Quantitative approximation theorems for elliptic operators

Let L(D) be an elliptic linear partial differential operator with constant coefficients and only highest order terms. For compact sets K subset of R(N) whose complements are John domains we prove a quantitative Runge theorem: if a function f satisfies L(D)f = 0 on a fixed neighborhood of K, we estimate the sup-norm distance from f to the polynomial solutions of degree at most n. The proof utilizes a two-constants theorem for solutions to elliptic equations. We then deduce versions of Jackson and Bernstein theorems for elliptic operators. (C) 1996 Academic Press, Inc.