\(C^{2,\alpha }\) estimates for nonlinear elliptic equations of twisted type

详细信息    查看全文

• 作者：Tristan C. Collins
• 关键词：35J60 ; 35B45 ; 35J70
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：55
• 期：1
• 全文大小：444 KB
• 参考文献：1.Bian, B., Guan, P.: A microscopic convexity principle for nonlinear partial differential equations. Invent. Math. 177(2), 307–335 (2009)MATH MathSciNet CrossRef
  2.Caffarelli, L., Cabré, X.: Fully nonlinear elliptic equations, vol. 43. American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (1995)
  3.Caffarelli, L., Cabré, X.: Interior \(C^{2,\alpha }\) regularity theory for a class of nonconvex fully nonlinear elliptic equations. J. Math. Pures Appl. 9(5), 573–612 (2003)
  4.Caffarelli, L., Yuan, Y.: A priori estimates for solutions of fully nonlinear equations with convex level sets. Indiana Univ. Math. J. 49(2), 681–695 (2000)MATH MathSciNet CrossRef
  5.Collins, T.C., Székelyhidi, G.: Convergence of the J-flow on toric manifolds. J. Differ. Geom. To appear
  6.Evans, L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35(3), 333–363 (1982)MATH CrossRef
  7.Fu, J., Yau, S.-T.: The theory of superstring with flux on non-Kähler manifolds and the complex Monge–Ampère equation. J. Differ. Geom. 78, 369–428 (2008)MATH MathSciNet
  8.Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, Reprint of the 1998 edition. Classics in mathematics. Springer, Berlin (2001)
  9.Han, Q., Lin, F.: Elliptic partial differential equations, 2nd edn. Courant Lecture Notes in Mathematics 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2011)
  10.Krylov, N.V.: Boundedly nonhomogeneous elliptic and parabolic equations. Izv. Akad. Nak. SSSR Ser. Mat. 46, 487–523 (1982). English transl. in Math. USSR Izv. 20 (1983), 459–492MATH
  11.Krylov, N.V.: Boundedly nonhomogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nak. SSSR Ser. Mat. 47, 75–108 (1983). English transl. in Math. USSR Izv. 22 (1984), 67–97
  12.Krylov, N.V., Safonov, N.V.: A certain property of the solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 44, 161–175 (1980). In Russian. English translation in Math. USSR Izv. 16 (1981), 151–164MATH MathSciNet
  13.Nadirashvili, N., Vlăduţ, S.: Nonclassical solutions of fully nonlinear elliptic equations. Geom. Funct. Anal. 17, 1283–1296 (2007)MATH MathSciNet CrossRef
  14.Nirenberg, L.: On nonlinear elliptic partial differential equations and Hölder continuity. Commun. Pure. Appl. Math. 6, 103–156 (1953)MATH MathSciNet CrossRef
  15.Savin, O.: Small perturbation solutions for elliptic equations. Commun. Partial Differ. Equ. 32(4–6), 557–578 (2007)MATH MathSciNet CrossRef
  16.Streets, J.: Pluriclosed flow on generalized Kähler manifolds with split tangent bundle, preprint, arXiv:​1405.​0727
  17.Streets, J., Tian, G.: A parabolic flow of pluriclosed metrics. Int. Math. Res. Not. 16, 3101–3133 (2010)MathSciNet
  18.Streets, J., Tian, G.: Regularity results for the pluriclosed flow. Geom. Top. 17, 2389–2429 (2013)MATH MathSciNet CrossRef
  19.Streets, J., Tian, G.: Generalized Kähler geometry and the pluriclosed flow. Nucl. Phys. B 858(2), 366–376 (2012)MATH MathSciNet CrossRef
  20.Streets, J., Warren, M.: Evans–Krylov estimates for a nonconvex Monge–Ampère equation, Math. Ann., to appear
  21.Tosatti, V., Wang, Y., Weinkove, B., Yang, X.: \(C^{2,\alpha }\) estimates for nonlinear elliptic equations in complex and almost complex geometry. Calc. Var. Partial Differ. Equ. 54(1), 431–453 (2015)
  22.Wang, Y.: On the \(C^{2,\alpha }\) regularity of the complex Monge–Ampère equation. Math. Res. Lett. 19(4), 939–946 (2012)
  23.Yuan, Y.: A priori estimates for solutions of fully nonlinear special Lagrangian equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 18(2), 261–270 (2001)MATH CrossRef
  24.Yuan, Y.: Global solutions to special Lagrangian equations. Proc. Am. Math. Soc. 134(5), 1355–1358 (2006)MATH CrossRef
  
• 作者单位：Tristan C. Collins (1)
  
  1. Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA, 02138, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

We prove a priori interior \(C^{2,\alpha }\) estimates for solutions of fully nonlinear elliptic equations of twisted type. For example, our estimates apply to equations of the type convex + concave. These results are particularly well suited to equations arising from elliptic regularization. As application, we obtain a new proof of an estimate of Streets and Warren on the twisted real Monge–Ampère equation. Mathematics Subject Classification 35J60 35B45 35J70