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Interior second derivative estimates for solutions to the linearized Monge-Ampère equation


Authors: Cristian E. Gutiérrez and Truyen Nguyen
Journal: Trans. Amer. Math. Soc. 367 (2015), 4537-4568
MSC (2010): Primary 35J60, 35J70, 35J96
DOI: https://doi.org/10.1090/S0002-9947-2015-06048-6
Published electronically: March 13, 2015
MathSciNet review: 3335393
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Abstract: Let $ \Omega \subset \mathbb{R}^n$ be a bounded convex domain and $ \phi \in C(\overline \Omega )$ be a convex function such that $ \phi $ is sufficiently smooth on $ \partial \Omega $ and the Monge-Ampère measure $ \det D^2\phi $ is bounded away from zero and infinity in $ \Omega $. The corresponding linearized Monge-Ampère equation is

$\displaystyle \mathrm {trace}(\Phi D^2 u) =f, $

where $ \Phi := \det D^2 \phi ~ (D^2\phi )^{-1}$ is the matrix of cofactors of $ D^2\phi $. We prove a conjecture about the relationship between $ L^p$ estimates for $ D^2 u$ and the closeness between $ \det D^2\phi $ and one. As a consequence, we obtain interior $ W^{2,p}$ estimates for solutions to such equations whenever the measure $ \det D^2\phi $ is given by a continuous density and the function $ f$ belongs to $ L^q(\Omega )$ for some $ q> \max {\{p,n\}}$.

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  • [C1] Luis A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189-213. MR 1005611 (90i:35046), https://doi.org/10.2307/1971480
  • [C2] L. A. Caffarelli, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math. (2) 131 (1990), no. 1, 129-134. MR 1038359 (91f:35058), https://doi.org/10.2307/1971509
  • [C3] Luis A. Caffarelli, Interior $ W^{2,p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2) 131 (1990), no. 1, 135-150. MR 1038360 (91f:35059), https://doi.org/10.2307/1971510
  • [C4] Luis A. Caffarelli, Some regularity properties of solutions of Monge Ampère equation, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 965-969. MR 1127042 (92h:35088), https://doi.org/10.1002/cpa.3160440809
  • [CC] Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007 (96h:35046)
  • [CG1] Luis A. Caffarelli and Cristian E. Gutiérrez, Real analysis related to the Monge-Ampère equation, Trans. Amer. Math. Soc. 348 (1996), no. 3, 1075-1092. MR 1321570 (96h:35047), https://doi.org/10.1090/S0002-9947-96-01473-0
  • [CG2] Luis A. Caffarelli and Cristian E. Gutiérrez, Properties of the solutions of the linearized Monge-Ampère equation, Amer. J. Math. 119 (1997), no. 2, 423-465. MR 1439555 (98e:35060)
  • [CNP] M. J. P. Cullen, J. Norbury, and R. J. Purser, Generalised Lagrangian solutions for atmospheric and oceanic flows, SIAM J. Appl. Math. 51 (1991), no. 1, 20-31. MR 1089128 (92g:76081), https://doi.org/10.1137/0151002
  • [dPF] Guido De Philippis and Alessio Figalli, $ W^{2,1}$ regularity for solutions of the Monge-Ampère equation, Invent. Math. 192 (2013), no. 1, 55-69. MR 3032325, https://doi.org/10.1007/s00222-012-0405-4
  • [D1] S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), no. 2, 289-349. MR 1988506 (2005c:32028)
  • [D2] S. K. Donaldson, Interior estimates for solutions of Abreu's equation, Collect. Math. 56 (2005), no. 2, 103-142. MR 2154300 (2006d:35035)
  • [D3] S. K. Donaldson, Extremal metrics on toric surfaces: a continuity method, J. Differential Geom. 79 (2008), no. 3, 389-432. MR 2433928 (2009j:58018)
  • [D4] Simon K. Donaldson, Constant scalar curvature metrics on toric surfaces, Geom. Funct. Anal. 19 (2009), no. 1, 83-136. MR 2507220 (2010j:32041), https://doi.org/10.1007/s00039-009-0714-y
  • [Es] Luis Escauriaza, $ W^{2,n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J. 42 (1993), no. 2, 413-423. MR 1237053 (94h:35022), https://doi.org/10.1512/iumj.1993.42.42019
  • [E] Lawrence C. Evans, Some estimates for nondivergence structure, second order elliptic equations, Trans. Amer. Math. Soc. 287 (1985), no. 2, 701-712. MR 768735 (86g:35056), https://doi.org/10.2307/1999671
  • [GiT] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364 (2001k:35004)
  • [G] Cristian E. Gutiérrez, The Monge-Ampère equation, Progress in Nonlinear Differential Equations and their Applications, 44, Birkhäuser Boston Inc., Boston, MA, 2001. MR 1829162 (2002e:35075)
  • [GH] Cristian E. Gutiérrez and Qingbo Huang, Geometric properties of the sections of solutions to the Monge-Ampère equation, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4381-4396. MR 1665332 (2000m:35060), https://doi.org/10.1090/S0002-9947-00-02491-0
  • [GN] Cristian E. Gutiérrez and Truyen Nguyen, Interior gradient estimates for solutions to the linearized Monge-Ampère equation, Adv. Math. 228 (2011), no. 4, 2034-2070. MR 2836113 (2012h:35093), https://doi.org/10.1016/j.aim.2011.06.035
  • [GT] Cristian E. Gutiérrez and Federico Tournier, $ W^{2,p}$-estimates for the linearized Monge-Ampère equation, Trans. Amer. Math. Soc. 358 (2006), no. 11, 4843-4872 (electronic). MR 2231875 (2007d:35046), https://doi.org/10.1090/S0002-9947-06-04189-4
  • [H] Qingbo Huang, Sharp regularity results on second derivatives of solutions to the Monge-Ampère equation with VMO type data, Comm. Pure Appl. Math. 62 (2009), no. 5, 677-705. MR 2494811 (2010j:35149), https://doi.org/10.1002/cpa.20272
  • [LS1] N. Q. Le and O. Savin, Boundary regularity for solutions to the linearized Monge-Ampère equations, Arch. Ration. Mech. Anal. 210 (2013), no. 3, 813-836. MR 3116005, https://doi.org/10.1007/s00205-013-0653-5
  • [LS2] N. Q. Le and O. Savin, Some minimization problems in the class of convex functions with prescribed determinant, Anal. PDE 6 (2013), no. 5, 1025-1050. MR 3125549, https://doi.org/10.2140/apde.2013.6.1025
  • [L] Fang-Hua Lin, Second derivative $ L^p$-estimates for elliptic equations of nondivergent type, Proc. Amer. Math. Soc. 96 (1986), no. 3, 447-451. MR 822437 (88b:35058), https://doi.org/10.2307/2046592
  • [Lo] G. Loeper, A fully nonlinear version of the incompressible Euler equations: the semigeostrophic system, SIAM J. Math. Anal. 38 (2006), no. 3, 795-823 (electronic). MR 2262943 (2007j:35173), https://doi.org/10.1137/050629070
  • [PT] Carlo Pucci and Giorgio Talenti, Elliptic (second-order) partial differential equations with measurable coefficients and approximating integral equations, Advances in Math. 19 (1976), no. 1, 48-105. MR 0419989 (54 #8006)
  • [S1] Ovidiu Savin, A Liouville theorem for solutions to the linearized Monge-Ampere equation, Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 865-873. MR 2644770 (2011e:35139), https://doi.org/10.3934/dcds.2010.28.865
  • [S2] O. Savin, Global $ W^{2,p}$ estimates for the Monge-Ampère equation, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3573-3578. MR 3080179, https://doi.org/10.1090/S0002-9939-2013-11748-X
  • [Sc] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521 (94d:52007)
  • [St] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44 #7280)
  • [Sw] Andrzej Świech, $ W^{1,p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations 2 (1997), no. 6, 1005-1027. MR 1606359 (99a:35098)
  • [TiW] Gu-Ji Tian and Xu-Jia Wang, A class of Sobolev type inequalities, Methods Appl. Anal. 15 (2008), no. 2, 263-276. MR 2481683 (2010e:35016)
  • [T] Neil S. Trudinger, Glimpses of nonlinear partial differential equations in the twentieth century: A priori estimates and the Bernstein problem, Challenges for the 21st century (Singapore, 2000) World Sci. Publ., River Edge, NJ, 2001, pp. 196-212. MR 1875020 (2003a:34037)
  • [TW1] Neil S. Trudinger and Xu-Jia Wang, The Bernstein problem for affine maximal hypersurfaces, Invent. Math. 140 (2000), no. 2, 399-422. MR 1757001 (2001h:53016), https://doi.org/10.1007/s002220000059
  • [TW2] Neil S. Trudinger and Xu-Jia Wang, The affine Plateau problem, J. Amer. Math. Soc. 18 (2005), no. 2, 253-289. MR 2137978 (2006e:53071), https://doi.org/10.1090/S0894-0347-05-00475-3
  • [TW3] Neil S. Trudinger and Xu-Jia Wang, Boundary regularity for the Monge-Ampère and affine maximal surface equations, Ann. of Math. (2) 167 (2008), no. 3, 993-1028. MR 2415390 (2010h:35168), https://doi.org/10.4007/annals.2008.167.993
  • [TW4] Neil S. Trudinger and Xu-Jia Wang, The Monge-Ampère equation and its geometric applications, Handbook of geometric analysis. No. 1, Adv. Lect. Math. (ALM), vol. 7, Int. Press, Somerville, MA, 2008, pp. 467-524. MR 2483373 (2010g:53065)
  • [U] N. N. Uralceva, The impossibility of $ W_{q}{}^{2}$ estimates for multidimensional elliptic equations with discontinuous coefficients, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5 (1967), 250-254 (Russian). MR 0226179 (37 #1769)
  • [WL] Li He Wang, A geometric approach to the Calderón-Zygmund estimates, Acta Math. Sin. (Engl. Ser.) 19 (2003), no. 2, 381-396. MR 1987802 (2004e:42033), https://doi.org/10.1007/s10114-003-0264-4
  • [W] Xu Jia Wang, Some counterexamples to the regularity of Monge-Ampère equations, Proc. Amer. Math. Soc. 123 (1995), no. 3, 841-845. MR 1223269 (95d:35025), https://doi.org/10.2307/2160809
  • [Z1] Bin Zhou, The first boundary value problem for Abreu's equation, Int. Math. Res. Not. IMRN 7 (2012), 1439-1484. MR 2913180
  • [Z2] Bin Zhou, The Bernstein theorem for a class of fourth order equations, Calc. Var. Partial Differential Equations 43 (2012), no. 1-2, 25-44. MR 2860401, https://doi.org/10.1007/s00526-011-0401-3

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Additional Information

Cristian E. Gutiérrez
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Email: gutierre@temple.edu

Truyen Nguyen
Affiliation: Department of Mathematics, The University of Akron, Akron, Ohio 44325
Email: tnguyen@uakron.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06048-6
Received by editor(s): August 13, 2012
Received by editor(s) in revised form: December 12, 2012
Published electronically: March 13, 2015
Additional Notes: The first author gratefully acknowledges the support provided by NSF grant DMS-1201401
The second author gratefully acknowledges the support provided by NSF grant DMS-0901449
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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