On uniform approximation by harmonic and almost harmonic vector fields

Abstract

Three- dimensional analogs of rational uniform approximation in $$ \mathbb{C}$$ are considered. These analogs are related to approximation properties of harmonic (i. e., curl-free and solenoidal) vector fields. The usual uniform approximation by fields harmonic near a given compact set K ⊂ $$ \mathbb{R}^3$$ is compared with the uniform approximation by smooth fields whose curls and divergences tends to zero uniformly on K. A similar two-dimensional modification of the uniform approximation by functions f that are complex analytic near a given compact set K ⊂ $$ \mathbb{C}$$ (when f is assumed to be in C 1 with $$\bar \partial {\kern 1pt}f$$ small on K) results in a problem equivalent to the original one. In the three-dimensional settings, the two problems (of harmonic and of almost harmonic approximation) are different. The first problem is nonlocal whereas the second one is local (i. e., an analog of the Bishop theorem on the locality of R(K) is still valid for almost harmonic approximation). Almost curl-free approximation is also considered. Bibliography: 7 titles.