Quantitative approximation theorems for elliptic operators (Q1914796)

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