$L^p$-norm approximation of h$\ddot{o}$lder functions by harmonic functions on some multidimensional compact sets

In this paper we consider the class of H$\ddot{o}$lder functions in the sense of $L^p$ norm on certain compacts in $R^m (m \geqslant 3)$ 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 $R^3$. 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 $L^p$-norm.