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On the local theory of prescribed Jacobian equations
1. | Centre for Mathematics and Its Applications, the Australian National University, Canberra, ACT 0200, Australia |
References:
[1] |
L. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104.
doi: 10.1090/S0894-0347-1992-1124980-8. |
[2] |
L. Caffarelli, Boundary regularity of maps with convex potentials II, Problems in mathematical analysis. No. 37. Ann. of Math., 144 (1996), 453-496.
doi: 10.2307/2118564. |
[3] |
L. Caffarelli and V. I. Oliker, Weak solutions of one inverse problem in geometric optics. J. Math. Sci. (N. Y.), 154 (2008), 39-49.
doi: 10.1007/s10958-008-9152-x. |
[4] |
Ph. Delanoë, Classical solvability in dimension two of the second boundary value problem associated with the Monge-Ampère operator, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 8 (1991), 443-457.
doi: 10.1016/j.anihpc.2007.03.001. |
[5] |
A. Figalli, Y.-H Kim and R. J. McCann, Hölder continuity and injectivity of optimal maps, Arch. Ration. Mech. Anal., 209 (2013), 747-795.
doi: 10.1007/s00205-013-0629-5. |
[6] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer-Verlag, Berlin, 1983. |
[7] |
T. Graf and V. Oliker, An optimal mass transport approach to the near-field reflector problem in optimal design, Inverse Problems, 28 (2012), 1-15.
doi: 10.1088/0266-5611/28/2/025001. |
[8] |
C. E. Gutierrez, "The Monge -Ampère Equation," Progress in Nonlinear Differential Equations and their Applications, 44. Birkhäuser Boston, Inc., Boston, MA, 2001.
doi: 10.1007/978-1-4612-0195-3. |
[9] |
F. Jiang, N. S. Trudinger and X.-P. Yang, On the Dirichlet Problem for Monge-Ampère type equations,, Calc. Var. Partial Differ. Equ., (): 00526.
|
[10] |
Y.-H. Kim and R. J. McCann, Continuity, curvature and the general covariance of optimal transportation, J. Eur. Math. Soc., 12 (2010), 1009-1040.
doi: 10.4171/JEMS/221. |
[11] |
S. A. Kochengin and V. I. Oliker, Determination of reflector surfaces from near-field scattering data, Inverse Problems, 13 (1997), 363-373.
doi: 10.1088/0266-5611/13/2/011. |
[12] |
A. Karakhanyan and X.-J. Wang, On the reflector shape design, J. Diff. Geom., 84 (2010), 561-610. |
[13] |
P.-L. Lions, N. S. Trudinger and J. Urbas, Neumann problem for equations of Monge-Ampère type, Comm. Pure Appl. Math., 39 (1986), 539-563.
doi: 10.1002/cpa.3160390405. |
[14] |
G. Loeper, On the regularity of solutions of optimal transportation problems, Acta Math., 202 (2009), 241-283.
doi: 10.1007/s11511-009-0037-8. |
[15] |
J.-K. Liu, Hölder regularity of optimal mappings in optimal transportation, Calc. Var. Partial Differ. Equ., 34 (2009), 435-451.
doi: 10.1007/s00526-008-0190-5. |
[16] |
J.-K. Liu, Light reflection is nonlinear optimization, Calc. Var. Partial Differ. Equ., 46 (2013), 861-878.
doi: 10.1007/s00526-012-0506-3. |
[17] |
J.-K. Liu, A class of nonlinear optimization problems with potentials, preprint (2012). |
[18] |
J.-K. Liu and N. S. Trudinger, On Pogorelov estimates for Monge-Ampère type equations, Discrete Contin. Dyn. Syst., 28, (2010), 1121-1135.
doi: 10.3934/dcds.2010.28.1121. |
[19] |
J.-K. Liu and N. S. Trudinger, On classical solutions of near field reflection problems, preprint (2013). |
[20] |
J.-K. Liu, N. S. Trudinger and X. J. Wang, Interior $C^{2,\alpha}$ regularity for potential functions in optimal transportation, Comm. Partial Differential Equations, 35 (2010), 165-184.
doi: 10.1080/03605300903236609. |
[21] |
X.-N. Ma, N. S. Trudinger and X.-J.Wang, Regularity of potential functions of the optimal transportation problem, Arch. Rat. Mech. Anal., 177 (2005), 151-183.
doi: 10.1007/s00205-005-0362-9. |
[22] |
S. T. Rachev and L. Ruschendorff, "Mass Transportation Problems," Springer, 1998. |
[23] |
N. S. Trudinger, Recent developments in elliptic partial differential equations of Monge-Ampère type, International Congress of Mathematicians. Eur. Math. Soc., III (2006), 291-301. |
[24] |
N. S. Trudinger, On the prescribed Jacobian equation, Gakuto Intl. Series, Math. Sci. Appl. 20 (2008), Proc. Intl. Conf. for the 25th Anniversary of Viscosity Solutions, 243-255. |
[25] |
N. S. Trudinger, On the local theory of prescribed Jacobian equations, Conference in honor of 60th birthday of C.-S. Lin, Taipei, 2011, (https://maths.anu.edu.au/~neilt/RecentPapers.html). |
[26] |
N. S. Trudinger, On generated prescribed Jacobian equations, Oberwolfach Reports, 38 (2011), 32-36. |
[27] |
N. S. Trudinger, The local theory of prescribed Jacobian equations revisited,, (in preparation)., ().
|
[28] |
N. S. Trudinger and X.-J. Wang, The Monge-Ampère equation and its geometric applications, in "Handbook of Geometric Analysis," International Press (2008), 467-524. |
[29] |
N. S. Trudinger and X.-J. Wang, On convexity notions in optimal transportation, preprint (2008). |
[30] |
N. S. Trudinger and X.-J. Wang , On the second boundary value problem for Monge-Ampère type equations and optimal transportation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Series (5), 8 (2009), 143-174. |
[31] |
N. S. Trudinger and X.-J.Wang, On strict convexity and continuous differentiability of potential functions in optimal transportation, Arch. Rat. Mech.. Anal., 192 (2009), 403-418.
doi: 10.1007/s00205-008-0147-z. |
[32] |
J. Urbas, On the second boundary value problem for equations of Monge-Ampère type, J. Reine Angew. Math., 487 (1997),115-124.
doi: 10.1515/crll.1997.487.115. |
[33] |
J. Vetois, Continuity and injectivity of optimal maps, preprint (2011). |
[34] |
C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. xvi+370 pp.
doi: 10.1007/b12016. |
[35] |
C. Villani, "Optimal Transportation, Old and New," Springer, 2008. |
show all references
References:
[1] |
L. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104.
doi: 10.1090/S0894-0347-1992-1124980-8. |
[2] |
L. Caffarelli, Boundary regularity of maps with convex potentials II, Problems in mathematical analysis. No. 37. Ann. of Math., 144 (1996), 453-496.
doi: 10.2307/2118564. |
[3] |
L. Caffarelli and V. I. Oliker, Weak solutions of one inverse problem in geometric optics. J. Math. Sci. (N. Y.), 154 (2008), 39-49.
doi: 10.1007/s10958-008-9152-x. |
[4] |
Ph. Delanoë, Classical solvability in dimension two of the second boundary value problem associated with the Monge-Ampère operator, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 8 (1991), 443-457.
doi: 10.1016/j.anihpc.2007.03.001. |
[5] |
A. Figalli, Y.-H Kim and R. J. McCann, Hölder continuity and injectivity of optimal maps, Arch. Ration. Mech. Anal., 209 (2013), 747-795.
doi: 10.1007/s00205-013-0629-5. |
[6] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer-Verlag, Berlin, 1983. |
[7] |
T. Graf and V. Oliker, An optimal mass transport approach to the near-field reflector problem in optimal design, Inverse Problems, 28 (2012), 1-15.
doi: 10.1088/0266-5611/28/2/025001. |
[8] |
C. E. Gutierrez, "The Monge -Ampère Equation," Progress in Nonlinear Differential Equations and their Applications, 44. Birkhäuser Boston, Inc., Boston, MA, 2001.
doi: 10.1007/978-1-4612-0195-3. |
[9] |
F. Jiang, N. S. Trudinger and X.-P. Yang, On the Dirichlet Problem for Monge-Ampère type equations,, Calc. Var. Partial Differ. Equ., (): 00526.
|
[10] |
Y.-H. Kim and R. J. McCann, Continuity, curvature and the general covariance of optimal transportation, J. Eur. Math. Soc., 12 (2010), 1009-1040.
doi: 10.4171/JEMS/221. |
[11] |
S. A. Kochengin and V. I. Oliker, Determination of reflector surfaces from near-field scattering data, Inverse Problems, 13 (1997), 363-373.
doi: 10.1088/0266-5611/13/2/011. |
[12] |
A. Karakhanyan and X.-J. Wang, On the reflector shape design, J. Diff. Geom., 84 (2010), 561-610. |
[13] |
P.-L. Lions, N. S. Trudinger and J. Urbas, Neumann problem for equations of Monge-Ampère type, Comm. Pure Appl. Math., 39 (1986), 539-563.
doi: 10.1002/cpa.3160390405. |
[14] |
G. Loeper, On the regularity of solutions of optimal transportation problems, Acta Math., 202 (2009), 241-283.
doi: 10.1007/s11511-009-0037-8. |
[15] |
J.-K. Liu, Hölder regularity of optimal mappings in optimal transportation, Calc. Var. Partial Differ. Equ., 34 (2009), 435-451.
doi: 10.1007/s00526-008-0190-5. |
[16] |
J.-K. Liu, Light reflection is nonlinear optimization, Calc. Var. Partial Differ. Equ., 46 (2013), 861-878.
doi: 10.1007/s00526-012-0506-3. |
[17] |
J.-K. Liu, A class of nonlinear optimization problems with potentials, preprint (2012). |
[18] |
J.-K. Liu and N. S. Trudinger, On Pogorelov estimates for Monge-Ampère type equations, Discrete Contin. Dyn. Syst., 28, (2010), 1121-1135.
doi: 10.3934/dcds.2010.28.1121. |
[19] |
J.-K. Liu and N. S. Trudinger, On classical solutions of near field reflection problems, preprint (2013). |
[20] |
J.-K. Liu, N. S. Trudinger and X. J. Wang, Interior $C^{2,\alpha}$ regularity for potential functions in optimal transportation, Comm. Partial Differential Equations, 35 (2010), 165-184.
doi: 10.1080/03605300903236609. |
[21] |
X.-N. Ma, N. S. Trudinger and X.-J.Wang, Regularity of potential functions of the optimal transportation problem, Arch. Rat. Mech. Anal., 177 (2005), 151-183.
doi: 10.1007/s00205-005-0362-9. |
[22] |
S. T. Rachev and L. Ruschendorff, "Mass Transportation Problems," Springer, 1998. |
[23] |
N. S. Trudinger, Recent developments in elliptic partial differential equations of Monge-Ampère type, International Congress of Mathematicians. Eur. Math. Soc., III (2006), 291-301. |
[24] |
N. S. Trudinger, On the prescribed Jacobian equation, Gakuto Intl. Series, Math. Sci. Appl. 20 (2008), Proc. Intl. Conf. for the 25th Anniversary of Viscosity Solutions, 243-255. |
[25] |
N. S. Trudinger, On the local theory of prescribed Jacobian equations, Conference in honor of 60th birthday of C.-S. Lin, Taipei, 2011, (https://maths.anu.edu.au/~neilt/RecentPapers.html). |
[26] |
N. S. Trudinger, On generated prescribed Jacobian equations, Oberwolfach Reports, 38 (2011), 32-36. |
[27] |
N. S. Trudinger, The local theory of prescribed Jacobian equations revisited,, (in preparation)., ().
|
[28] |
N. S. Trudinger and X.-J. Wang, The Monge-Ampère equation and its geometric applications, in "Handbook of Geometric Analysis," International Press (2008), 467-524. |
[29] |
N. S. Trudinger and X.-J. Wang, On convexity notions in optimal transportation, preprint (2008). |
[30] |
N. S. Trudinger and X.-J. Wang , On the second boundary value problem for Monge-Ampère type equations and optimal transportation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Series (5), 8 (2009), 143-174. |
[31] |
N. S. Trudinger and X.-J.Wang, On strict convexity and continuous differentiability of potential functions in optimal transportation, Arch. Rat. Mech.. Anal., 192 (2009), 403-418.
doi: 10.1007/s00205-008-0147-z. |
[32] |
J. Urbas, On the second boundary value problem for equations of Monge-Ampère type, J. Reine Angew. Math., 487 (1997),115-124.
doi: 10.1515/crll.1997.487.115. |
[33] |
J. Vetois, Continuity and injectivity of optimal maps, preprint (2011). |
[34] |
C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. xvi+370 pp.
doi: 10.1007/b12016. |
[35] |
C. Villani, "Optimal Transportation, Old and New," Springer, 2008. |
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