Structure of prescribed gradient domains for non-integrable vector fields

Abstract

Let $$F \in C^1(\Omega ,{\mathbb {R}}^{n})$$ and $$f\in C^2(\Omega )$$ , where $$\Omega $$ is an open subset of $${\mathbb {R}}^{n}$$ with n even. We describe the structure of the set of points in $$\Omega $$ at which the equality $$D f = F$$ and a certain non-integrability condition on F hold. This result generalizes the second statement of Balogh (J Reine Angew Math 564:63–83, 2003, Theorem 3.1).