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Regularity of solutions of two-dimensional Monge-Ampère equations


Authors: Friedmar Schulz and Liang Yuan Liao
Journal: Trans. Amer. Math. Soc. 307 (1988), 271-277
MSC: Primary 35J60; Secondary 35B65
MathSciNet review: 936816
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Abstract: In the paper we investigate the regularity of solutions $ z(x,\,y) \in {C^{1,1}}(\Omega )$, resp. $ {C^{1,1}}(\overline \Omega )$ of elliptic Monge-Ampére equations of the form

$\displaystyle Ar + 2Bs + Ct + (rt - {s^2}) = E.$

It is shown that $ z(x,\,y) \in {C^{2,\alpha }}(\Omega )$, resp. $ {C^{2,\alpha }}(\overline \Omega )$, with corresponding a priori estimates, if $ A,\,B,\,C,\,E \in {C^\alpha }(\Omega \times {{\mathbf{R}}^3})$. The results are deduced via the Campanato technique for equations of variational structure invoking a Legendre-like transformation.

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  • [1] A. D. Alexandrow, Die innere Geometrie der konvexen Flächen, Akademie-Verlag, Berlin, 1955 (German). MR 0071041
  • [2] I. Ya. Bakel′man, Generalized solutions of Monge-Ampère equations, Dokl. Akad. Nauk SSSR (N.S.) 114 (1957), 1143–1145 (Russian). MR 0095481
  • [3] Sergio Campanato, Equazioni ellittiche del 𝐼𝐼deg ordine espazi 𝔏^{(2,𝜆)}, Ann. Mat. Pura Appl. (4) 69 (1965), 321–381 (Italian). MR 0192168
  • [4] -, Sistemi ellittici in forma divergenza. Regolarità all'interno, Quaderni, Scuola Normale Superiore Pisa, Pisa, 1980.
  • [5] Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
  • [6] Erhard Heinz, Interior estimates for solutions of elliptic Monge-Ampère equations, Proc. Sympos. Pure Math., Vol. IV, American Mathematical Society, Providence, R.I., 1961, pp. 149–155. MR 0157100
  • [7] Hans Lewy, A priori limitations for solutions of Monge-Ampère equations, Trans. Amer. Math. Soc. 37 (1935), no. 3, 417–434. MR 1501794, 10.1090/S0002-9947-1935-1501794-9
  • [8] Louis Nirenberg, On nonlinear elliptic partial differential equations and Hölder continuity, Comm. Pure Appl. Math. 6 (1953), 103–156; addendum, 395. MR 0064986
  • [9] A. V. Pogorelov, Monge-Ampère equations of elliptic type, Translated from the first Russian edition by Leo F. Boron with the assistance of Albert L. Rabenstein and Richard C. Bollinger, P. Noordhoff, Ltd., Groningen, 1964. MR 0180763
  • [10] I. Kh. Sabitov, The regularity of convex regions with a metric that is regular in the Hölder classes, Siberian Math. J. 17 (1976), 681-687.
  • [11] M. V. Safonov, On the classical solution of Bellman's elliptic equation, Soviet Math. Dokl. 30 (1984), 482-485.
  • [12] F. Schulz, Über elliptische Monge-Ampéresche Differentialgleichungen mit einer Bemerkung zum Weylschen Einbettungsproblem, Nachr. Akad. Wiss. Göttingen II: Math. Phys. Kl. 1981, 93-108.
  • [13] Friedmar Schulz, Über die Differentialgleichung 𝑟𝑡-𝑠²=𝑓 und das Weylsche Einbettungsproblem, Math. Z. 179 (1982), no. 1, 1–10 (German). MR 643043, 10.1007/BF01173911
  • [14] Friedmar Schulz, A priori estimates for solutions of Monge-Ampère equations, Arch. Rational Mech. Anal. 89 (1985), no. 2, 123–133. MR 786542, 10.1007/BF00282328
  • [15] Friedmar Schulz, Boundary estimates for solutions of Monge-Ampère equations in the plane, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 3, 431–440. MR 785620
  • [16] Friedmar Schulz, Über nichtlineare, konkave elliptische Differentialgleichungen, Math. Z. 191 (1986), no. 3, 429–448 (German). MR 824444, 10.1007/BF01162718
  • [17] Neil S. Trudinger, Regularity of solutions of fully nonlinear elliptic equations, Boll. Un. Mat. Ital. A (6) 3 (1984), no. 3, 421–430. MR 769173

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DOI: https://doi.org/10.1090/S0002-9947-1988-0936816-1
Article copyright: © Copyright 1988 American Mathematical Society