We prove the existence of entire solutions of the Monge–Ampère equations with prescribed asymptotic behavior at infinity of the plane, which was left unsolved by Caffarelli–Li in 2003. The special difficulty of the problem in dimension two is due to the global logarithmic term in the asymptotic expansion of solutions at infinity. Furthermore, we give a PDE proof of the characterization of the space of solutions of the Monge–Ampère equation \(\det \nabla ^2 u=1\) with \(k\ge 2\) singular points, which was established by Gálvez–Martínez–Mira in 2005. We also obtain the existence of solutions in higher dimensional cases with general right hand sides.
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Bao, J., Li, H., Zhang, L.: Monge–Ampère equation on exterior domains. Calc. Var. Partial Differ. Equ. 52, 39–63 (2015)
Caffarelli, L.A.: Topics in PDEs: The Monge–Ampère equation. Graduate course. Courant Institute, New York University (1995)
Caffarelli, L.A., Li, Y.Y.: An extension to a theorem of Jörgens, Calabi, and Pogorelov. Commun. Pure Appl. Math. 56, 549–583 (2003)
Caffarelli, L.A., Li, Y.Y.: A Liouville theorem for solutions of the Monge–Ampère equation with periodic data. Ann. Inst. H. Poincar Anal. Non Linaire 21, 97–120 (2004)
Calabi, E.: Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Mich. Math. J. 5, 105–126 (1958)
Cheng, S.Y., Yau, S.T.: Complete affine hypersurfaces. I. The completeness of affine metrics. Commun. Pure Appl. Math. 39(6), 839–866 (1986)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
Ferrer, L., Martínez, A., Milán, F.: The space of parabolic affine spheres with fixed compact boundary. Monatsh. Math. 130(1), 19–27 (2000)
Figalli, A.: The Monge–Ampère Equation and its Applications. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2017)
Gálvez, J.A., Martínez, A., Mira, P.: The space of solutions to the Hessian one equation in the finitely punctured plane. J. Math. Pures Appl. (9) 84(12), 1744–1757 (2005)
Gutierrez, C.E.: The Monge–Ampère Equation. Progress in Nonlinear Differential Equations and Applications, vol. 44. Birkhauser Boston Inc., Boston (2001)
Jörgens, K.: Über die Lösungen der Differentialgleichung \(rt-s^2=1\). Math. Ann. 127, 130–134 (1954)
Jörgens, K.: Harmonische Abbildungen und die Differentialgleichung \(rt-s^2=1\). Math. Ann. 129, 330–344 (1955)
Jost, J., Xin, Y.L.: Some aspects of the global geometry of entire space-like submanifolds. Results Math. 40, 233–245 (2001)
Jin, T., Xiong, J.: A Liouville theorem for solutions of degenerate Monge–Ampère equations. Commun. Partial Differ. Equ. 39, 306–320 (2014)
Jin, T., Xiong, J.: Solutions of some Monge–Ampère equations with isolated and line singularities. Adv. Math. 289, 114–141 (2016)
Li, Y.Y.: Some existence results of fully nonlinear elliptic equations of Monge–Ampère type. Commun. Pure Appl. Math. 43, 233–271 (1990)
Li, Y.Y., Lu, S.: Existence and nonexistence to exterior Dirichlet problem for Monge–Ampère equation. To appear in Calculus of Variatoins and PDEs
Nitsche, J.C.C.: Elementary proof of Bernsteins theorem on minimal surfaces. Ann. Math. 66, 543–544 (1957)
Pogorelov, A.V.: On the improper convex affine hyperspheres. Geom. Dedicata 1, 33–46 (1972)
Teixeira, E.V., Zhang, L.: Global Monge–Ampère equation with asymptotically periodic data. Indiana Univ. Math. J. 65, 399–422 (2016)
Communicated by O. Savin.
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All authors are supported in part by the key project NSFC 11631002, and J. Xiong is also supported in part by NSFC 11501034 and 11571019.
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Bao, J., Xiong, J. & Zhou, Z. Existence of entire solutions of Monge–Ampère equations with prescribed asymptotic behavior. Calc. Var. 58, 193 (2019). https://doi.org/10.1007/s00526-019-1639-4