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Download Asymptotic Theory of Elliptic Boundary Value Problems in by Vladimir Maz'ya, Serguei Nazarov, Boris Plamenevskij, B. PDF

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 common 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 better dimensions in addition to nonlocal perturbations.
At the middle of this publication are ideas of elliptic boundary worth difficulties by way of asymptotic growth in powers of a small parameter that characterizes the perturbation of the area. particularly, it treats the real certain circumstances of skinny domain names, domain names with small cavities, inclusions or ligaments, rounded corners and edges, and issues of speedy oscillations of the boundary or the coefficients of the differential operator. The tools offered the following capitalize at the thought of elliptic boundary price issues of nonsmooth boundary that has been built long ago thirty years.
Moreover, a research at the homogenization of differential and distinction equations on periodic grids and lattices is given. a lot recognition is paid to concrete difficulties in mathematical physics, fairly elasticity thought and electrostatics.
To a wide quantity the e-book is predicated at the authors’ paintings and has no major overlap with different books at the idea of elliptic boundary worth problems.

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

Example text

Let JI Td s be the torus {x log lyll- 2 Iu(x)1 2 Iyl- 2dx :::; Proof. 4, JI dRdT = 4- 1 log lyll- 2 Iu(xW lyl- 2dx:::; J Iv JI where X is the cut-off function (11) and J lu(x)1 2 Ixl- 2dx. 3:

1 on the problem in a domain with a point deleted. Therefore, the condition I + max{O, 1 - 7f19 01 } < (3 < I + 1 is replaced by I < (3 < I + 1. 6. 7. Let I be a nonnegative integer and (3 E (l, 1+ 1). Let also f E V~,i3(O; M), ip E W;+3/2 (aO). 8. 3 is true. ) 1+3/2 (::> A .. V- 1+2 uO ) . 3 The Asymptotics Near an Edge of Solutions to Boundary Value Problems for the Laplace Operator In this section we derive the asymptotic formulas for solutions to the boundary value problems considered in the two preceding sections.

We seek vel) and W(l) that give the second approximation v(O) (x) + E(x(r)w(O) (

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