Singularly perturbed and irregularly degenerate elliptic problems: common approachesстатья
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Дата последнего поиска статьи во внешних источниках: 6 декабря 2018 г.
Аннотация:On the example of two problems it is shown that theregularization method of singular perturbations, developed for theconstruction of regularized asymptotic solutions of singularlyperturbed problems, can be successfully applied to the constructionof solutions of irregularly degenerate elliptic problems.In both cases, the spectrum of the limit operator is used todescribe the characteristics of the problem. New variables (countingnumber) are entered and a new problem is written in the space ofinfinite number of dimensions. The resulting task will already beregular. Narrowing its solution is the solution of the originalproblem.In the case of a problem with a small parameter at the highestderivative, the solution of the newly obtained problem is sought bythe method of the classical perturbation theory in a special spaceof nonresonant solutions. Theorems on existence of formal andasymptotic solution of the problem are given.In the case of a degenerate elliptic equation, an extended problemis solved. Statements about existence of formal and classicalsolutions of the considered problem are given. Estimates of the rateof decrease of the components of solutions are given. It is shownthat, as in the classical Cauchy - Kowalewsky theorem, the solutionpartly inherits the analytic properties of the coefficients and theright-hand side of the differential equation. Possiblegeneralizations of the obtained result are discussed.An example of a problem for which all the conditions of the theoremon the existence of a classical solution of the original problem aresatisfied is given. Its solution is written out, clearlydemonstrating the preservation of the analytical data of theproblem. In solving the initial problem and the problem from theexample it is necessary to solve the problem of small denominators.