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 price 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 e-book are ideas of elliptic boundary worth difficulties via asymptotic growth in powers of a small parameter that characterizes the perturbation of the area. particularly, it treats the $64000 designated circumstances of skinny domain names, domain names with small cavities, inclusions or ligaments, rounded corners and edges, and issues of swift oscillations of the boundary or the coefficients of the differential operator. The equipment provided right here capitalize at the idea of elliptic boundary worth issues of nonsmooth boundary that has been built some time past 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, quite elasticity conception and electrostatics.

To a wide quantity the booklet 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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M) 'P E Vl+ 2,/'1 U~ G, (7) max b, f3 - I}. , Z ) E Vi2,/'1 (G) ' I +3 / 2 (aG) (. z) E V 2,/'1 'P, for almost all z EM. 7 on the asymptotics of solutions to the Dirichlet problem in a plane domain with corner point. 9 0]' and V1(·,Z) E V~;/,~ -6 (G) for almost all z and small () > O. If 1'1 2 1+ 1 - 7r /fJ0, then C1 (z) = 0 in (8). 9). 1, aujaz E V~;I(D; M). Hence, aujaz E V~;I(G) for almost all z E M. Repeating the above reasoning to aujaz, we obtain (ajaz)u(y, z) = d(z) r 7r /{}o sin(m9/fJo) + w(y, z), (9) where w(·, z) E V~;~l -6 (G) for almost all z E M and d E L2 (M).

3; (39) 32 12. Asymptotics of Solutions of Classical Boundary Value Problems k _~ryw(k) (tp, T), log E) = L L j (T)2, 0/ OT)2, 0/ otp) ( w(k-j) (tp, T), log E) j=l (40) j-1 Lw~k-j)(tp,'l9,logp,logE)), - T)Effi. 2 \W. q=O We have shown how to determine v(k) and w(k) in (33). When solving (39), one has to know w(O), ... ,W(k-1). The functions w(O), ... ,w(k-1) and v(O), ... ,v(k) must be known in (40) and (38). 2. Notice that vJq) (tp, 'l9, log p + log E, log E) and wJ P )(tp,'l9,logr -logE,logE) are polynomials in log E.

The role of ((x) uo(z) is successively played by the terms ((x) rk sin(k'l9) in the asymptotic series of the solution. Thus 0 the operator A is not Fredholm for any /3. 4 Sobolev Problems We state without proof some theorems on the solvability of the Sobolev problems in weighted classes and give asymptotic formulas for solutions. These assertions generalize the results obtained in Section 2 and 3 for the Laplace operator and will be of use in Chapter 13. 1 Statement of the problem Let 0 0 be a domain in ]Rn with smooth (n - 1)-dimensional boundary 00 0 and M a smooth d-dimensional submanifold in ]Rn without boundary while M C 0 0 and o < d < n -1.