On the uniform approximation by harmonic and almost harmonic vector fields

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 $K\subset\mathbb R^3$ is compared with the uniform aproximation by smooth fields whose curls and divergences tend to zero uniformly on $K$. A similar two-dimensional modification of the uniform approximation by functions $f$ complex analytic near a given compact set $K\subset\mathbb C$ (when $f$ is assumed to be in $C^1$ with $\bar\partial f$ small on $K$) results in a problem equivalent to the original one. In the three-dimensional setting, the two problems (of harmonic and of almost harmonic approximation) are different. The first is nonlocal whereas the second 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.