On uniform approximation by harmonic and almost harmonic vector fields

Three- dimensional analogs of rational uniform approximation in mathbbC 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 ⊂ mathbbR³ 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 ⊂ mathbbC mathbb{C} (when f is assumed to be in C ¹ with [`(?)] fbar 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.