In the last decade the study of asymptotic methods for approximative solution of differential equations depending on parameters has rapidly evolved. This found its expression in a series of monographs to this subject, DE GROEN (1976), ECKHAUS (1979), LIONS (1973), O´MALLEY (1974), WASILJEWA/BUTUSOW (1973, 1978), WASOW (1968). Whereas with the monograph of LIONS a certain closing of the theory of asymptotic approximation of solutions of differential equation problems in the sense of the integral-norm was given, the achieved progress in constructing and proving uniformly valid asymptotic approximation (in the sense of the maximum norm) of solutions of singular perturbed elliptic and parabolic problems has not been comprehensively presented until now. The present paper aims at doing this. Here we restrict our considerations to such problems (important for applications) for which the formal global problem (also called reduced problem) is of first order.

The first chapter has proper importance. Here the basic ideas and relations are stated and formal construction methods for the asymptotic study of perturbed problems are given. The necessity to prove the asymptotic of formal solutions is emphasized. Existence and estimate theorems for elliptic and parabolic boundary value problems are given in Chapter two and their application to singularly perturbed problems is investigated. Statements concerning existence, uniqueness and regularity of the solution of the formal global problem are contained in the third chapter.

The main part of the present material is summarized in Chapter four. in order not to interrupt the representation by numerous details, we have stated the analytical properties of the treated boundary layer equations in an appendix. In 4.1 for problems of second order, the question is studied, in which subdomains the solution of the formal global problem is already an uniform asymptotic solution. For linear problems in simple cases, the asymptotic of formal solution is constructed and proved in 4.2 and 4.3. Such simple cases appear if the solution of the formal global problem is sufficiently regular and the characteristics behave uniformly in a neighbourhood of the boundary. Two kinds of problems of second order are investigated, for which the global formal solution is not sufficiently regular. On the one hand, results are given in 4.4 in which the characteristics are tangential to the boundary of the domain and, on the other hand, special problems with turning points and singular lines are studied in 4.5. In 4.6, finally, some results of 4.2 are extended to nonlinear elliptic problems of second order.

In order not to exceed the extent of the book, we did not deal with the application of the present theory to physical and technical problems here. The reader interested in this subject can consult for instance GOERING (1977), NAYFEH (1973), VAN DYKE (1975), COLE (1968), KEVORKIAN/COLE (1981). Likewise we have neglected aspects of the numerical treatment of singular perturbed problems. A first monograph (DOOLAN/MILLER/SCHILDERS (1980)) to difference methods for one-dimensional problems of the kind considered here has already been published, for other problems we refer to conference materials (HEMKER/MILLER (1979) and MILLER (1980)).

For understanding the results in Chapter 2 - 4 it is not necessary to study Chapter 1 in detail.On pages 32 - 34 - beginning with formula (1.41a/b) - the reader interested in applications of the results can find a summary of the principles for the asymptotic study of partial differential equations which is independent of the other statements of Chapter 1. Then, it is possible to continue with Chapter 2.

For orientation in the book we have numbered the Lemmata, Theorems and formulas within a chapter and, additionally, equipped with the number of the corresponding chapter. Essential assumptions have been designated for instance by (H8) and refer to the corresponding chapter. In the running text we tried to dispense with references to the literature. Quotations to the literature and further remarks can be found in the comments following each chapter.