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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

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

Received  November 2012 Revised  May 2013 Published  October 2013

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.
Citation: Neil S. Trudinger. On the local theory of prescribed Jacobian equations. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.1090/S0894-0347-1992-1124980-8. Google Scholar [2] L. Caffarelli, Boundary regularity of maps with convex potentials II,, Problems in mathematical analysis. No. 37. Ann. of Math., 144 (1996), 453. doi: 10.2307/2118564. Google Scholar [3] L. Caffarelli and V. I. Oliker, Weak solutions of one inverse problem in geometric optics., J. Math. Sci. (N. Y.), 154 (2008), 39. doi: 10.1007/s10958-008-9152-x. Google Scholar [4] Ph. Delanoë, Classical solvability in dimension two of the second boundary value problem associated with the Monge-Ampère operator,, Ann. Inst. Henri Poincaré, 8 (1991), 443. doi: 10.1016/j.anihpc.2007.03.001. Google Scholar [5] A. Figalli, Y.-H Kim and R. J. McCann, Hölder continuity and injectivity of optimal maps,, Arch. Ration. Mech. Anal., 209 (2013), 747. doi: 10.1007/s00205-013-0629-5. Google Scholar [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], (1983). Google Scholar [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. doi: 10.1088/0266-5611/28/2/025001. Google Scholar [8] C. E. Gutierrez, "The Monge -Ampère Equation,", Progress in Nonlinear Differential Equations and their Applications, (2001). doi: 10.1007/978-1-4612-0195-3. Google Scholar [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. Google Scholar [10] Y.-H. Kim and R. J. McCann, Continuity, curvature and the general covariance of optimal transportation,, J. Eur. Math. Soc., 12 (2010), 1009. doi: 10.4171/JEMS/221. Google Scholar [11] S. A. Kochengin and V. I. Oliker, Determination of reflector surfaces from near-field scattering data,, Inverse Problems, 13 (1997), 363. doi: 10.1088/0266-5611/13/2/011. Google Scholar [12] A. Karakhanyan and X.-J. Wang, On the reflector shape design,, J. Diff. Geom., 84 (2010), 561. Google Scholar [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. doi: 10.1002/cpa.3160390405. Google Scholar [14] G. Loeper, On the regularity of solutions of optimal transportation problems,, Acta Math., 202 (2009), 241. doi: 10.1007/s11511-009-0037-8. Google Scholar [15] J.-K. Liu, Hölder regularity of optimal mappings in optimal transportation,, Calc. Var. Partial Differ. Equ., 34 (2009), 435. doi: 10.1007/s00526-008-0190-5. Google Scholar [16] J.-K. Liu, Light reflection is nonlinear optimization,, Calc. Var. Partial Differ. Equ., 46 (2013), 861. doi: 10.1007/s00526-012-0506-3. Google Scholar [17] J.-K. Liu, A class of nonlinear optimization problems with potentials,, preprint (2012)., (2012). Google Scholar [18] J.-K. Liu and N. S. Trudinger, On Pogorelov estimates for Monge-Ampère type equations,, Discrete Contin. Dyn. Syst., 28 (2010), 1121. doi: 10.3934/dcds.2010.28.1121. Google Scholar [19] J.-K. Liu and N. S. Trudinger, On classical solutions of near field reflection problems,, preprint (2013)., (2013). Google Scholar [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. doi: 10.1080/03605300903236609. Google Scholar [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. doi: 10.1007/s00205-005-0362-9. Google Scholar [22] S. T. Rachev and L. Ruschendorff, "Mass Transportation Problems,", Springer, (1998). Google Scholar [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. Google Scholar [24] N. S. Trudinger, On the prescribed Jacobian equation,, Gakuto Intl. Series, 20 (2008), 243. Google Scholar [25] N. S. Trudinger, On the local theory of prescribed Jacobian equations,, Conference in honor of 60th birthday of C.-S. Lin, (2011). Google Scholar [26] N. S. Trudinger, On generated prescribed Jacobian equations,, Oberwolfach Reports, 38 (2011), 32. Google Scholar [27] N. S. Trudinger, The local theory of prescribed Jacobian equations revisited,, (in preparation)., (). Google Scholar [28] N. S. Trudinger and X.-J. Wang, The Monge-Ampère equation and its geometric applications,, in, (2008), 467. Google Scholar [29] N. S. Trudinger and X.-J. Wang, On convexity notions in optimal transportation,, preprint (2008)., (2008). Google Scholar [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., 8 (2009), 143. Google Scholar [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. doi: 10.1007/s00205-008-0147-z. Google Scholar [32] J. Urbas, On the second boundary value problem for equations of Monge-Ampère type,, J. Reine Angew. Math., 487 (1997), 115. doi: 10.1515/crll.1997.487.115. Google Scholar [33] J. Vetois, Continuity and injectivity of optimal maps,, preprint (2011)., (2011). Google Scholar [34] C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, (2003). doi: 10.1007/b12016. Google Scholar [35] C. Villani, "Optimal Transportation, Old and New,", Springer, (2008). Google Scholar

show all references

##### References:
 [1] L. Caffarelli, The regularity of mappings with a convex potential,, J. Amer. Math. Soc., 5 (1992), 99. doi: 10.1090/S0894-0347-1992-1124980-8. Google Scholar [2] L. Caffarelli, Boundary regularity of maps with convex potentials II,, Problems in mathematical analysis. No. 37. Ann. of Math., 144 (1996), 453. doi: 10.2307/2118564. Google Scholar [3] L. Caffarelli and V. I. Oliker, Weak solutions of one inverse problem in geometric optics., J. Math. Sci. (N. Y.), 154 (2008), 39. doi: 10.1007/s10958-008-9152-x. Google Scholar [4] Ph. Delanoë, Classical solvability in dimension two of the second boundary value problem associated with the Monge-Ampère operator,, Ann. Inst. Henri Poincaré, 8 (1991), 443. doi: 10.1016/j.anihpc.2007.03.001. Google Scholar [5] A. Figalli, Y.-H Kim and R. J. McCann, Hölder continuity and injectivity of optimal maps,, Arch. Ration. Mech. Anal., 209 (2013), 747. doi: 10.1007/s00205-013-0629-5. Google Scholar [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], (1983). Google Scholar [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. doi: 10.1088/0266-5611/28/2/025001. Google Scholar [8] C. E. Gutierrez, "The Monge -Ampère Equation,", Progress in Nonlinear Differential Equations and their Applications, (2001). doi: 10.1007/978-1-4612-0195-3. Google Scholar [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. Google Scholar [10] Y.-H. Kim and R. J. McCann, Continuity, curvature and the general covariance of optimal transportation,, J. Eur. Math. Soc., 12 (2010), 1009. doi: 10.4171/JEMS/221. Google Scholar [11] S. A. Kochengin and V. I. Oliker, Determination of reflector surfaces from near-field scattering data,, Inverse Problems, 13 (1997), 363. doi: 10.1088/0266-5611/13/2/011. Google Scholar [12] A. Karakhanyan and X.-J. Wang, On the reflector shape design,, J. Diff. Geom., 84 (2010), 561. Google Scholar [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. doi: 10.1002/cpa.3160390405. Google Scholar [14] G. Loeper, On the regularity of solutions of optimal transportation problems,, Acta Math., 202 (2009), 241. doi: 10.1007/s11511-009-0037-8. Google Scholar [15] J.-K. Liu, Hölder regularity of optimal mappings in optimal transportation,, Calc. Var. Partial Differ. Equ., 34 (2009), 435. doi: 10.1007/s00526-008-0190-5. Google Scholar [16] J.-K. Liu, Light reflection is nonlinear optimization,, Calc. Var. Partial Differ. Equ., 46 (2013), 861. doi: 10.1007/s00526-012-0506-3. Google Scholar [17] J.-K. Liu, A class of nonlinear optimization problems with potentials,, preprint (2012)., (2012). Google Scholar [18] J.-K. Liu and N. S. Trudinger, On Pogorelov estimates for Monge-Ampère type equations,, Discrete Contin. Dyn. Syst., 28 (2010), 1121. doi: 10.3934/dcds.2010.28.1121. Google Scholar [19] J.-K. Liu and N. S. Trudinger, On classical solutions of near field reflection problems,, preprint (2013)., (2013). Google Scholar [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. doi: 10.1080/03605300903236609. Google Scholar [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. doi: 10.1007/s00205-005-0362-9. Google Scholar [22] S. T. Rachev and L. Ruschendorff, "Mass Transportation Problems,", Springer, (1998). Google Scholar [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. Google Scholar [24] N. S. Trudinger, On the prescribed Jacobian equation,, Gakuto Intl. Series, 20 (2008), 243. Google Scholar [25] N. S. Trudinger, On the local theory of prescribed Jacobian equations,, Conference in honor of 60th birthday of C.-S. Lin, (2011). Google Scholar [26] N. S. Trudinger, On generated prescribed Jacobian equations,, Oberwolfach Reports, 38 (2011), 32. Google Scholar [27] N. S. Trudinger, The local theory of prescribed Jacobian equations revisited,, (in preparation)., (). Google Scholar [28] N. S. Trudinger and X.-J. Wang, The Monge-Ampère equation and its geometric applications,, in, (2008), 467. Google Scholar [29] N. S. Trudinger and X.-J. Wang, On convexity notions in optimal transportation,, preprint (2008)., (2008). Google Scholar [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., 8 (2009), 143. Google Scholar [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. doi: 10.1007/s00205-008-0147-z. Google Scholar [32] J. Urbas, On the second boundary value problem for equations of Monge-Ampère type,, J. Reine Angew. Math., 487 (1997), 115. doi: 10.1515/crll.1997.487.115. Google Scholar [33] J. Vetois, Continuity and injectivity of optimal maps,, preprint (2011)., (2011). Google Scholar [34] C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, (2003). doi: 10.1007/b12016. Google Scholar [35] C. Villani, "Optimal Transportation, Old and New,", Springer, (2008). Google Scholar
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