American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations
  • DCDS Home
  • This Issue
  • Next Article
    Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces
February  2016, 36(2): 895-916. doi: 10.3934/dcds.2016.36.895

On the classical solvability of near field reflector problems

Jiakun Liu 1, and  Neil S. Trudinger 2,

1. 

School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia
2. 

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

Received  July 2014 Revised  March 2015 Published  August 2015

In this paper we prove the existence of classical solutions to near field reflector problems, both for a point light source and for a parallel light source, with planar receivers. These problems involve Monge-Ampère type equations, subject to nonlinear oblique boundary conditions. Our approach builds on earlier work in the optimal transportation case by Trudinger and Wang and makes use of a recent extension of degree theory to oblique boundary conditions by Li, Liu and Nguyen.
Keywords: existence, obliqueness, Monge-Ampère, regularity., Reflector.
Mathematics Subject Classification: Primary: 35J60; Secondary: 78A0.
Citation: Jiakun Liu, Neil S. Trudinger. On the classical solvability of near field reflector problems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 895-916. doi: 10.3934/dcds.2016.36.895
References:
[1]

L. A. Caffarelli, Boundary regularity of maps with convex potentials II,, Ann. of Math., 144 (1996), 453.  doi: 10.2307/2118564.  Google Scholar

[2]

L. A. Caffarelli and V. Oliker, Weak solutions of one inverse problem in geometric optics,, J. Math. Sci., 154 (2008), 37.  doi: 10.1007/s10958-008-9152-x.  Google Scholar

[3]

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

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, $2^{nd}$ edition, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[5]

P. Guan and X.-J. Wang, On a Monge-Ampère equation arising in geometric optics,, J. Diff. Geom., 48 (1998), 205.   Google Scholar

[6]

C. Gutiérrez and F. Tournier, The parallel refractor,, Developments in Mathematics, 28 (2013), 325.  doi: 10.1007/978-1-4614-4075-8_14.  Google Scholar

[7]

F. Jiang and N. S. Trudinger, On Pogorelov estimates in optimal transportation and geometric optics,, Bull. Math. Sci., 4 (2014), 407.  doi: 10.1007/s13373-014-0055-5.  Google Scholar

[8]

A. Karakhanyan, Existence and regularity of the reflector surfaces in $\mathbbR^{n+1}$,, Arch. Ration. Mech. Anal., 213 (2014), 833.  doi: 10.1007/s00205-014-0743-z.  Google Scholar

[9]

A. Karakhanyan and X.-J. Wang, On the reflector shape design,, J. Diff. Geom., 84 (2010), 561.   Google Scholar

[10]

S. Kochengin and V. 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

[11]

Y. Li, J. Liu and L. Nguyen, A degree theory for second order nonlinear elliptic operators with nonlinear oblique boundary conditions,, preprint, (): 26.   Google Scholar

[12]

G. M. Lieberman and N. S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations., Trans. Amer. Math. Soc., 295 (1986), 509.  doi: 10.1090/S0002-9947-1986-0833695-6.  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]

J. Liu, Light reflection is nonlinear optimization,, Calc. Var. PDE, 46 (2013), 861.  doi: 10.1007/s00526-012-0506-3.  Google Scholar

[15]

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

[16]

V. Oliker, Differential equations for design of a freeform single lens with prescribed irradiance properties,, Optical Engineering, 53 (2014).  doi: 10.1117/1.OE.53.3.031302.  Google Scholar

[17]

V. Oliker, J. Rubinstein and G. Wolansky, Supporting quadric method in optical design of freeform lenses for precise illumination control of a collimated light beam,, Advances in Appl. Math., 62 (2015), 160.  doi: 10.1016/j.aam.2014.09.009.  Google Scholar

[18]

N. S. Trudinger, On the Dirichlet problem for Hessian equations,, Acta Math., 175 (1995), 151.  doi: 10.1007/BF02393303.  Google Scholar

[19]

N. S. Trudinger, Recent developments in elliptic partial differential equations of Monge-Ampère type,, Proc. Int. Cong. Math., 3 (2006), 291.   Google Scholar

[20]

N. S. Trudinger, On the prescribed Jacobian equation,, Gakuto Intl. Series, XX (2008), 243.   Google Scholar

[21]

N. S. Trudinger, On the local theory of prescribed Jacobian equations,, Discrete Contin. Dyn. Syst., 34 (2014), 1663.  doi: 10.3934/dcds.2014.34.1663.  Google Scholar

[22]

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

[23]

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

[24]

J. Urbas, Mass Transfer Problems,, Lecture Notes, (1998).   Google Scholar

[25]

J. Urbas, Oblique boundary value problems for equations of Monge-Ampère type,, Calc. Var. PDE, 7 (1998), 19.  doi: 10.1007/s005260050097.  Google Scholar

[26]

G. von Nessi, On the second boundary value problem for a class of modified-Hessian equations,, Comm. Partial Differential Equations, 35 (2010), 745.  doi: 10.1080/03605301003632317.  Google Scholar

[27]

X.-J. Wang, On the design of a reflector antenna,, Inverse problems, 12 (1996), 351.  doi: 10.1088/0266-5611/12/3/013.  Google Scholar

[28]

X.-J. Wang, On the design of a reflector antenna II,, Calc. Var. PDE, 20 (2004), 329.  doi: 10.1007/s00526-003-0239-4.  Google Scholar

show all references

References:
[1]

L. A. Caffarelli, Boundary regularity of maps with convex potentials II,, Ann. of Math., 144 (1996), 453.  doi: 10.2307/2118564.  Google Scholar

[2]

L. A. Caffarelli and V. Oliker, Weak solutions of one inverse problem in geometric optics,, J. Math. Sci., 154 (2008), 37.  doi: 10.1007/s10958-008-9152-x.  Google Scholar

[3]

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

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, $2^{nd}$ edition, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[5]

P. Guan and X.-J. Wang, On a Monge-Ampère equation arising in geometric optics,, J. Diff. Geom., 48 (1998), 205.   Google Scholar

[6]

C. Gutiérrez and F. Tournier, The parallel refractor,, Developments in Mathematics, 28 (2013), 325.  doi: 10.1007/978-1-4614-4075-8_14.  Google Scholar

[7]

F. Jiang and N. S. Trudinger, On Pogorelov estimates in optimal transportation and geometric optics,, Bull. Math. Sci., 4 (2014), 407.  doi: 10.1007/s13373-014-0055-5.  Google Scholar

[8]

A. Karakhanyan, Existence and regularity of the reflector surfaces in $\mathbbR^{n+1}$,, Arch. Ration. Mech. Anal., 213 (2014), 833.  doi: 10.1007/s00205-014-0743-z.  Google Scholar

[9]

A. Karakhanyan and X.-J. Wang, On the reflector shape design,, J. Diff. Geom., 84 (2010), 561.   Google Scholar

[10]

S. Kochengin and V. 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

[11]

Y. Li, J. Liu and L. Nguyen, A degree theory for second order nonlinear elliptic operators with nonlinear oblique boundary conditions,, preprint, (): 26.   Google Scholar

[12]

G. M. Lieberman and N. S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations., Trans. Amer. Math. Soc., 295 (1986), 509.  doi: 10.1090/S0002-9947-1986-0833695-6.  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]

J. Liu, Light reflection is nonlinear optimization,, Calc. Var. PDE, 46 (2013), 861.  doi: 10.1007/s00526-012-0506-3.  Google Scholar

[15]

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

[16]

V. Oliker, Differential equations for design of a freeform single lens with prescribed irradiance properties,, Optical Engineering, 53 (2014).  doi: 10.1117/1.OE.53.3.031302.  Google Scholar

[17]

V. Oliker, J. Rubinstein and G. Wolansky, Supporting quadric method in optical design of freeform lenses for precise illumination control of a collimated light beam,, Advances in Appl. Math., 62 (2015), 160.  doi: 10.1016/j.aam.2014.09.009.  Google Scholar

[18]

N. S. Trudinger, On the Dirichlet problem for Hessian equations,, Acta Math., 175 (1995), 151.  doi: 10.1007/BF02393303.  Google Scholar

[19]

N. S. Trudinger, Recent developments in elliptic partial differential equations of Monge-Ampère type,, Proc. Int. Cong. Math., 3 (2006), 291.   Google Scholar

[20]

N. S. Trudinger, On the prescribed Jacobian equation,, Gakuto Intl. Series, XX (2008), 243.   Google Scholar

[21]

N. S. Trudinger, On the local theory of prescribed Jacobian equations,, Discrete Contin. Dyn. Syst., 34 (2014), 1663.  doi: 10.3934/dcds.2014.34.1663.  Google Scholar

[22]

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

[23]

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

[24]

J. Urbas, Mass Transfer Problems,, Lecture Notes, (1998).   Google Scholar

[25]

J. Urbas, Oblique boundary value problems for equations of Monge-Ampère type,, Calc. Var. PDE, 7 (1998), 19.  doi: 10.1007/s005260050097.  Google Scholar

[26]

G. von Nessi, On the second boundary value problem for a class of modified-Hessian equations,, Comm. Partial Differential Equations, 35 (2010), 745.  doi: 10.1080/03605301003632317.  Google Scholar

[27]

X.-J. Wang, On the design of a reflector antenna,, Inverse problems, 12 (1996), 351.  doi: 10.1088/0266-5611/12/3/013.  Google Scholar

[28]

X.-J. Wang, On the design of a reflector antenna II,, Calc. Var. PDE, 20 (2004), 329.  doi: 10.1007/s00526-003-0239-4.  Google Scholar

[1]

Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069
[2]

Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559
[3]

Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825
[4]

Jiakun Liu, Neil S. Trudinger. On Pogorelov estimates for Monge-Ampère type equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1121-1135. doi: 10.3934/dcds.2010.28.1121
[5]

Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991
[6]

Fan Cui, Huaiyu Jian. Symmetry of solutions to a class of Monge-Ampère equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1247-1259. doi: 10.3934/cpaa.2019060
[7]

Luca Codenotti, Marta Lewicka. Visualization of the convex integration solutions to the Monge-Ampère equation. Evolution Equations & Control Theory, 2019, 8 (2) : 273-300. doi: 10.3934/eect.2019015
[8]

Limei Dai, Hongyu Li. Entire subsolutions of Monge-Ampère type equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 19-30. doi: 10.3934/cpaa.2020002
[9]

Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221
[10]

Limei Dai. Multi-valued solutions to a class of parabolic Monge-Ampère equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1061-1074. doi: 10.3934/cpaa.2014.13.1061
[11]

Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058
[12]

Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59
[13]

Cristian Enache. Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1347-1359. doi: 10.3934/cpaa.2014.13.1347
[14]

Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705
[15]

Zhijun Zhang. Optimal global asymptotic behavior of the solution to a singular monge-ampère equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1129-1145. doi: 10.3934/cpaa.2020053
[16]

Gregorio Díaz, Jesús Ildefonso Díaz. On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447
[17]

Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure & Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697
[18]

Geng Chen, Yannan Shen. Existence and regularity of solutions in nonlinear wave equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3327-3342. doi: 10.3934/dcds.2015.35.3327
[19]

Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Local existence with mild regularity for the Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 1011-1041. doi: 10.3934/krm.2013.6.1011
[20]

Claudianor O. Alves, Vincenzo Ambrosio, Teresa Isernia. Existence, multiplicity and concentration for a class of fractional $ p \& q $ Laplacian problems in $ \mathbb{R} ^{N} $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2009-2045. doi: 10.3934/cpaa.2019091

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences