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December  2015, 35(12): 6069-6084. doi: 10.3934/dcds.2015.35.6069

## Regularity of the homogeneous Monge-Ampère equation

 1 Centre for Mathematics and Its Applications, the Australian National University, Canberra, ACT 0200, Australia 2 Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT 0200

Received  September 2013 Revised  February 2014 Published  May 2015

In this paper, we study the regularity of convex solutions to the Dirichlet problem of the homogeneous Monge-Ampère equation $\det D^2 u=0$. We prove that if the domain is a strip region and the boundary functions are locally uniformly convex and $C^{k+2,\alpha}$ smooth, then the solution is $C^{k+2,\alpha}$ smooth up to boundary. By an example, we show the solution may fail to be $C^{2}$ smooth if boundary functions are not locally uniformly convex. Similar results have also been obtained for the Dirichlet problem on bounded convex domains.
Citation: Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069
##### References:
 [1] J. Benoist and J. B. Hiriart-Urruty, What is the subdifferential of the closed convex hull of a function?,, SIAM J. Math. Anal., 27 (1996), 1661. doi: 10.1137/S0036141094265936. Google Scholar [2] L. Caffarelli, Interior $W^{2,p}$ estimates for solutions of Monge-Ampère equations,, Ann. Math., 131 (1990), 135. doi: 10.2307/1971510. Google Scholar [3] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation,, Revista Math. Iberoamericana, 2 (1986), 19. doi: 10.4171/RMI/23. Google Scholar [4] X. Chen, Complex Monge-Ampère and symplectic manifolds,, J. Diff. Geom., 56 (2000), 189. Google Scholar [5] X. Chen and W. He, The space of volume forms,, Int. Math. Res. Not. IMRN, (2011), 967. doi: 10.1093/imrn/rnq099. Google Scholar [6] G. De Philippis and A. Figallli, Optimal regularity of the convex envelope,, Trans. Amer. Math. Soc., 367 (2015), 4407. doi: 10.1090/S0002-9947-2014-06306-X. Google Scholar [7] S. K. Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics,, Northern California Symplectic Geometry Seminar, 196 (1999), 13. Google Scholar [8] P. Guan, Regularity of a class of quasilinear degenerate elliptic equations,, Advances in Math., 132 (1997), 24. doi: 10.1006/aima.1997.1677. Google Scholar [9] P. Guan and E. T. Sawyer, Regularity of subelliptic Monge-Ampère equations in the plane,, Trans. Amer. Math. Soc., 361 (2009), 4581. doi: 10.1090/S0002-9947-09-04640-6. Google Scholar [10] P. Guan, N. S. Trudinger and X.-J. Wang, On the Dirichlet problem for degenerate Monge-Ampère equations,, Acta. Math., 182 (1999), 87. doi: 10.1007/BF02392824. Google Scholar [11] J. X. Hong, Dirichlet problems for general Monge-Ampère equations,, Math. Z., 209 (1992), 289. doi: 10.1007/BF02570835. Google Scholar [12] J. X. Hong, G. Huang and W. Wang, Existence of global smooth solutions to Dirichlet problem for degenrate elliptic Monge-Ampère equations,, Comm. PDE, 36 (2011), 635. doi: 10.1080/03605302.2010.514171. Google Scholar [13] B. Kirchheim and J. Kristensen, Differentiability of convex envelopes,, C. R. Acad. Sci. Paris Ser. I Math., 333 (2001), 725. doi: 10.1016/S0764-4442(01)02117-6. Google Scholar [14] A. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope,, Trans. Amer. Math. Soc., 363 (2011), 5871. doi: 10.1090/S0002-9947-2011-05240-2. Google Scholar [15] A. V. Pogorelov, The Minkowski Multidimensional Problem,, J. Wiley, (1978). Google Scholar [16] J. Rauch and B. A. Taylor, The Dirichlet problem for the multi-dimensional Monge-Ampère equation,, Rocky Mountain J. Math., 7 (1977), 345. doi: 10.1216/RMJ-1977-7-2-345. Google Scholar [17] E. T. Sawyer and R. L. Wheeden, Hölder continuity of weak solutions to subelliptic equations with rough coefficients,, Mem. Amer. Math. Soc., 180 (2006). doi: 10.1090/memo/0847. Google Scholar [18] O. Savin, Pointwise $C^{2,\alpha}$ estimates at the boundary for the Monge-Ampère equation,, J. Amer. Math. Soc., 26 (2013), 63. doi: 10.1090/S0894-0347-2012-00747-4. Google Scholar [19] O. Savin, Global $W^{2,p}$ estimates for the Monge-Ampère equation,, Proc. Amer. Math. Soc., 141 (2013), 3573. doi: 10.1090/S0002-9939-2013-11748-X. Google Scholar [20] O. Savin, A localisation theorem and boundary regularity for a class of degenerate Monge-Ampère equations,, J. Differential Equations, 256 (2014), 327. doi: 10.1016/j.jde.2013.08.019. Google Scholar [21] S. Semmes, Complex Monge-Ampère and symplectic manifolds,, Amer. J. Math., 114 (1992), 495. doi: 10.2307/2374768. Google Scholar [22] C. Rios, E. T. Sawyer and R. L. Wheeden, A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations,, Adv. Math., 193 (2005), 373. doi: 10.1016/j.aim.2004.05.009. Google Scholar [23] C. Rios, E. T. Sawyer and R. L. Wheeden, Regularity of subelliptic Monge-Ampère equations,, Advances in Math., 217 (2008), 967. doi: 10.1016/j.aim.2007.07.004. Google Scholar [24] N. S. Trudinger and J. Urbas, On the second derivative estimates for equations of Monge-Ampère type,, Bull. Austral. Math. Soc., 30 (1984), 321. doi: 10.1017/S0004972700002069. Google Scholar [25] N. S. Trudinger and X. J. Wang, Boundary regularity for the Monge-Ampère and affine maximal surface equations,, Ann. of Math. (2), 167 (2008), 993. doi: 10.4007/annals.2008.167.993. Google Scholar [26] N. S. Trudinger and X. J. Wang, The Monge-Ampère equation and its geometric applications,, Handbook of Geometric Analysis, 7 (2008), 467. Google Scholar [27] X. J. Wang, Some counterexamples to the regularity of Monge-Ampère equations,, Proc. Amer. Math. Soc., 123 (1995), 841. doi: 10.2307/2160809. Google Scholar

show all references

##### References:
 [1] J. Benoist and J. B. Hiriart-Urruty, What is the subdifferential of the closed convex hull of a function?,, SIAM J. Math. Anal., 27 (1996), 1661. doi: 10.1137/S0036141094265936. Google Scholar [2] L. Caffarelli, Interior $W^{2,p}$ estimates for solutions of Monge-Ampère equations,, Ann. Math., 131 (1990), 135. doi: 10.2307/1971510. Google Scholar [3] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation,, Revista Math. Iberoamericana, 2 (1986), 19. doi: 10.4171/RMI/23. Google Scholar [4] X. Chen, Complex Monge-Ampère and symplectic manifolds,, J. Diff. Geom., 56 (2000), 189. Google Scholar [5] X. Chen and W. He, The space of volume forms,, Int. Math. Res. Not. IMRN, (2011), 967. doi: 10.1093/imrn/rnq099. Google Scholar [6] G. De Philippis and A. Figallli, Optimal regularity of the convex envelope,, Trans. Amer. Math. Soc., 367 (2015), 4407. doi: 10.1090/S0002-9947-2014-06306-X. Google Scholar [7] S. K. Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics,, Northern California Symplectic Geometry Seminar, 196 (1999), 13. Google Scholar [8] P. Guan, Regularity of a class of quasilinear degenerate elliptic equations,, Advances in Math., 132 (1997), 24. doi: 10.1006/aima.1997.1677. Google Scholar [9] P. Guan and E. T. Sawyer, Regularity of subelliptic Monge-Ampère equations in the plane,, Trans. Amer. Math. Soc., 361 (2009), 4581. doi: 10.1090/S0002-9947-09-04640-6. Google Scholar [10] P. Guan, N. S. Trudinger and X.-J. Wang, On the Dirichlet problem for degenerate Monge-Ampère equations,, Acta. Math., 182 (1999), 87. doi: 10.1007/BF02392824. Google Scholar [11] J. X. Hong, Dirichlet problems for general Monge-Ampère equations,, Math. Z., 209 (1992), 289. doi: 10.1007/BF02570835. Google Scholar [12] J. X. Hong, G. Huang and W. Wang, Existence of global smooth solutions to Dirichlet problem for degenrate elliptic Monge-Ampère equations,, Comm. PDE, 36 (2011), 635. doi: 10.1080/03605302.2010.514171. Google Scholar [13] B. Kirchheim and J. Kristensen, Differentiability of convex envelopes,, C. R. Acad. Sci. Paris Ser. I Math., 333 (2001), 725. doi: 10.1016/S0764-4442(01)02117-6. Google Scholar [14] A. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope,, Trans. Amer. Math. Soc., 363 (2011), 5871. doi: 10.1090/S0002-9947-2011-05240-2. Google Scholar [15] A. V. Pogorelov, The Minkowski Multidimensional Problem,, J. Wiley, (1978). Google Scholar [16] J. Rauch and B. A. Taylor, The Dirichlet problem for the multi-dimensional Monge-Ampère equation,, Rocky Mountain J. Math., 7 (1977), 345. doi: 10.1216/RMJ-1977-7-2-345. Google Scholar [17] E. T. Sawyer and R. L. Wheeden, Hölder continuity of weak solutions to subelliptic equations with rough coefficients,, Mem. Amer. Math. Soc., 180 (2006). doi: 10.1090/memo/0847. Google Scholar [18] O. Savin, Pointwise $C^{2,\alpha}$ estimates at the boundary for the Monge-Ampère equation,, J. Amer. Math. Soc., 26 (2013), 63. doi: 10.1090/S0894-0347-2012-00747-4. Google Scholar [19] O. Savin, Global $W^{2,p}$ estimates for the Monge-Ampère equation,, Proc. Amer. Math. Soc., 141 (2013), 3573. doi: 10.1090/S0002-9939-2013-11748-X. Google Scholar [20] O. Savin, A localisation theorem and boundary regularity for a class of degenerate Monge-Ampère equations,, J. Differential Equations, 256 (2014), 327. doi: 10.1016/j.jde.2013.08.019. Google Scholar [21] S. Semmes, Complex Monge-Ampère and symplectic manifolds,, Amer. J. Math., 114 (1992), 495. doi: 10.2307/2374768. Google Scholar [22] C. Rios, E. T. Sawyer and R. L. Wheeden, A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations,, Adv. Math., 193 (2005), 373. doi: 10.1016/j.aim.2004.05.009. Google Scholar [23] C. Rios, E. T. Sawyer and R. L. Wheeden, Regularity of subelliptic Monge-Ampère equations,, Advances in Math., 217 (2008), 967. doi: 10.1016/j.aim.2007.07.004. Google Scholar [24] N. S. Trudinger and J. Urbas, On the second derivative estimates for equations of Monge-Ampère type,, Bull. Austral. Math. Soc., 30 (1984), 321. doi: 10.1017/S0004972700002069. Google Scholar [25] N. S. Trudinger and X. J. Wang, Boundary regularity for the Monge-Ampère and affine maximal surface equations,, Ann. of Math. (2), 167 (2008), 993. doi: 10.4007/annals.2008.167.993. Google Scholar [26] N. S. Trudinger and X. J. Wang, The Monge-Ampère equation and its geometric applications,, Handbook of Geometric Analysis, 7 (2008), 467. Google Scholar [27] X. J. Wang, Some counterexamples to the regularity of Monge-Ampère equations,, Proc. Amer. Math. Soc., 123 (1995), 841. doi: 10.2307/2160809. Google Scholar
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