On asymptotic approximations of solutions of an equation with a small parameter

Abstract

Abstract. A second order elliptic equation with a small parameter at one of the highest order derivatives is considered in a three-dimensional domain. The limiting equation is a collection of two-dimensional elliptic equations in two-dimensional domains depending on one parameter. By the method of matching of asymptotic expansions, a uniform asymptotic approximation of the solution of a boundary-value problem is constructed and justified up to an arbitrary power of a small parameter. §1. IntroductionProblems for elliptic equations with small parameters at highest order derivatives are the subject of intense studies. We mention only the earliest papers [1]-[4] on this subject and the extensive survey [5]. To treat more complicated problems of this type, we need to exploit the method of matching of asymptotic expansions, also called the matching method. In [6]-[8] and [10] this method was developed and applied to various problems, including elliptic equations.In the majority of papers on elliptic equations with small parameters at highest derivatives, the reduction of order is applied, in which the limiting equation has an order smaller than the perturbed equation. In particular, this was the reason for the singularity of the problem. At the same time, it turns out that even if the order reduction is not used, the problem often continues to be singular, and more involved methods are necessary for its solution.Some cases in which the limiting equation is an ordinary differential equation were considered in the recent papers [11]-[13].In the present paper, we also assume that the limiting equation is a second order elliptic equation but with a smaller number of independent variables. We consider the perturbed equation in a three-dimensional domain. The limiting equation is a collection of two-dimensional equations in domains depending on one parameter. The asymptotic behavior of a solution of a boundary-value problem depends substantially on the structure of the domain at singular points related to the degeneration type of the operator. In this regard, the situation is similar to the case where degeneration causes order reduction. In general, our method is close to that used in the above-mentioned cases. However, the auxiliary problems that arise in passing require other methods. Sometimes, they are more involved. Even the formal asymptotic expansion is not constructed in all cases of 2010 Mathematics Subject Classification. Primary 35J47. Key words and phrases. Asymptotic, boundary value problem, small parameter, matching of asymptotic expansions.

Asymptotics for solutions of problem on optimally distributed control in convex domain with small parameter at one of higher derivatives

“…Asymptotics of the solutions of the Dirichlet problem for such elliptic equations in such domains was studied in [3], [4]. An asymptotics of the distributed control for an operator with a small coefficient at the Laplace operator in an essentially different domain was considered in [8], [9], while the case of a similar domain was treated in [10].…”