Quantitative Approximation Theorems for Elliptic Operators

Abstract

LetL(D) be an elliptic linear partial differential operator with constant coefficients and only highest order terms. For compact setsK⊂RNwhose complements are John domains we prove a quantitative Runge theorem: if a functionfsatisfiesL(D)f=0 on a fixed neighborhood ofK, we estimate the sup-norm distance fromfto the polynomial solutions of degree at mostn. The proof utilizes a two-constants theorem for solutions to elliptic equations. We then deduce versions of Jackson and Bernstein theorems for elliptic operators.