Approximation of continuous functions on a compact set by solutions of elliptic equations. Quantitative results

Abstract We establish that a generalized Hölder continuous function on an (m−2)-Ahlfors regular compact set K in R m can be uniformly on K approximated by solutions of a second order elliptic equation with mildly regular coefficients, with the rate of the approximation and the quality of the approximating function determined by the continuity modulus of the function.