Approximation numbers of Sobolev imbeddings over unbounded domains

AbstractLet Ω ⊂ RN be an open set with dist(x, ∂Ω) = O(¦ x ¦−l) for x ϵ Ω and some l > 0 satisfying an additional regularity condition. We give asymptotic estimates for the approximation numbers αn of Sobolev imbeddings over these quasibounded domains Ω. Here denotes the Sobolev space obtained by completing C0staggered∞(Ω) under the usual Sobolev norm. We prove αn(Ip,qm) $̌n−γ, where . There are quasibounded domains of this type where γ is the exact order of decay, in the case p ⩽ q under the additional assumption that either 1 ⩽ p ⩽ q ⩽ 2 or 2 ⩽ p ⩽ q ⩽ ∞. This generalizes the known results for bounded domains which correspond to l = ∞. Similar results are indicated for the Kolmogorov and Gelfand numbers of Ip,qm. As an application we give the rate of growth of the eigenvalues of certain elliptic differential operators with Dirichlet boundary conditions in L2(Ω), where Ω is a quasibounded domain of the above type