Abstract
In this paper, we derive a priori interior Hessian estimates for the Lagrangian mean curvature equation if the Lagrangian phase is supercritical and has bounded second derivatives.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Harvey, R., Lawson, H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)
Hitchin, N.J.: The moduli space of special Lagrangian submanifolds. Ann. Sc. Norm. Super. Pisa-Classe Sci. 25(3–4), 503–515 (1997)
Warren, M.: Calibrations associated to Monge–Ampère equations. Trans. Am. Math. Soc. 362(8), 3947–3962 (2010)
Mealy, J.G.: Calibrations on Semi-Riemannian Manifolds. Thesis (Ph.D.) Rice University (1989)
Yuan, Y.: Global solutions to special Lagrangian equations. Proc. Am. Math. Soc. 134, 1355–1358 (2006)
Warren, M., Yuan, Y.: Hessian estimates for the sigma-2 equation in dimension 3. Commun. Pure Appl. Math. 62(3), 305–321 (2009)
Warren, M., Yuan, Y.: Hessian and gradient estimates for three dimensional special Lagrangian equations with large phase. Am. J. Math. 132(3), 751–770 (2010)
Wang, D., Yuan, Y.: Hessian estimates for special Lagrangian equations with critical and supercritical phases in general dimensions. Am. J. Math. 136(2), 481–499 (2014)
Nikolai, N., Serge, V.: Singular solution to special Lagrangian equations. Ann. l’I.H.P. Anal. Non Linéaire 27(5), 1179–1188 (2010)
Wang, D., Yuan, Y.: Singular solutions to special Lagrangian equations with subcritical phases and minimal surface systems. Am. J. Math. 135(5), 1157–1177 (2013)
Bhattacharya, A.: The Dirichlet problem for Lagrangian mean curvature equation. Preprint arXiv:2005.14420 (2020)
Bhattacharya, A., Shankar, R.: Regularity for convex viscosity solutions of Lagrangian mean curvature equation. Preprint arXiv:2006.02030 (2020)
Heinz, E.: On elliptic Monge–Ampère equations and Weyls embedding problem. J. D’Anal. Math. 7(1), 1–52 (1959)
Pogorelov, A.V.: In: Rabenstein, Albert L., Richard, C., Bollinger, P. (eds.) Monge–Ampère equations of elliptic type. Translated from the first Russian edition by Leo F. Boron with the assistance of. Noordhoff Ltd, Groningen (1964)
Pogorelov, A.V.: The Minkowski multidimensional problem, vol. H. Winston & Sons, Washington, D.C. (1978). Translated from the Russian by Vladimir Oliker, Introduction by Louis Nirenberg, Scripta Series in Mathematics
John, U.: On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations. Indiana Univ. Math. J. 39, 355–382 (1990)
Gregori, G.: Compactness and gradient bounds for solutions of the mean curvature system in two independent variables. J. Geom. Anal. 4(3), 327–360 (1994)
Chou, K.-S., Wang, X.-J.: A variational theory of the Hessian equation. Commun. Pure Appl. Math. J. Issued Courant Inst. Math. Sci. 54(9), 1029–1064 (2001)
Trudinger, N.S.: Regularity of solutions of fully nonlinear elliptic equations. Boll. Un. Mat. Ital. A (6) 3(3), 421–430 (1984)
Urbas, J.: Some interior regularity results for solutions of Hessian equations. Calc. Var. Partial Differ. Equ. 11(1), 1–31 (2000)
Urbas, J.: An interior second derivative bound for solutions of Hessian equations. Calc. Var. Partial Differ. Equ. 12(4), 417–431 (2001)
Bao, J., Chen, J.: Optimal regularity for convex strong solutions of special Lagrangian equations in dimension 3. Indiana Univ. Math. J. 52, 1231–1249 (2003)
Bao, J., Chen, J., Guan, B., Ji, M.: Liouville property and regularity of a Hessian quotient equation. Am. J. Math. 125(2), 301–316 (2003)
Qiu, G.: Interior Hessian estimates for sigma-2 equations in dimension three. Preprint arXiv:1711.00948 (2017)
Guan, P., Qiu, G.: Interior \( c^2\) regularity of convex solutions to prescribing scalar curvature equations. Duke Math. J. 168(9), 1641–1663 (2019)
Chen, J., Yuan, Y., Warren, M.: A priori estimate for convex solutions to special Lagrangian equations and its application. Commun. Pure Appl. Math. 62(4), 583–595 (2009)
Chen, J., Shankar, R., Yuan, Y.: Regularity for convex viscosity solutions of special Lagrangian equation. Preprint arXiv:1911.05452 (2019)
Shankar, R., Yuan, Y.: Hessian estimate for semiconvex solutions to the sigma-2 equation. Calc. Var. Partial Differ. Equ. 59(1), 30 (2020)
Shankar, R., Yuan, Y.: Regularity for almost convex viscosity solutions of the sigma-2 equation. J. Math. Study 54(2), 164–170 (2020)
Ishii, H.: On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. Funkc. Ekvac 38(1), 101–120 (1995)
Michael, J.H., Simon, L.M.: Sobolev and mean-value inequalities on generalized submanifolds of \({mathbb{R}}^{n}\). Commun. Pure Appl. Math. 26(3), 361–379 (1973)
Yuan, Yu.: A Bernstein problem for special Lagrangian equations. Invent. Math. 150, 117–125 (2002)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Classics in Mathematics. Springer, Berlin (2001). (Reprint of the 1998 edition)
Acknowledgements
The author is grateful to Yu Yuan for his guidance, support, and many useful discussions. The author is grateful to Ravi Shankar and Micah Warren for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N.S. Trudinger.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
