Lipschitz solvability of prescribed Jacobian and divergence for singular measures
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| Format: | Preprint |
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2026
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| _version_ | 1866908923580121088 |
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| author | De Masi, Luigi Marchese, Andrea |
| author_facet | De Masi, Luigi Marchese, Andrea |
| contents | Let $μ$ be a finite Radon measure on an open set $Ω\subset\mathbb{R}^d$, 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 $f \colon Ω\to \mathbb{R}$ there exists a vector field $V\in C^1_c(Ω;\mathbb{R}^d)$ such that $\operatorname{div} V=f$ on a compact set $K\subsetΩ$ with $μ(Ω\setminus K)<\varepsilon$, and $\operatorname{Lip}(V)\le (1+\varepsilon)\|f\|_{L^\infty(Ω,μ)}$. Similarly, for every Borel datum $g\colon Ω\to \mathbb{R}$ there exists a map $Φ$ with $Φ-\operatorname{Id}\in C^1_c(Ω;\mathbb{R}^d)$ such that $\det DΦ=g$ on a compact set $K\subsetΩ$ with $μ(Ω\setminus K)<\varepsilon$, and $\operatorname{Lip}(Φ-\operatorname{Id})\le (1+\varepsilon)\|g-1\|_{L^\infty(Ω,μ)}$. The maps $V$ and $Φ-\operatorname{Id}$ can be chosen arbitrarily small in supremum norm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_28912 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lipschitz solvability of prescribed Jacobian and divergence for singular measures De Masi, Luigi Marchese, Andrea Analysis of PDEs Functional Analysis 49Q15, 26B05, 28A75 Let $μ$ be a finite Radon measure on an open set $Ω\subset\mathbb{R}^d$, 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 $f \colon Ω\to \mathbb{R}$ there exists a vector field $V\in C^1_c(Ω;\mathbb{R}^d)$ such that $\operatorname{div} V=f$ on a compact set $K\subsetΩ$ with $μ(Ω\setminus K)<\varepsilon$, and $\operatorname{Lip}(V)\le (1+\varepsilon)\|f\|_{L^\infty(Ω,μ)}$. Similarly, for every Borel datum $g\colon Ω\to \mathbb{R}$ there exists a map $Φ$ with $Φ-\operatorname{Id}\in C^1_c(Ω;\mathbb{R}^d)$ such that $\det DΦ=g$ on a compact set $K\subsetΩ$ with $μ(Ω\setminus K)<\varepsilon$, and $\operatorname{Lip}(Φ-\operatorname{Id})\le (1+\varepsilon)\|g-1\|_{L^\infty(Ω,μ)}$. The maps $V$ and $Φ-\operatorname{Id}$ can be chosen arbitrarily small in supremum norm. |
| title | Lipschitz solvability of prescribed Jacobian and divergence for singular measures |
| topic | Analysis of PDEs Functional Analysis 49Q15, 26B05, 28A75 |
| url | https://arxiv.org/abs/2603.28912 |