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Curvature estimates for a class of fully nonlinear elliptic equations with general right hand sides

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In this paper, we establish the curvature estimates for a class of curvature equations with general right hand sides depending on the gradient. We show an existence result by using the continuity method based on a priori estimates. We also derive interior curvature bounds for solutions of a class of curvature equations subject to affine Dirichlet data.

Abstract

In this paper, we establish the curvature estimates for a class of curvature equations with general right hand sides depending on the gradient. We show an existence result by using the continuity method based on a priori estimates. We also derive interior curvature bounds for solutions of a class of curvature equations subject to affine Dirichlet data.

참고문헌 (37)

  1. I. J. Bakelman, "Existence of a hypersurface homeomorphic to the sphere in Euclidean space with a given mean curvature", Geometry and topology, (2), pp.3~10 (1974), Leningrad. Gos. Ped. Inst.
  2. L. Caffarelli, "The Dirichlet problem for nonlinear secondorder elliptic equations. I. Monge-Ampere equation", Comm. Pure Appl. Math, 37(3), pp.369~402 (1984)[DOI]
  3. L. Caffarelli, "The Dirichlet problem for nonlinear secondorder elliptic equations. III. Functions of the eigenvalues of the Hessian", Acta Math, 155(3-4), pp.261~301 (1985)[DOI]
  4. L. Caffarelli, "Nonlinear second order elliptic equations. IV. Starshaped compact Weingarten hypersurfaces", Current topics in partial differential equations, pp.1~26 (1986)
  5. C. Chen, "Interior Hessian estimates for a class of Hessian type equations", Calc. Var. Partial Differential Equations, 62(2), pp.52~ (2023)[DOI]
  6. X. Chen, "A class of Hessian quotient equations in Euclidean space", J. Differential Equations, 269(12), pp.11172~11194 (2020)[DOI]
  7. L. Chen, "Pogorelov type estimates for a class of Hessian quotient equations", J. Differential Equations, 282, pp.272~284 (2021)[DOI]
  8. J. Chu, "Curvature estimates for a class of Hessian type equations", Calc. Var. Partial Differential Equations, 60(3, Paper No. 90), pp.18~ (2021)[DOI]
  9. W. Dong, "Curvature estimates for p-convex hypersurfaces of prescribed curvature", Rev. Mat. Iberoam, 39(3), pp.1039~1058 (2023)[DOI]
  10. L. C. Evans, "Classical solutions of fully nonlinear, convex, second-order elliptic equations", Comm. Pure Appl. Math, 35(3), pp.333~363 (1982)[DOI]
  11. P. Gauduchon, "La 1-forme de torsion d’une vari´et´e hermitienne compacte", Math. Ann, 267(4), pp.495~518 (1984)[DOI]
  12. C. Gerhardt, "Closed Weingarten hypersurfaces in Riemannian manifolds", J. Differential Geom, 43(3), pp.612~641 (1996)
  13. B. Guan, "The Dirichlet problem for Hessian equations on Riemannian manifolds", Calc. Var. Partial Differential Equations, 8(1), pp.45~69 (1999)[DOI]
  14. B. Guan, "Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds", Duke Math. J, 163(8), pp.1491~1524 (2014)[DOI]
  15. B. Guan, "Convex hypersurfaces of prescribed curvatures", Ann. of Math, 156(2), pp.655~673 (2002)[DOI]
  16. P. Guan, "C1,1 estimates for solutions of a problem of Alexandrov", Comm. Pure Appl. Math, 50(8), pp.789~811 (1997)
  17. P. Guan, "Hypersurfaces of prescribed curvature measure", Duke Math. J, 161(10), pp.1927~1942 (2012)[DOI]
  18. P. Guan, "The existence of convex body with prescribed curvature measures", Int. Math. Res. Not. IMRN, 2009(11), pp.1947~1975 (2009)[DOI]
  19. P. Guan, "Interior C2 regularity of convex solutions to prescribing scalar curvature equations", Duke Math. J, 168(9), pp.1641~1663 (2019)[DOI]
  20. P. Guan, "Global C2-estimates for convex solutions of curvature equations", Comm. Pure Appl. Math, 68(8), pp.1287~1325 (2015)[DOI]
  21. N. M. Ivochkina, "Solution of the Dirichlet problem for curvature equations of order m", Math. USSR-Sb, 67(2), pp.317~339 (1990)[DOI]
  22. N. M. Ivochkina, "The Dirichlet problem for the equations of curvature of order m", Leningrad Math. J, 2(3), pp.631~654 (1991)
  23. Y. Y. Li, "Degree theory for second order nonlinear elliptic operators and its applications", Comm. Partial Differential Equations, 14(11), pp.1541~1578 (1989)[DOI]
  24. V. I. Oliker, "Hypersurfaces in Rn+1 with prescribed Gaussian curvature and related equations of Monge-Amp`ere type", Comm. Partial Differential Equations, 9(8), pp.807~838 (1984)[DOI]
  25. A. Pogorelov, "The Minkowski multidimensional problem" (1978), Scripta Series in Mathematics
  26. C. Ren, "On the curvature estimates for Hessian equations", Amer. J. Math, 141(5), pp.1281~1315 (2019)[DOI]
  27. W. Sheng, "Interior curvature bounds for a class of curvature equations", Duke Math. J, 123(2), pp.235~264 (2004)[DOI]
  28. J. Spruck, "A note on star-shaped compact hypersurfaces with prescribed scalar curvature in space forms", Rev. Mat. Iberoam, 33(2), pp.547~554 (2017)[DOI]
  29. G. Sz´ekelyhidi, "Gauduchon metrics with prescribed volume form", Acta Math, 219(1), pp.181~211 (2017)[DOI]
  30. A. E. Treibergs, "Embedded hyperspheres with prescribed mean curvature", J. Differential Geom, 18(3), pp.513~521 (1983)
  31. J. Urbas, "On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations", Indiana Univ. Math. J, 39(2), pp.355~382 (1990)[DOI]
  32. J. Urbas, "An interior curvature bound for hypersurfaces of prescribed k-th mean curvature", J. Reine Angew. Math, 519, pp.41~57 (2000)[DOI]
  33. J. D. Zhou, "k-Hessian curvature type equations in space forms", Electron. J. Differential Equations, (18), pp.14~ (2022)

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