On two parameter singular perturbation of linear boundary value problems

Abstract

A two parameter perturbation estimate for solutions of a functional equation in Hilbert space is derived. The estimate is applied to two parameter singular perturbation of elliptic boundary value problems with homogeneous Difichlet boundary data. 1. Consider the boundary value problem eAu + ACu + Bu = ff,,f for 0 2m over a bounded domain D; and C is a linear differential operator of order fo as e J 0 and IA J 0. In particular, bounds of the form |u-uol mDD=O(ET)+o(,) in the Bessel potential space Pm(D) will be derived, assuming a like bound for the P-m(D) norm of f-fo. For the one parameter problem obtained by setting C = 0 above, corresponding bounds have been obtained by Friedman [4], Greenlee [5] and Huet [8], [9]. Extensive studies in multiparameter singular perturbation theory have been carried out by O'Malley, cf. [12]. In Greenlee [6] a two parameter perturbation problem analogous to the above, but with C a quasilinear differential operator of order ?<2m, was considered. In this paper a perturbation theorem for a functional equation in abstract Hilbert space is proven. The theorem is then applied to differential problems of the type described above. The notation and methods of this paper are similar to those used in [5] and [6]. 2. Let V and VO be complex Hilbert spaces with VCeVo, and V dense in VO. Denote by I vI v, (v, W)V, I v O the norms and inner products in V and VO respectively. Let a(v, w), c(v, w) be continuous Hermitian bilinear (sesquilinear) forms on V and let b(v, w) be a continuous Hermitian bilinear form on VO. Further assume that: Received by the editors March 6, 1970. AMS 1969 subject classifications. Primary 3514, 4748; Secondary 3545.