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February  2016, 36(2): 895-916. doi: 10.3934/dcds.2016.36.895

## On the classical solvability of near field reflector problems

 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.
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
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