By Vladimir Maz'ya, Serguei Nazarov, Boris Plamenevskij, B. Plamenevskij

For the 1st time within the mathematical literature this two-volume paintings introduces a unified and normal method of the asymptotic research of elliptic boundary worth difficulties in singularly perturbed domain names. whereas the 1st quantity is dedicated to perturbations of the boundary close to remoted singular issues, this moment quantity treats singularities of the boundary in larger dimensions in addition to nonlocal perturbations.

At the center of this booklet are options of elliptic boundary price difficulties by way of asymptotic growth in powers of a small parameter that characterizes the perturbation of the area. particularly, it treats the real targeted circumstances of skinny domain names, domain names with small cavities, inclusions or ligaments, rounded corners and edges, and issues of fast oscillations of the boundary or the coefficients of the differential operator. The equipment awarded the following capitalize at the concept of elliptic boundary worth issues of nonsmooth boundary that has been constructed long ago thirty years.

Moreover, a examine at the homogenization of differential and distinction equations on periodic grids and lattices is given. a lot consciousness is paid to concrete difficulties in mathematical physics, really elasticity thought and electrostatics.

To a wide volume the publication relies at the authors’ paintings and has no major overlap with different books at the idea of elliptic boundary worth problems.

**Read or Download Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains: Volume II PDF**

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**Extra info for Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains: Volume II**

**Sample text**

Ix - ~1-llyl-l d'Pdy. 1. If Izl ~ 1, then from (29) it follows that IZ2(C,~)1 ::; C6 c. In the case Izl < 1 the last integral in (29) does not exceed JJ C 7c M (Iy - zl + 1'1' - TI)-llyl-l d'Pdy ::; Csc J (1 + Ilog Iy - zll) Iyl-l dy. B3/ 2 B3/2 The integral on the right can be represented as the sum h + 12 of the integrals over the sets B3/2 n {y: Iy - zl ::; Izl/2} and B3/2 \ {y: Iy - zl ::; Izl/2}. 2 .... the Dirichlet Problem in the Exterior of a Thin Tube 41 Since the origin of the coordinate Y = (Yl, Y2) belongs to We' the inequality Ilog Iz II :S Cd log el holds for z E B2 \ ne' Using the above estimates, we arrive at (26).

This solution is determined up to an additive constant c and admits the representation v(1]) = cdogl1]1 +C+Vl(1]), with Cl = cl((F, (W). We take C = O. 6, ICII + Ilvl; V;;-J(G)II :::; c(IIF; Let UI = VI E V~;-J(G) (VI E~,i3(K)11 + Ilw; E;;-J/ 2(oK)II). UI (1]) - UI (1]), (ojov)U2(1]) = W(1]) - (ojov)UI (1]), 1] E 1] E oK. K; (13) 12 11. Boundary Value Problems in Domains with Edges on the Boundary It is clear that F + ~UI = 0 in K near the point 0 and W - BUt! 8v = 0 in 8K \ {O} near O. According to (12), IlUI; E~;J(K)II + IIF; E&,/1(K) II + II*; E;~;/2(8K)11 :::; c(IIF;E&,/1(K)11 + Ilw;E;~;/2(8K)II), where F and stand for the right-hand side of problem (13). *

1, satisfy (12) and (13). 1, admits the representation J u(y, z) = L Uj (19, logr, z) rAJ + uJ (y, z) j=O with any nonnegative integer J and polynomials Uj in log r whose coefficients smoothly depend on 19 and z. The remainder uJ is subject to (8 k uJ/8z k ) (', z) E V{t~l-AJG) and the sequence {Aj} consists of numbers of the form +h ttj where j, h or kJr't9r;l + h, (13) = 0,1, ... , k = 1,2, .... 2. 3. Assume that f satisfies (10) and (12) and'IjJ admits the expanswn J 'IjJ(y,z) = L 'ljJj ('t9,logr,z)r iLJ - 1 + ¢J (y,z), j=O while (8 k ;j; J /8z k ) (-, z) E V;~~LiLJ (8G) and the coefficients 'ljJj possess the same properties as