Lipschitz solvability of prescribed Jacobian and divergence for singular measures

Abstract

Let $\mu$ be a finite Radon measure on an open set $\Omega \subset {\mathbb{R}}^d$, singular with respect to the Lebesgue measure. We prove Lusin-type solvability results for the prescribed divergence equation and the prescribed Jacobian equation with Lipschitz solutions. More precisely, for every $\epsilon >0$ and every Borel datum $f:\Omega \to \mathbb{R}$ there exists a vector field $V\in C_c^1(\Omega ;{\mathbb{R}}^d)$ such that $\mathrm{div}V=f$ on a compact set $K\subset \Omega$ with $\mu (\Omega \setminus K)<\epsilon$, and $\mathrm{Lip}(V)\le (1+\epsilon )\Vert f{\Vert }_{L^{\infty }(\Omega ,\mu )}$. Similarly, for every Borel datum $g:\Omega \to \mathbb{R}$ there exists a map $\Phi$ with $\Phi -\mathrm{Id}\in C_c^1(\Omega ;{\mathbb{R}}^d)$ such that $\det D\Phi =g$ on a compact set $K\subset \Omega$ with $\mu (\Omega \setminus K)<\epsilon$, and $\mathrm{Lip}(\Phi -\mathrm{Id})\le (1+\epsilon )\Vert g-1{\Vert }_{L^{\infty }(\Omega ,\mu )}$. The maps $V$ and $\Phi -\mathrm{Id}$ can be chosen arbitrarily small in supremum norm.