Abstract
In this paper, we investigate the Monge–Ampère equation \(\text{ det }D^2u=f \) in \({\mathbb {R}}^n_+\), where f is bounded, positive and \(f(x)=1+O(|x|^{-\beta })\) for some \(\beta >2\) at infinity. If u is a quadratic polynomial on \(\{x_n=0\}\) and satisfies \( \mu |x|^2\le u\le \mu ^{-1}|x|^2\) for some \(0<\mu \le \frac{1}{2}\) at infinity, then u tends to a quadratic polynomial at infinity with at least \(O((\frac{x_n}{|x|^{n}})^\delta )\) decay rate, where \(\delta >0\) is some constant depending only on \(\beta \) and n. Meanwhile, the existence and uniqueness of viscosity solutions of the Dirichlet problem with prescribed asymptotic behavior at infinity will be concerned. The condition \(\beta >2\) is sharp.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.G. Bao, H.G. Li, and L. Zhang. Monge-Ampère equation on exterior domains. Calc. Var. Partial Differential Equations, 52(1–2), 39–63, 2015
L.A. Caffarelli. Interior \(W^{2,p}\) estimates for solutions of the Monge-Ampère equation. Ann. of Math., 131(1):135–150, 1990
L.A. Caffarelli. A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity. Ann. of Math., 131(1):129–134, 1990
Caffarelli, L.A.: Topics in PDEs: The Monge–Ampère equation. Graduate course. Courant Institute, New York University (1995)
L.A. Caffarelli and Y.Y. Li. An extension to a theorem of Jörgens, Calabi, and Pogorelov. Comm. Pure Appl. Math., 56(5):549–583, 2003
E. Calabi. Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Michigan Math. J., 5:105–126, 1958
L. Ferrer, A. Martínez, and F. Milán. An extension of a theorem by K Jörgens and a maximum principle at infinity for parabolic affine spheres. Math. Z., 230(3):471–486, 1999
L. Ferrer, A. Martínez, and F. Milán. The space of parabolic affine spheres with fixed compact boundary. Monatsh. Math., 130(1):19–27, 2000
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition
Jia, X.B., Li, D.S.: The asymptotic behavior of viscosity solutions of Monge-Ampère equations in half space. Nonlinear Anal. 206, 112229 (2021)
X.B. Jia, D.S. Li, and Z.S. Li. Asymptotic behavior at infinity of solutions of Monge-Ampère equations in half spaces. J. Differential Equations, 269(1), 326–348, 2020
H.Y. Jian and X.-J. Wang. Existence of entire solutions to the Monge-Ampère equation. Amer. J. Math., 136(4):1093–1106, 2014
K. Jörgens. über die Lösungen der Differentialgleichung \(rt-s^2=1\). Math. Ann., 127:130–134, 1954
Mooney, C.: Monge–Ampère equation. https://people.math.ethz.ch/~mooneyc/
A.V. Pogorelov. On the improper convex affine hyperspheres. Geometriae Dedicata, 1(1), 33–46, 1972
O. Savin. Pointwise \(C^{2,\alpha }\) estimates at the boundary for the Monge-Ampère equation. J. Amer. Math. Soc., 26(1):63–99, 2013
O. Savin. A localization theorem and boundary regularity for a class of degenerate Monge-Ampere equations. J. Differential Equations, 256(2), 327–388, 2014
Trudinger, N.S., Wang, X.-J.: The Monge–Ampère equation and its geometric applications. In: Handbook of Geometric Analysis. No. 1, Volume 7 of Adv. Lect. Math. (ALM), pp. 467–524. International Press, Somerville (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Jia, X. On the existence and asymptotic behavior of viscosity solutions of Monge–Ampère equations in half spaces. manuscripta math. 170, 19–33 (2023). https://doi.org/10.1007/s00229-021-01354-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-021-01354-y