operators

Kamotski, I. V.; Назаров, С. А.

doi:10.1090/trans2/199/04

Cited by 30 publications

(33 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The known results are given in particular for singular perturbations of isolated points of the boundary (small holes in the domain, see [14], [15], [5], [1], [13], [35] and others), perturbations of straight boundaries including perturbations by changing the type of boundary conditions (cf. [2]- [3]), and the dependence of the obtained results in more general geometrical domains on the curvature is clarified in [8,22,23] in the case of scalar equations.…”

Section: Asymptotic Analysis Of Eigenvalues In Singularly Perturbed Dmentioning

confidence: 99%

See 1 more Smart Citation

On asymptotic analysis of spectral problems in elasticity

Sokołowski

2011

Lat. Am. j. solids struct.

View full text Add to dashboard Cite

The three-dimensional spectral elasticity problem is studied in an anisotropic and inhomogeneous solid with small defects, i.e., inclusions, voids, and microcracks. Asymptotics of eigenfrequencies and the corresponding elastic eigenmodes are constructed and justified. New technicalities of the asymptotic analysis are related to variable coefficients of differential operators, vectorial setting of the problem, and usage of intrinsic integral characteristics of defects. The asymptotic formulae are developed in a form convenient for application in shape optimization and inverse problems. Keywords singular perturbations; spectral problem; asymptotics of eigenfunctions and eignevalues; elasticity boundary value problem.

show abstract

Section: Asymptotic Analysis Of Eigenvalues In Singularly Perturbed Dmentioning

confidence: 99%

“…The coefficients M j kp in (25) form the (6 × 6)-matrix M j which is called the polarization matrix of the elastic inclusion ω j (see [28,41] and also [[20]; Ch. 6], [5], [21]). Some properties of the polarization matrix, and some comments on the solvability of problem (23) are given in section 4.…”

Section: Formal Construction Of Asymptoticsmentioning

confidence: 99%

On asymptotic analysis of spectral problems in elasticity

Sokołowski

2011

Lat. Am. j. solids struct.

View full text Add to dashboard Cite

show abstract

“…We consider the functional r Asymptotic structures for specific problems with the logarithmic growth of fundamental solutions, turn out to be quite complex and therefore, of limited practical interest for analysis of functional (9). The main particularity of the asymptotic analysis, beside the presence of boundary layers near the arcs yk, ..., ^(hI, is the form of asymptotic terms which are rational functions of the large parameter I In hl.…”

mentioning

confidence: 99%

“…On the other hand, the proposed singularities are not included in the energy class H 1 however the resulting singular solutions are still in the space L q ( 0 ) , for q 2 I. The main profit from our point of view for such modeling is the possibility, with the singular solutions, for asymptotically exact approximation of functional (9), under the condition that for some q E [I, CQ) and for any u, v E ~~( 0 )~ the following inequality is valid with the constant CF independent of h E (0, ho] and u, v. Modeling defects in media by an application of extensions of differential operators which give rise to the singular solutions comes back to the work [3] and is developped in [20], [16], [MI, [17], [9] and in other publications, for problems of mathematical physics and general elliptic systems. There are two possibilities for realization of such ideas.…”

mentioning

confidence: 99%

“…In this way the domain of the adjoint becomes wider, and finally the selfadjoint operator is selected in the form of an intermediate operator. Under the proper choice of extension parameters, the selfadjoint operator asymptotically acquires the attributes of singularly perturbed problem such as the energy functional and the spectre (see [17], [9], [19]). Since the selected operator is selfadjoint, the classical semigroup theory can be used to construct solutions for the associated evolution problems.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Modeling of Topology Variations in Elasticity

Назаров

Sokołowski

IFIP International Federation for Information Processing

View full text Add to dashboard Cite

Two approaches are proposed for the modeling of deformation of elastic solids with small geometrical defects. The first approach is based on the theory of self adjoint extensions of differential operators. In the second approach function spaces with separated asymptotics and point asymptotic conditions are introduced, and the variational formulation is established. For both approaches the accuracy estimates are derived. Finally, the spectral problems are considered and the error estimates for eigenvalues are given.

show abstract

Nonlinear artificial boundary conditions for the Navier‐Stokes equations in an aperture domain

Назаров

Specovius–Neugebauer

Videman³

2004

Mathematische Nachrichten

View full text Add to dashboard Cite

We consider the Dirichlet problem for the stationary Navier-Stokes system in a plane domain Ω, with two angular outlets to infinity. It is known that, under appropriate decay and smallness assumptions, this problem admits solutions with main asymptotic terms in Jeffrey-Hamel form. We will approach these solutions by constructing an approximating problem in the domain ΩR, which is the intersection of Ω with a sufficiently large circle. The main difficulty, in contrast to the corresponding linear problem, arises from the fact that the main asymptotic term is not known explicitly. Here, we create nonlinear, but local, artificial boundary conditions which involve second order differential operators on the truncation arcs. Unlike for the analogous three-dimensional exterior problem, we are able to show the existence of weak solutions to the approximating problem without smoothness nor smallness assumptions. For small data, we prove that the solutions of the approximating problem are unique and regular. Finally, we reach the main goal of this work, i.e. we obtain error estimates in weighted Hölder spaces which are asymptotically precise as R tends to infinity. Formulation of the problems and preliminary description of the resultsLet Ω be a plane domain with two angular outlets to infinity. More precisely, let us suppose that outside a circle B R 0 of radius R 0 > 0, Ω coincides with the union of anglesHere (r i , ϕ i ) denote polar coordinates centered at the vertex of the angle K i and θ i ∈ (0, π] is the opening of K i . We assume that the boundary ∂Ω consists of two simple smooth infinite curves Γ + and Γ −

show abstract

Spectral problems in singularly perturbed domains and selfadjoint extensions of differential operators

Cited by 30 publications

References 0 publications

On asymptotic analysis of spectral problems in elasticity

On asymptotic analysis of spectral problems in elasticity

Modeling of Topology Variations in Elasticity

Nonlinear artificial boundary conditions for the Navier‐Stokes equations in an aperture domain

Contact Info

Product

Resources

About