Bernstein Theorems for Elliptic Equations

Abstract

We study solutions of the equation L(D) u = 0, where L(D) is an elliptic linear partial differential operator with constant coefficients and only highest order terms. For compact sets K ⊂ RN with connected complement we prove a Bernstein theorem: if a function ƒ on K can be extended to a solution of the equation on an open neighborhood of K, then the supnorm distance from ƒ to the polynomial solutions of degree ≤ n decays exponentially with n. We give two proofs: a proof by duality which makes use of the theory of functions of several complex variables, and an elementary constructive proof using generalized Laurent expansions for solutions of elliptic equations. Finally, we discuss the use of orthogonal polynomial expansions, and the use of interpolation schemes, for the construction of polynomial approximations with asymptotically optimal behavior.