The resolvent of a singularly perturbed elliptic operator

AbstractLet A and B be uniformly elliptic operators of orders 2m and 2n, respectively, m > n. We consider the Dirichlet problems for the equations (ϵ2(m − n)A + B + λ2nI)uϵ = f and (B + λ2nI)u = f in a bounded domain Ω in Rk with a smooth boundary ∂Ω. The estimate ∥ uϵ − u ∥L2(Ω) ⩽ Cϵ ¦ λ ¦−2n + 1(1 + ϵ ¦ λ ¦)−1 ∥ f ∥L2(Ω) is derived. This result extends the results of [7, 9, 10, 12, 14, 15, 18]by giving estimates up to the boundary, improving the rate of convergence in ϵ, using lower norms, and considering operators of higher order with variable coefficients. An application to a parabolic boundary value problem is given