Lp-norm approximation of Holder functions by harmonic functions on some multidimensional compact sets

Abstract

In this paper we consider the class of H¨older functions in the sense of Lp norm on certain compacts in Rm (m >= 3) and prove theorems on approximation by functions harmonic in neighborhoods of these compacrs. These compacts are a generalization to the higher dimensions of the concept of chord-arc curve in R3. The size of the neighborhood decreases along with an increase in the accuracy of the approximation. Estimates of the approximation rate and the gradient of the approximation functions are made in the same Lp-norm.