Singularly perturbed and irregularly degenerate elliptic problems: common approaches

Abstract

ABSTRACTBy an example of two problems it is shown that the regularization method of singular perturbations, developed for the construction of regularized asymptotic solutions of singularly perturbed problems, can be successfully applied to the construction of solutions of irregularly degenerate elliptic problems. In both cases, the spectrum of the limit operator is used to describe the characteristics of the problem. New variables (countable many) are introduced and a new problem is written in a space of infinite dimension. The resulting task will already be regular. Narrowing its solution is the solution of the original problem. In the case of a problem with a small parameter at the highest derivative, the solution of the newly obtained problem is sought by the method of the classical perturbation theory in a special space of nonresonant solutions. Theorems on existence of formal and asymptotic solution of the problem are given. In the case of a degenerate elliptic equation, an extended problem is solved. Statements about existence of formal and classical solutions of the considered problem are given. Estimates of the rate of decrease of the components of solutions are given.