On the local theory of prescribed Jacobian equations

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April 2014, 34(4): 1663-1681. doi: 10.3934/dcds.2014.34.1663

On the local theory of prescribed Jacobian equations

Neil S. Trudinger ^1,

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

Received November 2012 Revised May 2013 Published October 2013

Abstract
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We develop the fundamentals of a local regularity theory for prescribed Jacobian equations which extend the corresponding results for optimal transportation equations. In this theory the cost function is extended to a generating function through dependence on an additional scalar variable. In particular we recover in this generality the local regularity theory for potentials of Ma, Trudinger and Wang, along with the subsequent development of the underlying convexity theory.

Keywords: Prescribed Jacobian equations, existence and regularity., convexity, duality, generating function.

Mathematics Subject Classification: Primary: 35J66, 35J25; Secondary: 78A0.

Citation: Neil S. Trudinger. On the local theory of prescribed Jacobian equations. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1663-1681. doi: 10.3934/dcds.2014.34.1663

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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