New PDF release: Asymptotic Theory of Elliptic Boundary Value Problems in

By Vladimir Maz'ya, Serguei Nazarov, Boris Plamenevskij

ISBN-10: 3034884346

ISBN-13: 9783034884341

ISBN-10: 3034895658

ISBN-13: 9783034895651

For the 1st time within the mathematical literature this two-volume paintings introduces a unified and basic method of the asymptotic research of elliptic boundary worth difficulties in singularly perturbed domain names. this primary quantity is dedicated to domain names whose boundary is tender in the community of finitely many conical issues. particularly, the idea encompasses the real case of domain names with small holes. the second one quantity, nonetheless, treats perturbations of the boundary in better dimensions in addition to nonlocal perturbations.
The center of this booklet involves the answer of normal elliptic boundary price difficulties by way of entire asymptotic growth in powers of a small parameter that characterizes the perturbation of the area. the development of this technique capitalizes at the idea of elliptic boundary price issues of nonsmooth boundary that has been constructed some time past thirty years.
Much cognizance is paid to concrete difficulties in mathematical physics, for instance in elasticity conception. specifically, a examine of the asymptotic habit of pressure depth elements, strength integrals and eigenvalues is presented.
To a wide quantity the ebook relies at the authors’ paintings and has no major overlap with different books at the idea of elliptic boundary price problems.

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Additional resources for Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains: Volume I

Example text

Can also be obtained by passing to the limit a --+ ao. We do not present the corresponding formulas; we only remark that the new expression (32), in the case of the functions z~+1), zii+ 2), ... , provides necessary differences that are required for compensation of the zero in denominator, which appears when passing to the limit. As a consequence for z~+h) (h EN), we obtain a representation of the form z~+k) = (ct,h log r + Dt,k)ri+ 2k sin(jfJ + k7r /2) with some constants ct,k and Di,k. 18, one can find the asymptotic decomposition of the solution of problem (23) also for a right-hand side of the form (21).

Here the point 0 has to be treated as the peak of the sector ]R2 \ {O}. In a similar manner, boundary value problems in the exterior of a bounded domain can be considered, for which the problem in ]R2 \ {O} serves also as a model. 1 Dirichlet and Neumann problems in a punctured planar domain Let n be a bounded planar domain containing the origin 0 8n. v(x) = f(x), x E with a smooth boundary n \ {O} (1) with boundary conditions v(x) = 0, (2) x E 8n or (fJ/fJv)v(x) = 0, (3) x E fJn. 1. The eigenvalues are Ak = ik (k E Z).

Ck(J), V + rJ k L Cq(J) (q7r)-1/2 rq7r/a sin(q7r(20 + a )/2a) E V~,'Y(O). (9) q=l Proof. We start from the relation Ll(rJv) = rJf + 2'VrJ'Vv + LlrJv = g E V~''Y(Ka). 2 with the right-hand side g belonging to 0 VL(Ka ). This yields relation (9). 8. The coefficients cq in the asymptotic formula (9) satisfy the condition Cq(J) = J f(x)(q(x)dx, (q = 1, ... , k). (11) n Proof. We apply Green's formula to the solution v E V~,,6(O) and the function (q in the domain 0 6 = {x EO: Ixl > 8}. Then we obtain f ((q(x)Llv(x) - v(x)Ll(q(x))dx = ~ f ((q(x)(8/8v)v(x) - v(x) (8/8v)(q (x))ds x .

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Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains: Volume I by Vladimir Maz'ya, Serguei Nazarov, Boris Plamenevskij


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