Abstract
In this paper, we consider the Neumann problem for special Lagrangian equations with critical phase. The global gradient and Hessian estimates are obtained. Using the method of continuity, we prove the existence of solutions to this problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brendle, S., Warren, M.: A boundary value problem for minimal Lagrangian graphs. Journal of Differential Geometry 84(2), 267–288 (2010)
Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations, iii: Functions of the eigenvalues of the Hessian. Acta Mathematica 155(1), 261–301 (1985)
Chen, C., Ma, X., Wei, W.: The Neumann boundary value problem for special Lagrangian equations with supercritical phase. Submitted
Chen, C., Ma, X., Zhang, D.: The Neumann problem for parabolic Hessian equations. Preprint
Chen, C., Zhang, D.: The Neumann problem of Hessian quotient equations. Submitted
Collins, T.C., Jacob, A., Yau, S.T.: (1, 1) forms with specified Lagrangian phase: A priori estimates and algebraic obstructions. arXiv preprint arXiv:1508.01934 (2015)
Collins, T.C., Picard, S., Wu, X.: Concavity of the Lagrangian phase operator and applications. Calculus of Variations and Partial Differential Equations 56(4), 89 (2017)
Harvey, R., Lawson, H.B.: Calibrated geometries. Acta Mathematica 148(1), 47–157 (1982)
Jacob, A., Yau, S.T.: A special Lagrangian type equation for holomorphic line bundles. Mathematische Annalen 369(1–2), 869–898 (2017)
Lieberman, G.: Oblique boundary value problems for elliptic equations, pp. xvi+509. World Scientific Publishing Co. Pte. Ltd., Hackensack (2013). https://doi.org/10.1142/8679
Lieberman, G.M., Trudinger, N.S.: Nonlinear oblique boundary value problems for nonlinear elliptic equations. Transactions of the American Mathematical Society 295(2), 509–546 (1986)
Lions, P.L., Trudinger, N.S., Urbas, J.I.: The Neumann problem for equations of Monge-Ampère type. Communications on Pure and Applied Mathematics 39(4), 539–563 (1986)
Ma, X., Qiu, G.: The Neumann problem for Hessian equations. arXiv preprint arXiv:1508.00196 (2015)
Ma, X., Wang, P., Wei, W.: Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains. Journal of Functional Analysis 274(1), 252–277 (2018)
Ma, X., Xu, J.: Gradient estimates of mean curvature equations with Neumann boundary value problems. Advances in Mathematics 290, 1010–1039 (2016)
Qiu, G., Xia, C.: Classical Neumann Problems for Hessian equations and Alexandrov-Fenchel’s inequalities. arXiv preprint arXiv:1607.03868 (2016)
Simon, L., Spruck, J.: Existence and regularity of a capillary surface with prescribed contact angle. Archive for Rational Mechanics and Analysis 61(1), 19–34 (1976)
Wang, D., Yuan, Y.: Hessian estimates for special Lagrangian equations with critical and supercritical phases in general dimensions. American Journal of Mathematics 136(2), 481–499 (2014)
Warren, M., Yuan, Y.: Hessian and gradient estimates for three dimensional special Lagrangian equations with large phase. American Journal of Mathematics 132(3), 751–770 (2010)
Yuan, Y.: Global solutions to special Lagrangian equations. Proceedings of the American Mathematical Society pp. 1355–1358 (2006)
Acknowledgements
The results of this paper are part of my doctoral dissertation. I would like to thank my supervisor Professor Xinan Ma for suggesting this problem to me and his guidance.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, J. The Neumann Problem for Special Lagrangian Equations with Critical Phase. Commun. Math. Stat. 7, 329–361 (2019). https://doi.org/10.1007/s40304-018-0157-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40304-018-0157-6