American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials
  • DCDS Home
  • This Issue
  • Next Article
    Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations
December  2015, 35(12): 6069-6084. doi: 10.3934/dcds.2015.35.6069

Regularity of the homogeneous Monge-Ampère equation

Qi-Rui Li 1, and  Xu-Jia Wang 2,

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.
Keywords: regularity., Degenerate Monge-Ampère equation.
Mathematics Subject Classification: Primary: 35J96; Secondary: 35J7.
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

[1]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
[2]

Ravi Anand, Dibyendu Roy, Santanu Sarkar. Some results on lightweight stream ciphers Fountain v1 & Lizard. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020128
[3]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597
[4]

Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149
[5]

Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341
[6]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044
[7]

Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks & Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021
[8]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[9]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213
[10]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
[11]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[12]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[13]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[14]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186
[15]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[16]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[17]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448
[18]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454
[19]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
[20]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (73)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences