Lipschitz solvability of prescribed Jacobian and divergence for singular measures

TL;DR

This paper establishes Lipschitz solvability results for prescribed divergence and Jacobian equations on measures singular to Lebesgue, with solutions approximating data on large measure sets.

Contribution

It provides new Lusin-type theorems ensuring Lipschitz solutions to divergence and Jacobian equations for singular measures, with precise control on solution norms and approximation sets.

Findings

Existence of Lipschitz vector fields solving divergence equations on large measure sets.

Construction of Lipschitz maps with prescribed Jacobian on large measure sets.

Solutions can be made arbitrarily close to data in supremum norm.

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 $\varepsilon>0$ and every Borel datum $f \colon \Omega \to \mathbb{R}$ there exists a vector field $V\in C^1_c(\Omega;\mathbb{R}^d)$ such that $\operatorname{div} V=f$ on a compact set $K\subset\Omega$ with $\mu(\Omega\setminus K)<\varepsilon$, and $\operatorname{Lip}(V)\le (1+\varepsilon)\|f\|_{L^\infty(\Omega,\mu)}$. Similarly, for every Borel datum $g\colon \Omega \to \mathbb{R}$ there exists a map $\Phi$ with $\Phi-\operatorname{Id}\in C^1_c(\Omega;\mathbb{R}^d)$ such that $\det D\Phi=g$ on a compact set $K\subset\Omega$ with $\mu(\Omega\setminus K)<\varepsilon$, and $\operatorname{Lip}(\Phi-\operatorname{Id})\le (1+\varepsilon)\|g-1\|_{L^\infty(\Omega,\mu)}$. The maps $V$ and $\Phi-\operatorname{Id}$ can be chosen arbitrarily small in supremum norm.