The Vishik–Lyusternik method in elliptic problems with a small parameter

Abstract

We consider boundary value problems where the operator defined in a domain and the boundary operators depend on a small parameter. Elliptic and properly elliptic problems with a small parameter are defined. It is proved that small parameter ellipticity is a necessary and sufficient condition for the existence of a priori estimates that are uniform with respect to the parameter. The proof of uniform estimates is based on the construction of the exponential boundary layer introduced in the classical paper by Vishik and Lyusternik. 0. Introduction The problem we study in this paper can be formulated as follows. On a manifold M with a smooth boundary ∂M we consider the following problem for an elliptic operator of order 2m: A(x,D, e)u(x) = f(x), x ∈ M, (0.1) Bj(x, D, e)u(x′) = gj(x), x′ ∈ ∂M, j = 1, . . . ,m, (0.2) where the operators in (0.1) and (0.2) depend polynomially on a small parameter e: A(x,D, e) := eA2m(x,D) + eA2m−1(x,D) + · · ·+A2μ(x,D), (0.3) Bj(x, D, e) := ejjBj,bj (x ′, D) + ejjBj,bj−1(x ′, D) + · · ·+Bj,βj (x′, D). (0.4) Here by A2m−j , j = 0, . . . , 2m − 2μ (μ < m) and Bj,bj−k, k = 0, . . . , bj − βj (βj < bj) we denote differential operators of orders 2m − j and bj − k, respectively, with leading homogeneous terms A2m−j and B 0 j,bj−k. Replacing in (0.3) A2m−j with A2m−j we obtain the leading term A (x,D, e) of the operator A(x,D, e). Similarly, replacing in (0.4) Bj,bj−k with B 0 j,bj−k we obtain the leading term B j (x ′, D, e) of the boundary operator (0.4). 0.1. Although specific problems of the type (0.1), (0.2) with operators of the form (0.3), (0.4) did appear quite some time ago in various areas of mathematical physics (especially in elasticity) and were well known for quite some time, a comprehensive theory of such operators had started up with a remarkable paper (essentially a monograph) by M. I. Vishik and L. A. Lyusternik [1]. In this paper the main condition on the dependence of the operator (0.1) on a small parameter was formulated and the asymptotics as e → 0 of the solution of the Dirichlet problem was constructed. In the proof of this asymptotics the condition of strong ellipticity of the operator (0.3) was used. The main feature of the problem (0.1), (0.2) is that at e = 0 we obtain an elliptic equation of order 2μ < 2m, which requires only μ boundary conditions. Therefore 2000 Mathematics Subject Classification. Primary 35B40. c ©2006 American Mathematical Society 87 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use