Uniform and $ C^1$-approximability of functions on compact subsets of $ \mathbb R^2$ by solutions of second-order elliptic equations

Abstract

Several necessary and sufficient conditions for the existence of uniform or C{sup 1}-approximation of functions on compact subsets of R{sup 2} by solutions of elliptic systems of the form c{sub 11}u{sub x{sub 1}}{sub x{sub 1}}+2c{sub 12}u{sub x{sub 1}}{sub x{sub 2}}+c{sub 22}u{sub x{sub 2}}{sub x{sub 2}}=0 with constant complex coefficients c{sub 11}, c{sub 12} and c{sub 22} are obtained. The proofs are based on a refinement of Vitushkin's localization method, in which one constructs localized approximating functions by 'gluing together' some special many-valued solutions of the above equations. The resulting conditions of approximation are of a topological and metric nature.