Singular perturbation problems for systems of partial differential equations of elliptic type

Abstract

Elliptic boundary value problems for systems of nonlinear partial differential equations of the form F i(x, u 1, u 2,…, u N, ∂u i ∂x j , ϵ p i ∂ 2u i ∂x j ∂x k ) = ƒ i(x), x ϵ R n , i = 1(1) N, j, k = 1(1) n, p i ⩾ 0, ϵ being a small parameter, with Dirichlet boundary conditions are considered. It is supposed that a formal approximation Z is given which satisfies the boundary conditions and the differential equations upto the order χ( ϵ) = o(1) in some norm. Then, using the theory of differential inequalities, it is shown that under certain conditions the difference between the exact solution u of the boundary value problem and the formal approximation Z, taken in the sense of a suitable norm, can be made small.