Solutions of the minimal surface equation and of the Monge–Ampère equation near infinity

References

[1] F. J. Almgren, Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem, Ann. of Math. (2) 84 (1966), 277–292. 10.2307/1970520Search in Google Scholar

[2] J. Bao, H. Li and L. Zhang, Monge–Ampère equation on exterior domains, Calc. Var. Partial Differential Equations 52 (2015), 39–63. 10.1007/s00526-013-0704-7Search in Google Scholar

[3] S. Bernstein, Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus, Math. Z. 26 (1927), 551–558. 10.1007/BF01475472Search in Google Scholar

[4] L. Bers, Isolated singularities of minimal surfaces, Ann. of Math. (2) 53 (1951), 364–386. 10.2307/1969547Search in Google Scholar

[5] E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. 10.1007/BF01404309Search in Google Scholar

[6] L. Caffarelli, Topics in PDEs: The Monge–Ampère equation, Graduate course, Courant Institute, New York University, 1995. Search in Google Scholar

[7] L. Caffarelli and Y. Li, An extension to a theorem of Jörgens, Calabi, and Pogorelov, Comm. Pure Appl. Math. 56 (2003), no. 5, 549–583. 10.1002/cpa.10067Search in Google Scholar

[8] E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J. 5 (1958), 105–126. 10.1307/mmj/1028998055Search in Google Scholar

[9] E. De Giorgi, Una estensione del teorema di Bernstein, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3) 19 (1965), 79–85. Search in Google Scholar

[10] 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 (1999), no. 3, 471–486. 10.1007/PL00004700Search in Google Scholar

[11] W. H. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2) 11 (1962), 69–90. 10.1007/BF02849427Search in Google Scholar

[12] Q. Han, Schauder estimates for elliptic operators with applications to nodal sets, J. Geom. Anal. 10 (2000), no. 3, 455–480. 10.1007/BF02921945Search in Google Scholar

[13] G. Hong and Y. Yuan, Maximal hypersurfaces over exterior domains, Comm. Pure Appl. Math. 74 (2021), no. 3, 589–614. 10.1002/cpa.21929Search in Google Scholar

[14] K. Jörgens, Über die Lösungen der Differentialgleichung r ⁢ t − s 2 = 1 , Math. Ann. 127 (1954), 130–134. 10.1007/BF01361114Search in Google Scholar

[15] D. Li, Z. Li and Y. Yuan, A Bernstein problem for special Lagrangian equations in exterior domains, Adv. Math. 361 (2020), Article ID 106927. 10.1016/j.aim.2019.106927Search in Google Scholar

[16] Z. Liu and J. Bao, Asymptotic expansion at infinity of solutions of Monge–Ampère type equations, Nonlinear Anal. 212 (2021), Article ID 112450. 10.1016/j.na.2021.112450Search in Google Scholar

[17] J. Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. 10.1002/cpa.3160140329Search in Google Scholar

[18] J. C. C. Nitsche, Vorlesungen über Minimalflächen, Grundlehren Math. Wiss. 199, Springer, Berlin 1975. 10.1007/978-3-642-65619-4Search in Google Scholar

[19] A. V. Pogorelov, On the improper convex affine hyperspheres, Geom. Dedicata 1 (1972), no. 1, 33–46. 10.1007/BF00147379Search in Google Scholar

[20] R. M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18 (1983), no. 4, 791–809. 10.4310/jdg/1214438183Search in Google Scholar

[21] L. Simon, Asymptotic behaviour of minimal graphs over exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), no. 3, 231–242. 10.1016/s0294-1449(16)30367-5Search in Google Scholar

[22] J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. 10.2307/1970556Search in Google Scholar