Lipschitz solvability of prescribed Jacobian and divergence for singular measures

Abstract: Let $μ$ be a finite Radon measure on an open set $Ω\subset\mathbb{R}<sup>d$, 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 $\varepsilon&gt;0$ and every Borel datum $f \colon Ω\to \mathbb{R}$ there exists a vector field $V\in C<sup>1_c(Ω;\mathbb{R}<sup>d)$ such that $\operatorname{div} V=f$ on a compact set $K\subsetΩ$ with $μ(Ω\setminus K)&lt;\varepsilon$, and $$\operatorname{Lip}(V)\le (1+\varepsilon)|f|<em>{L<sup>\infty(Ω,μ)}$$. Similarly, for every Borel datum $g\colon Ω\to \mathbb{R}$ there exists a map $Φ$ with $$Φ-\operatorname{Id}\in C<sup>1_c(Ω;\mathbb{R}<sup>d)$$ such that $\det DΦ=g$ on a compact set $K\subsetΩ$ with $μ(Ω\setminus K)&lt;\varepsilon$, and $$\operatorname{Lip}(Φ-\operatorname{Id})\le (1+\varepsilon)|g-1|</em>{L<sup>\infty(Ω,μ)}$$. The maps $V$ and $Φ-\operatorname{Id}$ can be chosen arbitrarily small in supremum norm.

• The paper proves that prescribed divergence and Jacobian equations have C¹ solutions exact on compact sets carrying arbitrarily large mass for any finite measure singular to Lebesgue measure.
• Its iterative directional method combines width functions, cone-null sets, and the decomposability bundle to control Lipschitz norms by the data’s L∞ norm while making the solutions arbitrarily small in supremum norm.
• The Jacobian result yields global diffeomorphisms when the perturbation is sufficiently small, while the divergence theorem provides a top-dimensional Lusin-solvability case relevant to the flat chain conjecture.

Overview

This paper, by Luigi De Masi and Andrea Marchese (2603.28912), establishes Lusin-type solvability results for two classical first-order PDEs — the prescribed divergence equation $\operatorname{div} V = f$ and the prescribed Jacobian equation $\det D\Phi = g$ — with $C^1$ solutions whose data are merely bounded Borel functions with respect to an arbitrary finite Radon measure $\mu$ singular with respect to the Lebesgue measure. The solutions are exact on compact sets carrying arbitrarily large $\mu$-mass and satisfy uniform Lipschitz bounds depending only on the $L^\infty(\Omega,\mu)$ norm of the datum. A distinctive feature is that both solutions can be taken arbitrarily small in supremum norm.

The work builds on the non-closability results of Alberti, Bate, and Marchese for differential operators from Lipschitz functions to $L^p(\mu)$ when $\mu \perp L^d$, which show that equi-Lipschitz sequences converging uniformly to zero can retain large derivatives on sets of positive $\mu$-measure. Rather than treating this phenomenon as a pathology, the authors exploit it constructively: the same geometric mechanism that produces non-closability also enables exact solvability on almost-full-measure sets with quantitative control.

Main results

Theorem (Lusin solvability for divergence). Let $\mu \perp L^d$ be a finite Radon measure on an open set $\det D\Phi = g$0, let $\det D\Phi = g$1 be Borel, and fix $\det D\Phi = g$2. For every $\det D\Phi = g$3 there exist a compact set $\det D\Phi = g$4 and a vector field $\det D\Phi = g$5 such that:

$\det D\Phi = g$6

with

$\det D\Phi = g$7

Theorem (Lusin solvability for the Jacobian). Under the same hypotheses on $\det D\Phi = g$8 and for a Borel datum $\det D\Phi = g$9, there exist a compact set $C^1$0 and a map $C^1$1 with $C^1$2 such that $C^1$3 for all $C^1$4, $C^1$5, and

$C^1$6

A notable corollary: if $C^1$7, then $C^1$8 is a global diffeomorphism of $C^1$9. This follows from a lower bound $\mu$0 together with a topological argument showing surjectivity via connectedness. The authors also prove a version perturbing any given diffeomorphism $\mu$1: one obtains $\mu$2 with $\mu$3 on a set of $\mu$4-measure at least $\mu$5 and

$\mu$6

The proof of this corollary pushes $\mu$7 forward by $\mu$8 (which preserves singularity), applies the identity-perturbation theorem in the target, and composes back; the datum transforms as $\mu$9.

The smallness of $\mu$0 has a further consequence recorded by the authors: applying the divergence theorem to the residual $\mu$1 for any background field $\mu$2 yields a field $\mu$3 with $\mu$4 outside an arbitrarily small exceptional set and $\mu$5 arbitrarily small. Thus the solvability statement extends to arbitrary perturbations of reference fields.

Method: the directional scalar lemma

The common core of both proofs is a local scalar result. Suppose $\mu$6 is a finite Radon measure on an open set $\mu$7, concentrated on a $\mu$8-null Borel set $\mu$9 (meaning every Lipschitz curve with velocity in the cone $L^\infty(\Omega,\mu)$0 meets $L^\infty(\Omega,\mu)$1 in $L^\infty(\Omega,\mu)$2-null sets). Then for every continuous bounded $L^\infty(\Omega,\mu)$3 and every $L^\infty(\Omega,\mu)$4, there exist a compact $L^\infty(\Omega,\mu)$5 and $L^\infty(\Omega,\mu)$6 with

$L^\infty(\Omega,\mu)$7

and

$L^\infty(\Omega,\mu)$8

where $L^\infty(\Omega,\mu)$9.

The proof is an iterative correction scheme. One builds a decreasing sequence of compact sets $L^p(\mu)$0 with controlled mass loss and a series of corrections $L^p(\mu)$1 such that the residual $L^p(\mu)$2 satisfies $L^p(\mu)$3 for a small parameter $L^p(\mu)$4. Each step partitions $L^p(\mu)$5 into finitely many pieces of small oscillation, selects compact subsets, and on each piece adds a localized bump built from a width function $L^p(\mu)$6 — a construction due to Alberti, Csörnyei, and Preiss — which satisfies $L^p(\mu)$7 on the piece, is bounded by $L^p(\mu)$8, and has transverse derivatives bounded by $L^p(\mu)$9. Because the piece lies inside a cone-null set, the width function lemma applies. Disjointness of supports ensures the gradient estimates add by maximum rather than summation, and the geometric series converges in $\mu \perp L^d$0. On the limit set $\mu \perp L^d$1, the residual vanishes pointwise, giving exact equality $\mu \perp L^d$2.

Two structural inputs convert this scalar lemma into the global theorems:

• Decomposability bundle. By the characterization of Alberti–Marchese, $\mu \perp L^d$3 for $\mu \perp L^d$4-a.e. $\mu \perp L^d$5 if and only if $\mu \perp L^d$6. Hence for singular $\mu \perp L^d$7, the bundle is a proper subspace $\mu \perp L^d$8-a.e.
• Finite directions lemma. For each opening angle $\mu \perp L^d$9, a finite family of unit vectors $\mu$0 exists (depending only on $\mu$1 and $\mu$2) such that every proper linear subspace $\mu$3 avoids at least one cone $\mu$4: $\mu$5. The proof uses a finite geodesic net on the sphere with mesh $\mu$6 and an elementary trigonometric estimate.

Combining these, $\mu$7 decomposes into finitely many Borel pieces $\mu$8 on which $\mu$9 is transverse to $\mu \perp L^d$0; inner regularity yields disjoint compact sets $\mu \perp L^d$1 covering all but $\mu \perp L^d$2 of the mass. On each $\mu \perp L^d$3, the concentration lemma for cone-null sets provides a full-$\mu \perp L^d$4-measure cone-null subset, so the scalar lemma applies with direction $\mu \perp L^d$5. Setting $\mu \perp L^d$6 (or $\mu \perp L^d$7), disjointness of supports gives $\mu \perp L^d$8 (respectively $\mu \perp L^d$9) on $\det D\Phi = g$00, since near each point only one summand is active and the Jacobian of $\det D\Phi = g$01 reduces under an orthonormal change of coordinates to $\det D\Phi = g$02.

Context and significance

Several aspects of the placement of this result deserve emphasis.

Relation to endpoint estimates. Ornstein's non-inequality shows that no $\det D\Phi = g$03-to-Lipschitz bound holds for first-order operators; for the divergence equation, counterexamples were given by Preiss and independently by McMullen, and Taškač's counterexample for the prescribed Jacobian equation invalidates Lang's formulation of the flat chain conjecture, which requires pointwise control. The present results show instead that Lusin-type solvability — imposing the constraint only on sets of almost-full singular-measure mass — is achievable with uniform Lipschitz bounds. In top dimension, the equation $\det D\Phi = g$04 for $\det D\Phi = g$05-forms reduces exactly to prescribed divergence upon identifying forms with vector fields. Since Marchese–Merlo showed that the Ambrosio–Kirchheim flat chain conjecture would follow from such an $\det D\Phi = g$06-to-Lipschitz Lusin solvability theory for prescribed differential forms satisfying an orthogonality condition relative to the bundle $\det D\Phi = g$07, the divergence theorem here supplies precisely the top-dimensional test case of that strategy and clarifies the mechanism by which the flat chain conjecture holds in dimension $\det D\Phi = g$08 despite the failure of pointwise estimates. The intermediate-dimensional cases $\det D\Phi = g$09 remain open.

Relation to rectifiability. When $\det D\Phi = g$10 is carried by a $\det D\Phi = g$11-rectifiable set with $\det D\Phi = g$12, approximate tangent planes exist locally and transverse directions are immediately available; the construction then becomes geometrically transparent. The substantive contribution is that the same transverse mechanism works for arbitrary singular measures, with the decomposability bundle playing the role of a measurable tangent structure even in the absence of any rectifiable support. This contrasts sharply with Lusin-type $\det D\Phi = g$13 approximation of Lipschitz functions, which Marchese–Merlo showed characterizes the much more rigid class of countably rectifiable-type measures. Rectifiability therefore plays a genuinely different role for prescription problems than for approximation problems: it simplifies geometry but does not change the analytic nature of the task.

Relation to Dacorogna–Moser. The classical Dacorogna–Moser theorem transports between absolutely continuous densities via diffeomorphisms solving $\det D\Phi = g$14. Theorem 2 is an ambient Jacobian statement and does not imply a push-forward relation $\det D\Phi = g$15 for singular $\det D\Phi = g$16; it constructs a global change of coordinates whose Jacobian matches the datum on almost-full $\det D\Phi = g$17-mass, not a transport of the measure itself. This distinction is stated explicitly by the authors as a boundary of the result.

Limitations and open questions

The results require $\det D\Phi = g$18 essentially through Proposition 2.4: for absolutely continuous $\det D\Phi = g$19, the decomposability bundle is full almost everywhere, no transverse direction exists, and the method breaks down. Whether analogous statements hold for general Radon measures (with the constraint imposed on the singular part, or with additional assumptions on the datum as in the gradient case treated previously) is not addressed here. The Lipschitz constant carries the dimensional factor $\det D\Phi = g$20, which blows up as $\det D\Phi = g$21; the trade-off between the cone opening and the final constant is inherent to the width-function technique. Finally, the connection to the flat chain conjecture is established only in top dimension; extending the Lusin solvability program to intermediate-degree differential forms — the route proposed toward cases $\det D\Phi = g$22 — remains open.

Conclusion

The paper converts a known obstruction — the failure of closability of first-order differential operators on singular measures — into a constructive tool, proving that prescribed divergence and prescribed Jacobian equations admit $\det D\Phi = g$23 solutions exact on sets of arbitrarily large $\det D\Phi = g$24-measure, with uniform Lipschitz bounds proportional to the $\det D\Phi = g$25 norm of the data and arbitrarily small amplitude. The proof rests on a directional scalar lemma combining width functions, cone-null sets, and the decomposability bundle, and its top-dimensional divergence case aligns exactly with the Lusin-type strategy proposed for the Ambrosio–Kirchheim flat chain conjecture.