Abstract
In this paper, we establish global \(W^{2,p}\) estimates for solutions to the linearized Monge–Ampère equations under natural assumptions on the domain, Monge–Ampère measures and boundary data. Our estimates are affine invariant analogues of the global \(W^{2,p}\) estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin’s global \(W^{2,p}\) estimates for the Monge–Ampère equations.
Similar content being viewed by others
References
Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44(4), 375–417 (1991)
Caffarelli, L.A.: Interior \(W^{2, p}\) estimates for solutions to the Monge-Ampère equation. Ann. Math. 131(1), 135–150 (1990)
Caffarelli, L.A., Cabré, X.: Fully nonlinear elliptic equations, vol. 43. American Mathematical Society Colloquium Publications (1995)
Caffarelli, L.A., Gutiérrez, C.E.: Properties of the solutions of the linearized Monge-Ampère equation. Am. J. Math. 119(2), 423–465 (1997)
Cullen, M.J.P., Norbury, J., Purser, R.J.: Generalized Lagrangian solutions for atmospheric and oceanic flows. SIAM J. Appl. Anal. 51(1), 20–31 (1991)
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(3), 369–402 (1984)
De Phillipis, G., Figalli, A.: \(W^{2,1}\) regularity for solutions of the Monge-Ampère equation. Invent. Math. 192(1), 55–69 (2013)
De Phillipis, G., Figalli, A.: Second order stability for the Monge-Ampère equation and strong Sobolev convergence of optimal transportation maps. Anal. PDE. arXiv:1202.5561v2 (math.AP) (to appear)
De Phillipis, G., Figalli, A., Savin, O.: A note on interior \(W^{2, 1+\epsilon }\) estimates for the Monge-Ampère equation. Math. Ann. 3571, 11–22 (2013)
Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62(2), 289–349 (2002)
Donaldson, S.K.: Interior estimates for solutions of Abreu’s equation. Collect. Math. 56(2), 103–142 (2005)
Donaldson, S.K.: Extremal metrics on toric surfaces: a continuity method. J. Differ. Geom. 79(3), 389–432 (2008)
Donaldson, S.K.: Constant scalar curvature metrics on toric surfaces. Geom. Funct. Anal. 19(1), 83–136 (2009)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, New York (2001)
Gutiérrez, C.E.: The Monge-Ampère Equation. Birkhaüser, Boston (2001)
Gutiérrez, C.E., Nguyen, T.V.: Interior gradient estimates for solutions to the linearized Monge-Ampère equation. Adv. Math. 228(4), 2034–2070 (2011)
Gutiérrez, C.E., Nguyen, T.V.: Interior second derivative estimates for solutions to the linearized Monge-Ampère equation. Trans. Am. Math. Soc. arXiv:1208.5097v1 (math.AP) (to appear)
Gutiérrez, C.E., Tournier, F.: \(W^{2, p}\)-estimates for the linearized Monge-Ampère equation. Trans. Am. Math. Soc. 358(11), 4843–4872 (2006)
Huang, Q.: Sharp regularity results on second derivatives of solutions to the Monge-Ampère equation with VMO type data. Commun. Pure. Appl. Math. 62(5), 677–705 (2009)
Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations in a domain. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 47(1):75–108 (1983)
Le, N.Q.: Global second derivative estimates for the second boundary value problem of the prescribed affine mean curvature and Abreu’s equations. Int. Math. Res. Not. 11, 2421–2438 (2013)
Le, N.Q., Nguyen, T.V.: Geometric properties of boundary sections of solutions to the Monge-Ampère equation and applications. J. Funct. Anal. 264(1), 337–361 (2013)
Le, N.Q., Savin, O.: Boundary regularity for solutions to the linearized Monge-Ampère equations. Arch. Ration. Mech. Anal. (2013). doi:10.1007/s00205-013-0653-5
Liu, J., Trudinger, N.S., Wang, X-J.: On asymptotic behavior and \(W^{2, p}\) regularity of potentials in Optimal Transportation (2010) (Preprint)
Loeper, G.: A fully nonlinear version of the incompressible euler equations: the semigeostrophic system. SIAM J. Math. Anal. 38(3), 795–823 (2006)
Savin, O.A.: Localization property at the boundary for the Monge-Ampère equation. In: Advances in Geometric Analysis. Adv. Lect. Math. (ALM), vol. 21, pp. 45–68. Int. Press, Somerville (2012)
Savin, O.: Pointwise \(C^{2,\alpha }\) estimates at the boundary for the Monge-Ampère equation. J. Am. Math. Soc. 26, 63–99 (2013)
Savin, O.: Global \(W^{2, p}\) estimates for the Monge-Ampère equations. Proc. Am. Math. Soc. 141(10), 3573–3578 (2013)
Schmidt, T.: \(W^{2, 1 +\varepsilon }\) estimates for the Monge-Ampère equation. Adv. Math. 240, 672–689 (2013)
Trudinger, N.S., Wang, X.J.: Boundary regularity for Monge-Ampère and affine maximal surface equations. Ann. Math. 167(3), 993–1028 (2008)
Trudinger, N.S., Wang, X.J.: The Bernstein problem for affine maximal hypersurfaces. Invent. Math. 140(2), 399–422 (2000)
Trudinger, N.S., Wang, X.J.: The affine plateau problem. J. Am. Math. Soc. 18, 253–289 (2005)
Trudinger, N.S., Wang, X.J.: The Monge-Ampère equation and its geometric applications. In: Handbook of Geometric Analysis. No. 1. Adv. Lect. Math. (ALM), vol. 7, pp. 467–524. Int. Press, Somerville (2008)
Wang, X.J.: Some counterexamples to the regularity of Monge-Ampère equations. Proc. Am. Math. Soc. 123(3), 841–845 (1995)
Winter, N.: \(W^{2, p}\) and \(W^{1, p}\)-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations. Z. Anal. Anwend. 28(2), 129–164 (2009)
Zhou, B.: The first boundary value problem for Abreu’s equation. Int. Math. Res. Not. 7, 1439–1484 (2012)
Acknowledgments
T. Nguyen gratefully acknowledges the support provided by NSF grant DMS-0901449.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Le, N.Q., Nguyen, T. Global \(W^{2,p}\) estimates for solutions to the linearized Monge–Ampère equations. Math. Ann. 358, 629–700 (2014). https://doi.org/10.1007/s00208-013-0974-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-013-0974-6