Bounded-Hessian Liouville rigidity in dimensions three and four
Let $n \in \{3, 4\}$ and let $F \in C^\infty(\operatorname{Sym}_n)$, a smooth function on the real symmetric $n \times n$ matrices, be uniformly elliptic: for some $0 < \lambda \le \Lambda$, $$\lambda \operatorname{tr} N \le F(M+N) - F(M) \le \Lambda \operatorname{tr} N \qquad (M, N \in \operatorname{Sym}_n,\ N \ge 0).$$ No convexity or concavity of $F$ or of its zero level set is assumed. In two dimensions every solution of $F(D^2 u) = 0$ on $\mathbb{R}^2$ with bounded Hessian is a quadratic polynomial, by work of Nirenberg, and in every dimension this holds when $\{F = 0\}$ is a convex hypersurface, by work of Evans and of Krylov. Mooney notes that, for solutions not assumed smooth, it fails in dimensions $n \ge 5$, citing a non-smooth solution in dimension five constructed by Nadirashvili, Tkachev and Vlăduţ. The theorem of Han, Nadirashvili and Yuan on homogeneous solutions of order one implies that solutions on $\mathbb{R}^3$ homogeneous of degree two are quadratic polynomials, and Nadirashvili and Vlăduţ proved the same on $\mathbb{R}^4$ for solutions analytic away from the origin.
Problem. If $u \in C^\infty(\mathbb{R}^n)$ satisfies $F(D^2 u) = 0$ on $\mathbb{R}^n$ and $\sup_{\mathbb{R}^n} |D^2 u| < \infty$, must $u$ be a polynomial of degree at most two? The question is asked for both $n = 3$ and $n = 4$.
Source: Mooney (2024), §2, after equation (4), who records it as open in dimensions three and four; stated for uniformly elliptic F by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, fully nonlinear elliptic equations, Liouville theorems
References
@article{mooney2024,
author = {Mooney, Connor},
title = {Bernstein theorems for nonlinear geometric {PDE}s},
journal = {Commun. Pure Appl. Anal.},
volume = {23},
number = {12},
year = {2024},
pages = {1958--1989},
doi = {10.3934/cpaa.2024072},
eprint = {2407.11903},
archivePrefix = {arXiv},
}
@article{nirenberg1953,
author = {Nirenberg, Louis},
title = {On nonlinear elliptic partial differential equations and {H}{\"o}lder continuity},
journal = {Comm. Pure Appl. Math.},
volume = {6},
number = {1},
year = {1953},
pages = {103--156},
doi = {10.1002/cpa.3160060105},
}
@article{evans1982,
author = {Evans, Lawrence C.},
title = {Classical solutions of fully nonlinear, convex, second-order elliptic equations},
journal = {Comm. Pure Appl. Math.},
volume = {35},
number = {3},
year = {1982},
pages = {333--363},
doi = {10.1002/cpa.3160350303},
}
@article{krylov1983,
author = {Krylov, N. V.},
title = {Boundedly nonhomogeneous elliptic and parabolic equations},
journal = {Math. USSR-Izv.},
volume = {20},
number = {3},
year = {1983},
pages = {459--492},
doi = {10.1070/IM1983v020n03ABEH001360},
}
@article{nadirashvili2012,
author = {Nadirashvili, Nikolai and Tkachev, Vladimir and Vl{\u{a}}du{\c{t}}, Serge},
title = {A non-classical solution to a {H}essian equation from {C}artan isoparametric cubic},
journal = {Adv. Math.},
volume = {231},
number = {3-4},
year = {2012},
pages = {1589--1597},
doi = {10.1016/j.aim.2012.07.005},
eprint = {1111.0329},
archivePrefix = {arXiv},
}
@article{han2003,
author = {Han, Qing and Nadirashvili, Nikolai and Yuan, Yu},
title = {Linearity of homogeneous order-one solutions to elliptic equations in dimension three},
journal = {Comm. Pure Appl. Math.},
volume = {56},
number = {4},
year = {2003},
pages = {425--432},
doi = {10.1002/cpa.10064},
}
@article{nadirashvili2013,
author = {Nadirashvili, Nikolai and Vl{\u{a}}du{\c{t}}, Serge},
title = {Homogeneous solutions of fully nonlinear elliptic equations in four dimensions},
journal = {Comm. Pure Appl. Math.},
volume = {66},
number = {10},
year = {2013},
pages = {1653--1662},
doi = {10.1002/cpa.21456},
eprint = {1201.1022},
archivePrefix = {arXiv},
}
@misc{aim-fully-nonlinear-bounded-hessian-liouville,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Bounded-Hessian Liouville rigidity in dimensions three and four}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/452-fully-nonlinear-bounded-hessian-liouville.md},
note = {Entry 452 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00596, "Bounded-Hessian Liouville rigidity in dimensions three and four", clopen.solutions.