Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten II

Mazja, W. G.; Nasarow, S. A.; Plamenewski, B. A.,

Berlin, Akademie Verlag, 1992.

The authors continue their investigations from Volume I [MR 92g:35059], employing the same general principles. Further, they consider elliptic boundary value problems the solutions of which are singularly perturbed caused by several degenerations of the domain. This concerns, especially, the following ones: (1) domains with narrow cavities, e.g., with tubes, notches, or cracks removed; (2) domains consisting of some parts which are connected merely by narrow passages; (3) narrow domains, i.e., the expansions in some directions are very small compared with the other ones (including, e.g., domains with closely neighbouring parallel cracks). Here "small" or "narrow" is measured by a positive parameter tex2html_wrap_inline3 on which the domain tex2html_wrap_inline5 in question depends; for tex2html_wrap_inline7, tex2html_wrap_inline5 will degenerate: e.g., tex2html_wrap_inline11 with a "normal" domain G in tex2html_wrap_inline15, and tex2html_wrap_inline17 a torus of thickness tex2html_wrap_inline3, will tend to tex2html_wrap_inline21 where M is a closed curve in G.

The main subject is the construction of asymptotic expansions, for tex2html_wrap_inline7, of the solutions of the problems posed in tex2html_wrap_inline5. This is done, as in Volume I, by iteration processes which consist of solving, in alternating fashion, certain limit problems (two, in general) where at least one of them depends on parameters: one problem in tex2html_wrap_inline31, and another one, e.g., in the complement of the model of the cavity (hole) removed from G. As usual in singular perturbation problems, boundary layers may arise.

The proofs of solvability of the limit problems, and of the asymptotic expansions of the solutions, rely essentially on results of the modern theory of elliptic boundary value problems in piecewise smoothly bounded domains. The constructions of the expansions here are mostly far more complicated then in the cases of singular perturbations of local type considered in Volume I. Fortunately, most problems are first presented in a simpler model setting (e.g., Poisson's equation in tex2html_wrap_inline35).

In the last part (Chapter VII) of the book, asymptotic expansions of solutions are, by similiar methods, constructed for elliptic boundary value problems with rapidly periodically oscillating coefficients, but also for differential or difference equations on tex2html_wrap_inline3-periodic curvilinear nets in tex2html_wrap_inline15; especially, the corresponding homogenized differential operators are derived.

Moreover, an interesting analysis of the paradoxa is given that may occur with the solution of plane boundary value problems if a smooth boundary is approximated by polygons.

Reviewed by Dietrich Gohde