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Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations
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 |
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), 1661-1679.
doi: 10.1137/S0036141094265936. |
[2] |
L. Caffarelli, Interior $W^{2,p}$ estimates for solutions of Monge-Ampère equations, Ann. Math., 131 (1990), 135-150.
doi: 10.2307/1971510. |
[3] |
L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation, Revista Math. Iberoamericana, 2 (1986), 19-27.
doi: 10.4171/RMI/23. |
[4] |
X. Chen, Complex Monge-Ampère and symplectic manifolds, J. Diff. Geom., 56 (2000), 189-234. |
[5] |
X. Chen and W. He, The space of volume forms, Int. Math. Res. Not. IMRN, (2011), 967-1009.
doi: 10.1093/imrn/rnq099. |
[6] |
G. De Philippis and A. Figallli, Optimal regularity of the convex envelope, Trans. Amer. Math. Soc., 367 (2015), 4407-4422.
doi: 10.1090/S0002-9947-2014-06306-X. |
[7] |
S. K. Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, 196 (1999), 13-33. |
[8] |
P. Guan, Regularity of a class of quasilinear degenerate elliptic equations, Advances in Math., 132 (1997), 24-45.
doi: 10.1006/aima.1997.1677. |
[9] |
P. Guan and E. T. Sawyer, Regularity of subelliptic Monge-Ampère equations in the plane, Trans. Amer. Math. Soc., 361 (2009), 4581-4591.
doi: 10.1090/S0002-9947-09-04640-6. |
[10] |
P. Guan, N. S. Trudinger and X.-J. Wang, On the Dirichlet problem for degenerate Monge-Ampère equations, Acta. Math., 182 (1999), 87-104.
doi: 10.1007/BF02392824. |
[11] |
J. X. Hong, Dirichlet problems for general Monge-Ampère equations, Math. Z., 209 (1992), 289-306.
doi: 10.1007/BF02570835. |
[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), 635-656.
doi: 10.1080/03605302.2010.514171. |
[13] |
B. Kirchheim and J. Kristensen, Differentiability of convex envelopes, C. R. Acad. Sci. Paris Ser. I Math., 333 (2001), 725-728.
doi: 10.1016/S0764-4442(01)02117-6. |
[14] |
A. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc., 363 (2011), 5871-5886.
doi: 10.1090/S0002-9947-2011-05240-2. |
[15] |
A. V. Pogorelov, The Minkowski Multidimensional Problem, J. Wiley, New York, 1978. |
[16] |
J. Rauch and B. A. Taylor, The Dirichlet problem for the multi-dimensional Monge-Ampère equation, Rocky Mountain J. Math., 7 (1977), 345-364.
doi: 10.1216/RMJ-1977-7-2-345. |
[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), x+157 pp.
doi: 10.1090/memo/0847. |
[18] |
O. Savin, Pointwise $C^{2,\alpha}$ estimates at the boundary for the Monge-Ampère equation, J. Amer. Math. Soc., 26 (2013), 63-99.
doi: 10.1090/S0894-0347-2012-00747-4. |
[19] |
O. Savin, Global $W^{2,p}$ estimates for the Monge-Ampère equation, Proc. Amer. Math. Soc., 141 (2013), 3573-3578.
doi: 10.1090/S0002-9939-2013-11748-X. |
[20] |
O. Savin, A localisation theorem and boundary regularity for a class of degenerate Monge-Ampère equations, J. Differential Equations, 256 (2014), 327-388.
doi: 10.1016/j.jde.2013.08.019. |
[21] |
S. Semmes, Complex Monge-Ampère and symplectic manifolds, Amer. J. Math., 114 (1992), 495-550.
doi: 10.2307/2374768. |
[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), 373-415.
doi: 10.1016/j.aim.2004.05.009. |
[23] |
C. Rios, E. T. Sawyer and R. L. Wheeden, Regularity of subelliptic Monge-Ampère equations, Advances in Math., 217 (2008), 967-1026.
doi: 10.1016/j.aim.2007.07.004. |
[24] |
N. S. Trudinger and J. Urbas, On the second derivative estimates for equations of Monge-Ampère type, Bull. Austral. Math. Soc., 30 (1984), 321-334.
doi: 10.1017/S0004972700002069. |
[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), 993-1028.
doi: 10.4007/annals.2008.167.993. |
[26] |
N. S. Trudinger and X. J. Wang, The Monge-Ampère equation and its geometric applications, Handbook of Geometric Analysis, International Press, 7 (2008), 467-524. |
[27] |
X. J. Wang, Some counterexamples to the regularity of Monge-Ampère equations, Proc. Amer. Math. Soc., 123 (1995), 841-845.
doi: 10.2307/2160809. |
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), 1661-1679.
doi: 10.1137/S0036141094265936. |
[2] |
L. Caffarelli, Interior $W^{2,p}$ estimates for solutions of Monge-Ampère equations, Ann. Math., 131 (1990), 135-150.
doi: 10.2307/1971510. |
[3] |
L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation, Revista Math. Iberoamericana, 2 (1986), 19-27.
doi: 10.4171/RMI/23. |
[4] |
X. Chen, Complex Monge-Ampère and symplectic manifolds, J. Diff. Geom., 56 (2000), 189-234. |
[5] |
X. Chen and W. He, The space of volume forms, Int. Math. Res. Not. IMRN, (2011), 967-1009.
doi: 10.1093/imrn/rnq099. |
[6] |
G. De Philippis and A. Figallli, Optimal regularity of the convex envelope, Trans. Amer. Math. Soc., 367 (2015), 4407-4422.
doi: 10.1090/S0002-9947-2014-06306-X. |
[7] |
S. K. Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, 196 (1999), 13-33. |
[8] |
P. Guan, Regularity of a class of quasilinear degenerate elliptic equations, Advances in Math., 132 (1997), 24-45.
doi: 10.1006/aima.1997.1677. |
[9] |
P. Guan and E. T. Sawyer, Regularity of subelliptic Monge-Ampère equations in the plane, Trans. Amer. Math. Soc., 361 (2009), 4581-4591.
doi: 10.1090/S0002-9947-09-04640-6. |
[10] |
P. Guan, N. S. Trudinger and X.-J. Wang, On the Dirichlet problem for degenerate Monge-Ampère equations, Acta. Math., 182 (1999), 87-104.
doi: 10.1007/BF02392824. |
[11] |
J. X. Hong, Dirichlet problems for general Monge-Ampère equations, Math. Z., 209 (1992), 289-306.
doi: 10.1007/BF02570835. |
[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), 635-656.
doi: 10.1080/03605302.2010.514171. |
[13] |
B. Kirchheim and J. Kristensen, Differentiability of convex envelopes, C. R. Acad. Sci. Paris Ser. I Math., 333 (2001), 725-728.
doi: 10.1016/S0764-4442(01)02117-6. |
[14] |
A. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc., 363 (2011), 5871-5886.
doi: 10.1090/S0002-9947-2011-05240-2. |
[15] |
A. V. Pogorelov, The Minkowski Multidimensional Problem, J. Wiley, New York, 1978. |
[16] |
J. Rauch and B. A. Taylor, The Dirichlet problem for the multi-dimensional Monge-Ampère equation, Rocky Mountain J. Math., 7 (1977), 345-364.
doi: 10.1216/RMJ-1977-7-2-345. |
[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), x+157 pp.
doi: 10.1090/memo/0847. |
[18] |
O. Savin, Pointwise $C^{2,\alpha}$ estimates at the boundary for the Monge-Ampère equation, J. Amer. Math. Soc., 26 (2013), 63-99.
doi: 10.1090/S0894-0347-2012-00747-4. |
[19] |
O. Savin, Global $W^{2,p}$ estimates for the Monge-Ampère equation, Proc. Amer. Math. Soc., 141 (2013), 3573-3578.
doi: 10.1090/S0002-9939-2013-11748-X. |
[20] |
O. Savin, A localisation theorem and boundary regularity for a class of degenerate Monge-Ampère equations, J. Differential Equations, 256 (2014), 327-388.
doi: 10.1016/j.jde.2013.08.019. |
[21] |
S. Semmes, Complex Monge-Ampère and symplectic manifolds, Amer. J. Math., 114 (1992), 495-550.
doi: 10.2307/2374768. |
[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), 373-415.
doi: 10.1016/j.aim.2004.05.009. |
[23] |
C. Rios, E. T. Sawyer and R. L. Wheeden, Regularity of subelliptic Monge-Ampère equations, Advances in Math., 217 (2008), 967-1026.
doi: 10.1016/j.aim.2007.07.004. |
[24] |
N. S. Trudinger and J. Urbas, On the second derivative estimates for equations of Monge-Ampère type, Bull. Austral. Math. Soc., 30 (1984), 321-334.
doi: 10.1017/S0004972700002069. |
[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), 993-1028.
doi: 10.4007/annals.2008.167.993. |
[26] |
N. S. Trudinger and X. J. Wang, The Monge-Ampère equation and its geometric applications, Handbook of Geometric Analysis, International Press, 7 (2008), 467-524. |
[27] |
X. J. Wang, Some counterexamples to the regularity of Monge-Ampère equations, Proc. Amer. Math. Soc., 123 (1995), 841-845.
doi: 10.2307/2160809. |
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