S. A. Nazarov

Asymptotic analysis of shape functionals

Abstract A family of boundary value problems is considered in domains Ω(e)=Ω⧹ ω e ⊂ R n , n⩾3, with cavities ωe depending on a small parameter e∈(0,e0]. An approximation U (e,x) , x∈Ω(e) , of the solution u(e,x), x∈Ω(e) , to the boundary value problem is obtained by an application of the methods of matched and compound asymptotic expansions. The asymptotic expansion is constructed with precise a priori estimates for solutions and remainders in Holder spaces, i.e., pointwise estimates are established as well. The asymptotic solution U (e,x) is used in order to derive the first term of the asymptotic expansion with respect to e for the shape functional J (Ξ(e))= J e (u)≅ J e ( U ) . In particular, we obtain the topological derivative T (x) of the shape functional J (Ξ) at a point x∈Ω . Volume and surface functionals are considered in the paper.

On the spectrum of the Steklov problem in a domain with a peak

We consider the spectral Steklov problem in a domain with a peak on the boundary. It is shown that the spectrum on the real nonnegative semi-axis can be either discrete or continuous depending on the sharpness of the exponent.

The Localization Effect for Eigenfunctions of the Mixed Boundary Value Problem in a Thin Cylinder with Distorted Ends

A simple sufficient condition on a curved end of a straight cylinder is found that provides a localization of the principal eigenfunction of the mixed boundary value for the Laplace operator with the Dirichlet conditions on the lateral side. Namely, when the small parameter, i.e., the ratio between the diameter and the length of the cylinder, tends to zero, the eigenfunction concentrates in the vicinity of the ends and decays exponentially in the interior. Similar effects are observed in the Dirichlet and Neumann problems, too.

Topological Derivatives for Semilinear Elliptic Equations

Topological Derivatives for Semilinear Elliptic Equations The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L∞ norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.

Polarization matrices in anisotropic heterogeneous elasticity

Polarization matrices (or tensors) are generalizations of mathematical objects like the harmonic capacity or the virtual mass tensor. They participate in many asymptotic formulae with broad applications to problems of structural mechanics. In the present paper polarization matrices for anisotropic heterogeneous elastic inclusions are investigated, the ambient anisotropic elastic space is allowed to be inhomogeneous near the inclusion as well. By variational arguments the existence of unique solutions to the corresponding transmission problems is proved. Using results about elliptic problems in domains with a compact complement, polarization matrices can be properly defined in terms of certain coefficients in the asymptotic expansion at infinity of the solution to the homogeneous transmission problem. Representation formulae are derived from which properties like positivity or negativity can be read of directly. Further the behavior of the polarization matrix is investigated under small changes of the interface.

A gap in the spectrum of the Neumann–Laplacian on a periodic waveguide

We study a spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder which contains a periodic arrangement of inclusions. On the boundary of the waveguide, we consider both Neumann and Dirichlet conditions. We prove that provided the diameter of the inclusion is small enough the spectrum of Laplace operator contains band gaps, i.e. there are frequencies that do not propagate through the waveguide. The existence of the band gaps is verified using the asymptotic analysis of elliptic operators.

Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary

The Neumann problem for the Poisson equation is considered in a domain Ωε ⊂ R with boundary components posed at a small distance ε > 0 so that in the limit, as ε → 0, the components touch each other at the point O with the tangency exponent 2m ≥ 2. Asymptotics of the solution uε and the Dirichlet integral ‖∇xuε; L(Ωε)‖ are evaluated and it is shown that main asymptotic term of uε and the existence of the finite limit of the integral depend on the relation between the spatial dimension n and the exponent 2m. For example, in the case n < 2m− 1 the main asymptotic term becomes of the boundary layer type and the Dirichlet integral has no finite limit. Some generalization are discussed and certain unsolved problems are formulated, in particular, non-integer exponents 2m and tangency of the boundary components along smooth curves. AMS subject classification: primary 35J25, secondary 46E35,35J20.

Asymptotic model of interaction of blood flow with vein walls and the surrounding muscular tissue

In this study, we propose a model of venous-mus� cular blood pumping. Contrary to an artery (see the asymptotic model (1)), where rapid and abundant blood flow is caused by a large pressure difference caused by the operation of the heart, the bloodreturn processes proceed in a sloweddown mode in veins at insignificant pressure differences (7-10 times lower than that in arteries). To a great degree, this takes place due to external actions: the gravitational forces (favor� ing the blood outflow from brain, but preventing its raising in lower limbs of orthograde species), the expansion-compression of lungs in the breast, and other muscular activity of human beings or animals (see (2)).

Korn's inequality for periodic solids and convergence rate of homogenization

In a three-dimensional solid with arbitrary periodic Lipschitz perforation the Korn inequality is proved with a constant independent of the perforation size. The convergence rate of homogenization as a function of the Sobolev–Slobodetskii smoothness of data is also estimated. We improve foregoing results in elasticity dropping customary restrictions on the shape of the periodicity cell and superfluous smoothness and smallness assumptions on the external forces and traction.

Artificial Boundary Conditions for the Stokes and Navier–Stokes Equations in Domains that are Layer-Like at Infinity

Artificial boundary conditions are presented to approximate solutions to Stokesand Navier-Stokes problems in domains that are layer-like at infinity. Based on results about existence and asymptotics of the solutions v∞, p∞ to the problems in the unbounded domain Ω the error v∞−vR, p∞−pR is estimated in H(ΩR) and L(ΩR), respectively. Here v, p are the approximating solutions on the truncated domain ΩR, the parameter R controls the exhausting of Ω. The artificial boundary conditions involve the Steklov-Poincare operator on a circle together with its inverse and thus turn out to be a combination of local and nonlocal boundary operators. Depending on the asymptotic decay of the data of the problems, in the linear case the error vanishes of order O(R−N ), where N can be arbitrarily large.