Junctions of singularly degenerating domains with different limit dimensions. II

This paper is aimed at finding asymptotic formulas for solutions to the mixed boundary problem for the Poisson equation in a domain obtained by joining singularly degenerating domains. In this paper, which is the second part of the work (the first part was published in No. 18), the main attention is given to three-dimensional problems in which a thin plate or a periodic family of thin rods is joined to a massive body (the distances between the rods are comparable with the diameters of their cross-sections). The distinctive feature of such problems is that an integral equation arises as one of the limit problems. Bibliography: 48 titles. Translated from Trudy Seminara imeni l. G. Petrovskogo, No. 20, pp. 155–195, 1997.     

Open waveguides in doubly periodic junctions of domains with different limit dimensions

Considering the spectral Neumann problem for the Laplace operator on a doubly periodic square grid of thin circular cylinders (of diameter ε ? 1) with nodes, which are sets of unit size, we show that by changing or removing one or several semi-infinite chains of nodes we can form additional spectral segments, the wave passage bands, in the essential spectrum of the original grid. The corresponding waveguide processes are localized in a neighborhood of the said chains, forming I-shaped, V-shaped, and L-shaped open waveguides. To derive the result, we use the asymptotic analysis of the eigenvalues of model problems on various periodicity cells.     

The incompressible limit of solutions of the two-dimensional compressible Euler system with degenerating initial data

Using Strichartz estimates, it is possible to pass to the limit in the weakly compressible 2-D Euler system, when the Mach number ε tends to zero, even if the initial data are not uniformly smooth. This leads to results of convergence to solutions of the incompressible Euler system whose regularity is critical, such as vortex patches or Yudovich solutions. To cite this article: A. Dutrifoy, T. Hmidi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Unique solvability of singularly perturbed boundary-value problems with unstable spectrum of the limit operator

The paper contains an asymptotic study of solutions of singularly perturbed boundary-value problems in the case of a limit operator with unstable spectrum. The main singularities of the solution are described and the principal term of the asymptotics is constructed.     

Asymptotic analysis of a singularly perturbed linear system with a singular limit pencil of matrices

With the help of perturbation methods and Newton diagrams, an asymptotic analysis is conducted of the general solution of a linear singularly perturbed system of ordinary differential equations in the case of degeneracy of a matrix multiplying the derivative in the approach of a small parameter to zero. It is assumed that the pencil of limit matrices of the system is singular and possesses a minimal index for rows and columns.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 106–122, January, 1992.     

Multiple positive solutions of singularly perturbed differential systems with different orders

In this paper, we consider the multiplicity of positive solutions for a class of singular higher-order perturbed differential systems with different orders. By employing a well-known fixed point theorem, some new existence results are given under the case where nonlinearity can be sign changing.     

Eigenvalues of Euclidean wedge domains in higher dimensions

In this paper, we use a weighted isoperimetric inequality to give a lower bound for the first Dirichlet eigenvalue of the Laplacian on a bounded domain inside a Euclidean cone. Our bound is sharp, in that only sectors realize it. This result generalizes a lower bound of Payne and Weinberger in two dimensions.     

The incompressible limit of solutions of the two‐dimensional compressible Euler system with degenerating initial data

Using Strichartz estimates, it is possible to pass to the limit in the weakly compressible 2‐D Euler system, when the Mach number ? tends to zero, even if the initial data are not uniformly smooth. More precisely, their norms in Sobolev spaces embedded in C¹ can be allowed to grow as small powers of ?^?1. This leads to results of convergence to solutions of the incompressible Euler system whose regularity is critical, such as vortex patches or Yudovich solutions. © 2000 Wiley Periodicals, Inc.     

Inverse singularly perturbed problems of the convection-diffusion type in quadrangular curvilinear domains

We have constructed an algorithm for the asymptotic approximation of the solutions of inverse singularly perturbed boundary-value problems of the convection-diffusion type with unknown diffusion coefficient, depending on the coordinates of a quadrangular curvilinear domain of filtration. The case of sufficient smoothness and consistency of the overdetermination, initial, and boundary conditions is considered. Unlike the construction of an algorithm for the solution of similar problems in doubly connected domains, here, in the corresponding relations, there appear corrections taking into account the influence of “lateral sources of pollution.” With the help of this algorithm, we have carried out a computer experiment, the results of which confirm the well-known fact of “strong sensitivity” of the model to assignment of the overdetermination condition. In particular, we have revealed the specific character of influence of this condition on the required diffusion coefficient depending on the filtration velocity.     

Concentration of solutions for a singularly perturbed Neumann problem in non-smooth domains

We consider the equation −?²Δu+u=up in a bounded domain Ω⊂R³ with edges. We impose Neumann boundary conditions, assuming 1<p<5, and prove concentration of solutions at suitable points of ∂Ω on the edges.     

On the structure of solutions of a system of singularly perturbed differential equations with nondiagonalizable limit operator

We construct uniform asymptotics of a solution of a heterogeneous system of singularly perturbed differential equations in the case of nondiagonalizable limit operator. We consider the case where the spectrum of the limit operator contains an unstable element at the point x = 0. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 867–876, July, 1998.     

Asymptotic behavior of the eigenvalues of the Steklov problem on a junction of domains of different limiting dimensions

The asymptotic behavior of eigenvalues and eigenfunctions of the Steklov problem on a junction of rectangles: a thin rectangle with a width of ? > 0 and a rectangle with unit dimensions, is studied. In addition to asymptotic formulas for the main series of eigenvalues (in the low-frequency region), other series with stable characteristics are found in the medium-frequency region and explicit formulas for the correction terms are derived. In the framework of the linear theory of surface waves, the results of this work describe the effect of wave localization in shallow water.     

On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries

We study the asymptotic behavior of solutions to the incompressible Navier-Stokes system considered on a sequence of spatial domains, whose boundaries exhibit fast oscillations with amplitude and characteristic wave length proportional to a small parameter. Imposing the complete slip boundary conditions we show that in the asymptotic limit the fluid sticks completely to the boundary provided the oscillations are non-degenerate, meaning not oriented in a single direction.     

The scaling limit of loop-erased random walk in three dimensions

We show that the scaling limit exists and is invariant under dilations and rotations. We give some tools that might be useful to show universality.     

Grid approximation of singularly perturbed parabolic reaction-diffusion equations on large domains with respect to the space and time variables

In an unbounded (with respect to x and t) domain (and in domains that can be arbitrarily large), an initial-boundary value problem for singularly perturbed parabolic reaction-diffusion equations with the perturbation parameter ε² multiplying the higher order derivative is considered. The parameter ε takes arbitrary values in the half-open interval (0, 1]. To solve this problem, difference schemes on grids with an infinite number of nodes (formal difference schemes) are constructed that converge ε-uniformly in the entire unbounded domain. To construct these schemes, the classical grid approximations of the problem on the grids that are refined in the boundary layer are used. Schemes on grids with a finite number of nodes (constructive difference schemes) are also constructed for the problem under examination. These schemes converge for fixed values of ε in the prescribed bounded subdomains that can expand as the number of grid points increases. As ε → 0, the accuracy of the solution provided by such schemes generally deteriorates and the size of the subdomains decreases. Using the condensing grid method, constructive difference schemes that converge ε-uniformly are constructed. In these schemes, the approximation accuracy and the size of the prescribed subdomains (where the schemes are convergent) are independent of ε and the subdomains may expand as the number of nodes in the underlying grids increases.     

Asymptotic analysis of problems in junctions of domains of different limit dimension. An elastic body pierced by thin rods

The leader term of an asymptotic expansion of a solution to the problem of elasticity about a body pierced by several parallel rods is considered. The role of parameters is played by the diameters (small parameters) of the rods and the ratios (large parameters) of the rigidity of the material of the rods to that of the body. The influence of the body on the rods and interactions between the rods are studied. The asymptotic formulas are justified by means of a special (weighted) Korn inequality. Bibliography: 37 titles. Illustrations: 1 figure. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 3–55.