Weak solutions of generated Jacobian equations

Feida Jiang; Feida Jiang

doi:10.3934/mine.2023064

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-20. doi: 10.3934/mine.2023064

Previous Article Next Article

Research article Special Issues

Weak solutions of generated Jacobian equations

Feida Jiang ^,

School of Mathematics and Shing-Tung Yau Center of Southeast University, Southeast University, Nanjing 211189, China

Received: 20 April 2022 Revised: 27 October 2022 Accepted: 20 November 2022 Published: 25 November 2022

We prove two groups of relationships for weak solutions to generated Jacobian equations under proper assumptions on the generating functions and the domains, which are generalizations for the optimal transportation case and the standard Monge-Ampère case respectively. One group of weak solutions is Aleksandrov solution, Brenier solution and $ C $-viscosity solution. The other group of weak solutions is Trudinger solution and $ L^p $-viscosity solution.
- generated Jacobian equations,
- Aleksandrov solutions,
- Brenier solutions,
- $ C $-viscosity solutions,
- Trudinger solutions,
- $ L^p $-viscosity solutions
Citation: Feida Jiang. Weak solutions of generated Jacobian equations[J]. Mathematics in Engineering, 2023, 5(3): 1-20. doi: 10.3934/mine.2023064

Related Papers:

Abstract

We prove two groups of relationships for weak solutions to generated Jacobian equations under proper assumptions on the generating functions and the domains, which are generalizations for the optimal transportation case and the standard Monge-Ampère case respectively. One group of weak solutions is Aleksandrov solution, Brenier solution and $ C $-viscosity solution. The other group of weak solutions is Trudinger solution and $ L^p $-viscosity solution.

References

[1]	F. Abedin, C. E. Gutiérrez, An iterative method for generated Jacobian equations, Calc. Var., 56 (2017), 101. http://doi.org/10.1007/s00526-017-1200-2 doi: 10.1007/s00526-017-1200-2
[2]	A. L. Amadori, B. Brandolini, C. Trombetti, Viscosity solutions of the Monge-Ampère equation with the right hand side in $L^p$, Rend. Lincei Mat. Appl., 18 (2007), 221–233. http://doi.org/10.4171/RLM/491 doi: 10.4171/RLM/491
[3]	G. Awanou, Computational nonimaging geometric optics: Monge-Ampère, Notices of the American Mathematical Society, 68 (2021), 186–193. http://doi.org/10.1090/noti2220 doi: 10.1090/noti2220
[4]	L. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99–104. http://doi.org/10.1090/S0894-0347-1992-1124980-8 doi: 10.1090/S0894-0347-1992-1124980-8
[5]	L. Caffarelli, M. G. Crandall, M. Kocan, A. Święch, On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., 49 (1996), 365–397. http://doi.org/10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A
[6]	B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 1–26. http://doi.org/10.1016/S0294-1449(16)30307-9 doi: 10.1016/S0294-1449(16)30307-9
[7]	A. Gallouét, Q. Mérigot, B. Thibert, A damped Newton algorithm for generated Jacobian equations, Calc. Var., 61 (2022), 49. http://doi.org/10.1007/s00526-021-02147-7 doi: 10.1007/s00526-021-02147-7
[8]	N. Guillen, A primer on generated Jacobian equations: geometry, optics, economics, Notices of the American Mathematical Society, 66 (2019), 1401–1411. http://doi.org/10.1090/noti1956 doi: 10.1090/noti1956
[9]	N. Guillen, J. Kitagawa, Pointwise estimates and regularity in geometric optics and other generated Jacobian equations, Commun. Pure Appl. Math., 70 (2017), 1146–1220. http://doi.org/10.1002/cpa.21691 doi: 10.1002/cpa.21691
[10]	H. Ishii, P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differ. Equations, 83 (1990), 26–78. http://doi.org/10.1016/0022-0396(90)90068-Z doi: 10.1016/0022-0396(90)90068-Z
[11]	S. Jeong, Local Hölder regularity of solutions to generated Jacobian equations, Pure and Applied Analysis, 3 (2021), 163–188. http://doi.org/10.2140/paa.2021.3.163 doi: 10.2140/paa.2021.3.163
[12]	Y. Jhaveri, Partial regularity of solutions to the second boundary value problem for generated Jacobian equations, Methods Appl. Anal., 24 (2017), 445–475. http://doi.org/10.4310/MAA.2017.v24.n4.a2 doi: 10.4310/MAA.2017.v24.n4.a2
[13]	F. Jiang, N. S. Trudinger, On Pogorelov estimates in optimal transportation and geometric optics, Bull. Math. Sci., 4 (2014), 407–431. http://doi.org/10.1007/s13373-014-0055-5 doi: 10.1007/s13373-014-0055-5
[14]	F. Jiang, N. S. Trudinger, On the second boundary value problem for Monge-Ampère type equations and geometric optics, Arch. Rational Mech. Anal., 229 (2018), 547–567. http://doi.org/10.1007/s00205-018-1222-8 doi: 10.1007/s00205-018-1222-8
[15]	F. Jiang, N. S. Trudinger, Oblique boundary value problems for augmented Hessian equations III, Commun. Part. Diff. Eq., 44 (2019), 708–748. http://doi.org/10.1080/03605302.2019.1597113 doi: 10.1080/03605302.2019.1597113
[16]	F. Jiang, X. P. Yang, Weak solutions of Monge-Ampère type equations in optimal transportation, Acta Math. Sci., 33 (2013), 950–962. http://doi.org/10.1016/S0252-9602(13)60054-5 doi: 10.1016/S0252-9602(13)60054-5
[17]	J. Liu, Monge-Ampère type equations and optimal transportation, PhD Thesis, Australian National University, 2010.
[18]	G. Loeper, On the regularity of solutions of optimal transportation problems, Acta Math., 202 (2009), 241–283. http://doi.org/10.1007/s11511-009-0037-8 doi: 10.1007/s11511-009-0037-8
[19]	G. Loeper, N. S. Trudinger, On the convexity theory of generating functions, arXiv: 2109.04585.
[20]	X.-N. Ma, N. S. Trudinger, X.-J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Rational Mech. Anal., 177 (2005), 151–183. http://doi.org/10.1007/s00205-005-0362-9 doi: 10.1007/s00205-005-0362-9
[21]	R. J. McCann, K. S. Zhang, On concavity of the monopolist's problem facing consumers with nonlinear price preferences, Commun. Pure Appl. Math., 72 (2019), 1386–1423. http://doi.org/10.1002/cpa.21817 doi: 10.1002/cpa.21817
[22]	G. Nöldeke, L. Samuelson, The implementation duality, Econometrica, 86 (2018), 1283–1324. http://doi.org/10.3982/ECTA13307
[23]	C. Rankin, Distinct solutions to generated Jacobian equations cannot intersect, Bull. Aust. Math. Soc., 102 (2020), 462–470. http://doi.org/10.1017/S0004972720000052 doi: 10.1017/S0004972720000052
[24]	C. Rankin, Strict convexity and $C^1$ regularity of solutions to generated Jacobian equations in dimension two, Calc. Var., 60 (2021), 221. http://doi.org/10.1007/s00526-021-02093-4 doi: 10.1007/s00526-021-02093-4
[25]	C. Rankin, Strict $g$-convexity for generated Jacobian equations with applications to global regularity, arXiv: 2111.00448.
[26]	C. Rankin, Regularity and uniqueness results for generated Jacobian equations, PhD Thesis, Australian National University, 2021.
[27]	C. Rankin, First and second derivative Hölder estimates for generated Jacobian equations, arXiv: 2204.07917.
[28]	L. B. Romijn, M. J. H. Anthonissen, J. H. M. ten Thije Boonkkamp, W. L. Ijzerman, Numerically solving generated Jacobian equations in freeform optical design, EPJ Web Conf., 238 (2020), 02001. http://doi.org/10.1051/epjconf/202023802001 doi: 10.1051/epjconf/202023802001
[29]	L. B. Romijn, J. H. M. ten Thije Boonkkamp, M. J. H. Anthonissen, W. L. Ijzerman, An iterative least-squares method for generated Jacobian equations in freeform optical design, SIAM J. Sci. Comput., 43 (2021), B298–B322. http://doi.org/10.1137/20M1338940 doi: 10.1137/20M1338940
[30]	N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal., 111 (1990), 153–179. http://doi.org/10.1007/BF00375406 doi: 10.1007/BF00375406
[31]	N. S. Trudinger, Weak solutions of Hessian equations, Commun. Part. Diff. Eq., 22 (1997), 25–54. http://doi.org/10.1080/03605309708821299 doi: 10.1080/03605309708821299
[32]	N. S. Trudinger, Recent developments in elliptic partial differential equations of Monge-Ampère type, In: International congress of mathematicians Madrid 2006 Volume III. Invited lectures, 2006 291–302. http://doi.org/10.4171/022-3/15
[33]	N. S. Trudinger, On the prescribed Jacobian equation, In: International conference for the 25th anniversary of viscosity solutions, Tokyo, Japan, 2008,243–255.
[34]	N. S. Trudinger, On generated prescribed Jacobian equations, Oberwolfach Reports, 8 (2011), 2194–2198.
[35]	N. S. Trudinger, On the local theory of prescribed Jacobian equations, Discrete Contin. Dyn. Syst., 34 (2014), 1663–1681. http://doi.org/10.3934/dcds.2014.34.1663 doi: 10.3934/dcds.2014.34.1663
[36]	N. S. Trudinger, On the local theory of prescribed Jacobian equations revisited, Mathematics in Engineering, 3 (2021), 1–17. http://doi.org/10.3934/mine.2021048 doi: 10.3934/mine.2021048
[37]	N. S. Trudinger, A note on second derivative estimates for Monge-Ampère type equations, arXiv: 2204.01039.
[38]	N. S. Trudinger, X.-J. Wang, On the second boundary value problem for Monge-Ampère type equations and optimal transportation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 8 (2009), 143–174. https://doi.org/10.2422/2036-2145.2009.1.07 doi: 10.2422/2036-2145.2009.1.07
[39]	X.-J. Wang, On the design of a refector antenna, Inverse Probl., 12 (1996), 351–375. http://doi.org/10.1088/0266-5611/12/3/013 doi: 10.1088/0266-5611/12/3/013

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)