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

Abstract

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.

On the Shapiro–Lopatinkii condition for elliptic problems

“…Volevich [12] also proved that estimate (1.3) holds true if the operator (A, B) satisfies the small parameter ellipticity condition, the Shapiro-Lopatinskii condition with small parameter, and the system of equations (1.1), (1.2) is correctly solvable. The problem (A, B) is called an elliptic problem with parameter if it satisfies all these conditions.…”

“…This problem was solved by Volevich [12] in the case where D is the half-space R n + = {(x , x n ) ∈ R n : x n > 0}. He used the norms for functions in D and their traces, which had been introduced in [5].…”

“…In all these works, the uniform estimates in norms depending on ε for solutions were found under some generalized coercivity condition, and the estimates justify the formal asymptotic series obtained by the Lyusternik-Vishik method. The comprehensive theory of elliptic equations with small parameter was constructed by Volevich in [12]. For the problem (A, B) in the half-space…”

Singular Perturbations of Elliptic Operators

“…Volevich [Vol06] was the first to present the small parameter theory as a part of general elliptic theory. …”

“…which is a generalization of the Agmon-Agranovich-Vishik condition of ellipticity with parameter corresponding to μ = 0, see [AV64], [Vol06] and the references given there.…”