No CrossRef data available.
Article contents
THE GROWTH OF SOLUTIONS OF MONGE–AMPÈRE EQUATIONS IN HALF SPACES AND ITS APPLICATION
Published online by Cambridge University Press: 31 March 2023
Abstract
We consider the growth of the convex viscosity solution of the Monge–Ampère equation 
$\det D^2u=1$ outside a bounded domain of the upper half space. We show that if u is a convex quadratic polynomial on the boundary 
$\{x_n=0\}$ and there exists some 
$\varepsilon>0$ such that 
$u=O(|x|^{3-\varepsilon })$ at infinity, then 
$u=O(|x|^2)$ at infinity. As an application, we improve the asymptotic result at infinity for viscosity solutions of Monge–Ampère equations in half spaces of Jia, Li and Li [‘Asymptotic behavior at infinity of solutions of Monge–Ampère equations in half spaces’, J. Differential Equations 269(1) (2020), 326–348].
MSC classification
Secondary:
35B40: Asymptotic behavior of solutions
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 109 , Issue 1 , February 2024 , pp. 125 - 137
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
The first author was supported by Natural Science Foundation of Henan Province (Grant No. 222300420321); the second author was supported by Natural Science Foundation of Henan Province (Grant No. 222300420232).
References
Caffarelli, L. A. and Li, Y. Y., ‘An extension to a theorem of Jörgens, Calabi and Pogorelov’, Comm. Pure Appl. Math. 56(5) (2003), 549–583.CrossRefGoogle Scholar
Figalli, A., The Monge–Ampère Equation and Its Applications, Zurich Lectures in Advanced Mathematics (European Mathematical Society (EMS), Zürich, 2017).CrossRefGoogle Scholar
Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 224 (Springer-Verlag, Berlin, 2001).CrossRefGoogle Scholar
Gutiérrez, C. E., The Monge–Ampère Equation, 2nd edn, Progress in Nonlinear Differential Equations and their Applications, 89 (Birkhäuser/Springer, Cham, 2016).CrossRefGoogle Scholar
Jia, X. B., Li, D. S. and Li, Z. S., ‘Asymptotic behavior at infinity of solutions of Monge–Ampère equations in half spaces’, J. Differential Equations 269(1) (2020), 326–348.CrossRefGoogle Scholar
Mooney, C., ‘Partial regularity for singular solutions to the Monge–Ampère equation’, Comm. Pure Appl. Math. 68(6) (2015), 1066–1084.CrossRefGoogle Scholar
Mooney, C., The Monge–Ampère equation, available online at https://www.math.uci.edu/~mooneycr/MongeAmpere_Notes.pdf.Google Scholar
Savin, O., ‘Pointwise
${C}^{2,\alpha }$
estimates at the boundary for the Monge–Ampère equation’, J. Amer. Math. Soc. 26(1) (2013), 63–99.CrossRefGoogle Scholar
${C}^{2,\alpha }$
estimates at the boundary for the Monge–Ampère equation’, J. Amer. Math. Soc. 26(1) (2013), 63–99.CrossRefGoogle ScholarSavin, O., ‘A localization theorem and boundary regularity for a class of degenerate Monge–Ampère equations’, J. Differential Equations 256(2) (2014), 327–388.CrossRefGoogle Scholar
Zhou, Z. W., Gong, S. Y. and Bao, J. G., ‘Ancient solutions of exterior problem of parabolic Monge–Ampère equations’, Ann. Mat. Pura Appl. (4) 200(4) (2021), 1605–1624.CrossRefGoogle Scholar