arXiv:2611.06978v1 Announce Type: new Abstract: 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 J\"{o}rgens, 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.
Continue to arXiv Mathematics →You are being forwarded in 6s — stay on this page