On a method of approximation by gradients

Abstract:

The subject of this study is the possibility of approximation of a continuous vector field on a compact set $K\subset \mathbb {R}^n$ by gradients of smooth functions defined on the entire $\mathbb {R}^n$. A method is obtained that yields either approximation or an obstruction for it. This method does not involve the Hahn–Banach theorem and is based on solving a quasilinear elliptic equation in partial derivatives. A discrete analog of the above problem is studied, namely, the problem of approximation by gradients on a finite oriented graph. A stepwise algorithm is suggested in this case.