Regularity Theory for the Equation det(D2u) = f(u) in Two Dimensions
Abstract
We study the interior regularity of convex solutions to det D2u = f(u) in planar domains, with no regularity or monotonicity assumptions on f. In particular, when f is bounded between positive constants, we show that u is locally C1, 1, and that its level sets near a critical point are asymptotically ellipses. Our analysis relies on a new monotone quantity for a second-order flow satisfied by the support functions of the level sets of u. As a byproduct of our methods, we obtain Liouville theorems that generalize a classical result of Jörgens, and include a complete classification of global solutions in 2D to an obstacle problem studied by Jin-Tu-Xiong. As another application, we obtain regularity and rigidity results for rotating-wave solutions of the semigeostrophic equations from meteorology.
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BibTeX
@misc{mooney2026regularity,
title = {{Regularity Theory for the Equation $\det(D^2u) = f(u)$ in Two Dimensions}},
author = {Connor Mooney and Aranya Sen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06978},
eprint = {2611.06978},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06978},
}
RIS
TY - UNPB
TI - Regularity Theory for the Equation det(D^2u) = f(u) in Two Dimensions
AU - Connor Mooney
AU - Aranya Sen
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06978
UR - https://arxiv.org/abs/2611.06978
AB - We study the interior regularity of convex solutions to det D^2u = f(u) in planar domains, with no regularity or monotonicity assumptions on f. In particular, when f is bounded between positive constants, we show that u is locally C^1, 1, and that its level sets near a critical point are asymptotically ellipses. Our analysis relies on a new monotone quantity for a second-order flow satisfied by the support functions of the level sets of u. As a byproduct of our methods, we obtain Liouville theorems that generalize a classical result of Jörgens, and include a complete classification of global solutions in 2D to an obstacle problem studied by Jin-Tu-Xiong. As another application, we obtain regularity and rigidity results for rotating-wave solutions of the semigeostrophic equations from meteorology.
ER -