Approximation in Uniform Norm by Solutions of Elliptic Differential Equations

Introduction. Let G be an open subset of the Euclidean w-space E, Gi an open subset with compact closure in G. If n = 2 and G is the whole of E, an important circle of theorems in the theory of analytic functions associated with the names of Walsh, HartogsRosenthal, Lavrentiev, Keldych, and Mergelyan deals with the possibility of approximating analytic functions on Gi continuous on its closure, uniformly on G\ by polynomials in the complex variable z. Mergelyan's theorem [ l ] , the most general of these results, asserts that if G\ does not disconnect E, then every such analytic function is uniformly approximable by polynomials on Gi. More generally, if we replace G\ by any compact subset K of E, Mergelyan's result asserts that if K does not disconnect E, then every continuous function on K which is analytic at every interior point of K is uniformly approximable on K by polynomials in z. In view of the classical theorem of Runge on uniform approximation of analytic functions on compact subsets of G\ by polynomials, Mergelyan's theorem is equivalent to the assertion that each function f(z) which is continuous on K and analytic in the interior of K may be approximated uniformly on K by functions analytic on a prescribed open set G containing K in its interior. From the point of view of differential equations, the class of analytic functions is merely the class of solutions of the homogeneous first-order linear elliptic differential equation with constant complex coefficients: