Skip to main content
SpringerLink
Log in

On the Second Boundary Value Problem for a Class of Fully Nonlinear Equations

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We proved the existence of convex solution to a class of fully nonlinear elliptic equations with second boundary condition on uniformly convex domains in \(\mathbb {R}^{n}\), and then applied it to solve a boundary value problem for minimal Lagrangian graphs in the pseudo-Euclidean space \(\mathbb {R}^{2n}_n\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brendle, S., Warren, M.: A boundary value problem for minimal Lagrangian graphs. J. Differ. Geom. 84, 267–287 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Huang, R.L.: On the second boundary value problem for Lagrangian mean curvature flow. J. Funct. Anal. 269, 1095–1114 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Warren, M.: Calibrations associated to Monge–Ampére equations. Trans. Am. Math. Soc. 362, 3947–3962 (2010)

    Article  MATH  Google Scholar 

  4. Delanoe, P.: Classical solvability in dimension two of the second boundary-value problem associated with the Monge–Amp‘ere operator. Ann. Inst. H. Poincaré Anal. Non Linéaire 8, 443–457 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caffarelli, L.: Boundary regularity of maps with convex potentials, II. Ann. Math. Stud. 144, 453–496 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Urbas, J.: On the second boundary value problems for equations of Monge–Ampère type. J. Reine Angew. Math. 487, 115–124 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Schnuere, O.C., Smoczyk, K.: Neumann and second boundary value problems for Hessian and Gauss curvature flows. Ann. Inst. H. Poincaré Anal. Non Linéaire. 20, 1043–1073 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1998)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Professor Xi-Nan Ma for useful discussions on this topic. They also would like to thank the referees for useful comments, which improved the paper. The first author was supported by National Nature Science Foundation of China (No. 11261008) and Guangxi Nature Science Foundation (2016GXNSFCA380010). The second author was supported by National Nature Science Foundation of China (No. 11061013) and by Guangxi Nature Science Foundation (2014GXNSFAA118028) and Guangxi Colleges and Universities Key Laboratory of Symbolic Computation and Engineering Data Processing.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Guangxi Normal University, Guilin, 541004, Guangxi, China

    Rongli Huang

  2. School of Science, Hezhou university, Hezhou, 542899, Guangxi, China

    Qianzhong Ou

Authors
  1. Rongli Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qianzhong Ou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rongli Huang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, R., Ou, Q. On the Second Boundary Value Problem for a Class of Fully Nonlinear Equations. J Geom Anal 27, 2601–2617 (2017). https://doi.org/10.1007/s12220-017-9774-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-017-9774-7

Keywords

Mathematics Subject Classification

Search

Navigation