Sharp weak integrability of planar Monge–Ampère Hessians
Let $0 < \lambda < 1$, let $\Omega \subset \mathbb{R}^2$ be a bounded convex domain, and let $u$ be a convex Alexandrov solution of $$\lambda \le \det D^2 u \le \lambda^{-1} \quad \text{in } \Omega,$$ that is, $\lambda |E| \le |\partial u(E)| \le \lambda^{-1} |E|$ for every Borel set $E \subset \Omega$, where $\partial u$ is the subdifferential. Put $$p_\lambda = \frac{1 + \lambda^2}{1 - \lambda^2}.$$ De Philippis, Figalli and Savin proved interior $W^{2,1+\varepsilon}$ estimates for strictly convex solutions with density between two positive constants, with $\varepsilon > 0$ depending only on the dimension and those constants. De Philippis and Figalli observed that for a smooth solution the map $\Phi = \Phi_1 \circ \Phi_2^{-1}$, where $\Phi_1(x, y) = (\partial_x u(x, y), y)$ and $\Phi_2(x, y) = (x, \partial_y u(x, y))$, is $1/\lambda^2$-quasiconformal, and the higher integrability theory of quasiconformal maps led them to the exponent $p_\lambda$.
Problem. Is the distributional Hessian of $u$ represented by a matrix field satisfying, for every $\Omega' \Subset \Omega$, $$\sup_{s > 0} s^{p_\lambda} \bigl|\{x \in \Omega' : |D^2 u(x)| > s\}\bigr| < \infty,$$ that is, the endpoint weak-$L^{p_\lambda}$ estimate rather than integrability with some exponent larger than one?
Source: De Philippis and Figalli (2014), §5.2, open problem (2), as a conjecture. Stated for bounded convex domains by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- CV
- Keywords
- AIM Open Applied Problems, Monge–Ampère equation, quasiconformal maps, Sobolev regularity
References
@article{dephilippis2014,
author = {De Philippis, Guido and Figalli, Alessio},
title = {The {M}onge--{A}mp{\`e}re equation and its link to optimal transportation},
journal = {Bull. Amer. Math. Soc. (N.S.)},
volume = {51},
number = {4},
year = {2014},
pages = {527--580},
doi = {10.1090/S0273-0979-2014-01459-4},
eprint = {1310.6167},
archivePrefix = {arXiv},
}
@article{dephilippis2013,
author = {De Philippis, G. and Figalli, A. and Savin, O.},
title = {A note on interior {$W^{2,1+\varepsilon}$} estimates for the {M}onge--{A}mp{\`e}re equation},
journal = {Math. Ann.},
volume = {357},
number = {1},
year = {2013},
pages = {11--22},
doi = {10.1007/s00208-012-0895-9},
eprint = {1202.5566},
archivePrefix = {arXiv},
}
@misc{aim-planar-monge-ampere-sharp-hessian-integrability,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp weak integrability of planar Monge–Ampère Hessians}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/390-planar-monge-ampere-sharp-hessian-integrability.md},
note = {Entry 390 at commit 24edaad, 2026-10-15},
}
Resolution claims
Let \(0<\lambda<1\) and let \(u\) be a convex Alexandrov solution on a bounded convex planar domain, with Monge--Ampère density between \(\lambda\) and \(\lambda^{-1}\). We prove that its full distributional Hessian belongs locally to strong \(L^{p_\lambda}\), where \(p_\lambda=(1+\lambda^2)/(1-\lambda^2)\). This implies the corresponding weak endpoint estimate posed in the Monge--Ampère regularity literature. The proof uses a change of variables with a Lipschitz …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00818, "Sharp weak integrability of planar Monge–Ampère Hessians", clopen.solutions.