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Curvature estimates for convex solutions of some fully nonlinear Hessian-type equations

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Abstract

In the research of fully nonlinear elliptic partial differential equations, the concavities of the nonlinear operators is always essential. The concavity is the key to establish the curvature estimates of these equations. However, very few concave operators are known. In the present paper, some new examples are provided, which are the sum Hessian operators and the linear combination of k Hessian operators. We obtain the concavities and the quotient concavities of these new operators. As an application of our concavities, we establish the curvature estimates of convex solutions for the equations with general right-hand side defined by these new operators.

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References

  1. Alexandrov, A.D.: Existence and uniqueness of a convex surface with a given integral curvature. Dokl. Acad. Nauk Kasah SSSR 36, 131–134 (1942)

    MathSciNet  Google Scholar 

  2. Alexandrov, A.D.: Uniqueness theorems for surfaces in the large. I (Russian). Vestnik Leningrad. Univ. 11, 5–17 (1956). (English translation: AMS Translations, series 2, 21, (1962), 341–354)

    MathSciNet  Google Scholar 

  3. Bakelman, I., Kantor, B.: Existence of spherically homeomorphic hypersurfaces in Euclidean space with prescribed mean curvature. Geom. Topol. Leningr. 1, 3–10 (1974)

    Google Scholar 

  4. Ball, J.: Differentiability properties of symmetric and isotropic functions. Duke Math. J. 51, 699–728 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bryan, P., Ivaki, M.N., Scheuer, J.: A Unified Flow Approach to Smooth, Even \(L_p\)-Minkowski Problems. arXiv:1608.02770

  6. Caffarelli, L.A., Guan, P., Ma, X.: A constant rank theorem for solutions of fully nonlinear elliptic equations. Commun. Pure Appl. Math. LX, 1769–1791 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caffarelli, L.A., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations I. Monge-Ampère equations. Commun. Pure Appl. Math. 37, 369–402 (1984)

    Article  MATH  Google Scholar 

  8. Caffarelli, L.A., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations, III: functions of the eigenvalues of the Hessian. Acta Math. 155, 261–301 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caffarelli, L.A., Nirenberg, L., Spruck, J.: Nonlinear second order elliptic equations IV: starshaped compact Weigarten hypersurfaces. In: Ohya, Y., Kasahara, K., Shimakura, N. (eds.) Current Topics in Partial Differential Equations, pp. 1–26. Kinokunize, Tokyo (1985)

    Google Scholar 

  10. Caffarelli, L.A., Nirenberg, L., Spruck, J.: Nonlinear second order elliptic equations. V. The Dirichlet problem for Weingarten hypersurfaces. Commun. Pure Appl. Math. 41, 41–70 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, D., Li, H., Wang, Z.: Starshaped compact hypersurfaces with prescribed Weingarden curvature in warped product manifolds. Calc. Var. Partial Differ. Equ. 57, 42 (2018). https://doi.org/10.1007/s00526-018-1314-1

    Article  MATH  Google Scholar 

  12. Cheng, S.Y., Yau, S.T.: On the regularity of the solution of the \(n\)-dimensional Minkowski problem. Commun. Pure Appl. Math. 29, 495–516 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chou, K.S., Wang, X.J.: A variational theory of the Hessian equation. Commun. Pure Appl. Math. 54, 1029–1064 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dong, H.: Hessian equations with elementary symmetric functions. Commun. Partial Differ. Equ. 31, 1005–1025 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Espinar, J.M., Gálvez, J.A., Mira, P.: Hypersurfaces in \(\mathbb{H}^{n+1}\) and conformally invariant equations: the generalized Christoffel and Nirenberg problems. J. Eur. Math. Soc. 11, 903–939 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Garding, L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959)

    MathSciNet  MATH  Google Scholar 

  17. Guan, B.: The Dirichlet problem for a class of fully nonlinear elliptic equations. Commun. Partial Differ. Equ. 19(3–4), 399–416 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guan, B.: The Dirichlet problem for Hessian equations on Riemannian manifolds. Calc. Var. Partial Differ. Equ. 8, 45–69 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guan, B.: Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds. Duke Math. J. 163, 1491–1524 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Guan, B., Guan, P.: Convex hypersurfaces of prescribed curvature. Ann. Math. 156, 655–674 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guan, P., Lin, C.-S.: Private note

  22. Guan, P., Lu, S.: Curvature estimates for immersed hypersurfaces in Riemannian manifolds. Invent. Math. 208(1), 191–215 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Guan, P., Zhang, X.: A class of curvature type equations. Private note (2013)

  24. Guan, P., Li, J., Li, Y.Y.: Hypersurfaces of prescribed curvature measure. Duke Math. J. 161, 1927–1942 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Guan, P., Lin, C.S., Ma, X.: The existence of convex body with prescribed curvature measures. Int. Math. Res. Not. 2009, 1947–1975 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Guan, P., Ren, C., Wang, Z.: Global \(C^2\) estimates for curvature equation of convex solution. Commun. Pure Appl. Math. LXVIII, 1287–1325 (2015)

    Article  MATH  Google Scholar 

  27. Guan, B., Spruck, J., Xiao, L.: Interior curvature estimates and the asymptotic plateau problem in hyperbolic space. J. Differ. Geom. 96(2), 201–222 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Huisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183(1), 45–70 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Harvey, F.R., Lawson Jr., H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  30. Harvey, F.R., Lawson, Jr., H.B.: Hyperbolic Polynomials and the Dirichlet Problem. arXiv:0912.5220

  31. Harvey, F.R., Lawson Jr., H.B.: Garding’s theory of hyperbolic polynomials. Commun. Pure Appl. Math. 66(7), 1102–1128 (2013)

    Article  MATH  Google Scholar 

  32. Ivochkina, N.: Solution of the Dirichlet problem for curvature equations of order m. Math. USSR Sb. 67, 317–339 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ivochkina, N.: The Dirichlet problem for the equations of curvature of order m. Leningr. Math. J. 2–3, 192–217 (1991)

    MATH  Google Scholar 

  34. Krylov, N.V.: On the general notion of fully nonlinear second order elliptic equation. Trans. Am. Math. Soc. 347(3), 857–895 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  35. Korevaar, N.J.: A priori interior gradient bounds for solutions to elliptic Weingarten equations. Ann. Inst. H. Poincaré, Anal. Non Linéaire 4, 405–421 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, M., Ren, C., Wang, Z.: An interior estimate for convex solutions and a rigidity theorem. J. Funct. Anal. 270, 2691–2714 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Li, Y.: Interior gradient estimates for solutions of certain fully nonlinear elliptic equations. J. Differ. Equ. 90(1), 172–185 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  38. Marcus, L., Lopes, L.: Inequalities for symmetric functions and Hermitian matrices. Can. J. Math. 9, 305–312 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  39. Nirenberg, L.: The Weyl and Minkowski problems in differential geometry in the large. Commun. Pure Appl. Math. 6, 337–394 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  40. Phong, D.H., Picard, S., Zhang, X.: The Fu–Yau equation with negative slope parameter. Invent. Math. 209(2), 541–576 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Phong, D.H., Picard, S., Zhang, X.: On estimates for the Fu–Yau generalization of a Strominger system. J. Reine Angew. Math. 751, 243–274 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  42. Phong, D.H., Picard, S., Zhang, X.: A second order estimates for general complex Hessian equations. Anal. PDE 9(7), 1693–1709 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. Pogorelov, A.V.: On the question of the existence of a convex surface with a given sum principal radii of curvature ( in Russian). Uspekhi Mat. Nauk. 8, 127–130 (1953)

    Google Scholar 

  44. Pogorelov, A.V.: The Minkowski Multidimensional Problem. Wiley, Hoboken (1978)

    MATH  Google Scholar 

  45. Ren, C.: The curvature estimates for sum type Hessian equations. preprint

  46. Ren, C., Wang, Z.: On the curvature estimates for Hessian equation. Am. J. Math. 141(5), 1281–1315 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sheng, W.M., Urbas, J., Wang, X.J.: Interior curvature bounds for a class of curvature equations. Duke Math. J. 123, 235–264 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. Spruck, J., Xiao, L.: A note on starshaped compact hypersurfaces with prescribed scalar curvature in space form. Rev. Mat. Iberoam. 33(2), 547–554 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  49. Treibergs, A., Wei, S.W.: Embedded hypersurfaces with prescribed mean curvature. J. Differ. Geom. 18, 513–521 (1983)

    Article  MATH  Google Scholar 

  50. Trudinger, N.S.: On degenerate fully nonlinear elliptic equations in balls. Bull. Aust. Math. Soc. 35(2), 299–307 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  51. Trudinger, N.S.: The Dirichlet problem for the prescribed curvature equations. Arch. Ration. Mech. Anal. 111, 153–170 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  52. Trudinger, N.S.: On the Dirichlet problem for Hessian equations. Acta Math. 175, 151–164 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  53. Wang, X.: The k-Hessian Equation. Geometric Analysis and PDEs. Lecture Notes in Mathematics (1977). Springer, Dordrecht (2009)

    Google Scholar 

  54. Xiao, L.: Motion of Level Set By General Curvature. arXiv:1602.0211

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Acknowledgements

The authors wish to thank Professor Pengfei Guan for introducing the problem and for his contribution with Professor C.-S. Lin on the admissible set. The authors are also grateful to Xiangwen Zhang for his interest in their work and for sharing the unpublished note [23] with them. The work started when the second and third authors were visiting the Shanghai Centre for Mathematical Sciences in 2014. They would like to thank the Centre for its support and hospitality. Part of the work was conducted while the first and last authors were visiting McGill University. They are grateful to the University for its hospitality and the support of the CSC program in 2014-2015. The authors also thank the anonymous referees for their valuable suggestions to improve the quality of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, China

    Chunhe Li

  2. School of Mathematical Sciences, Jilin University, Changchun, China

    Changyu Ren

  3. School of Mathematical Sciences, Fudan University, Shanghai, China

    Zhizhang Wang

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  1. Chunhe Li
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  2. Changyu Ren
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  3. Zhizhang Wang
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Correspondence to Zhizhang Wang.

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Communicated by A. Chang.

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Research of the Chunhe Li is supported by FRFCU Grant No. ZYGX2016J135, the Changyu Ren is supported by NSFC Grant No. 11871243 and the Zhizhang Wang is partially supported by NSFC Grant Nos. 11871161, 11671090 and 11771103.

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Li, C., Ren, C. & Wang, Z. Curvature estimates for convex solutions of some fully nonlinear Hessian-type equations. Calc. Var. 58, 188 (2019). https://doi.org/10.1007/s00526-019-1623-z

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