Lipschitz solvability of prescribed Jacobian and divergence for singular measures

Let $\mu$ be a finite Radon measure on an open set $\mathrm{\Omega }\subset {\mathbb{R}}^d$, 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 $f\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to \mathbb{R}$ there exists a vector field $V\in C_c^1(\mathrm{\Omega };{\mathbb{R}}^d)$ such that $\mathrm{div}V=f$ on a compact set $K\subset \mathrm{\Omega }$ with $\mu (\mathrm{\Omega }\setminus K)<\epsilon$, and $\mathrm{Lip}(V)\le (1+\epsilon )\mathrm{\parallel }f{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega },\mu )}$. Similarly, for every Borel datum $g\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to \mathbb{R}$ there exists a map $\mathrm{\Phi }$ with $\mathrm{\Phi }-\mathrm{Id}\in C_c^1(\mathrm{\Omega };{\mathbb{R}}^d)$ such that $\det D\mathrm{\Phi }=g$ on a compact set $K\subset \mathrm{\Omega }$ with $\mu (\mathrm{\Omega }\setminus K)<\epsilon$, and $\mathrm{Lip}(\mathrm{\Phi }-\mathrm{Id})\le (1+\epsilon )\mathrm{\parallel }g-1{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega },\mu )}$. The maps $V$ and $\mathrm{\Phi }-\mathrm{Id}$ can be chosen arbitrarily small in supremum norm.