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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 and Continuous Dynamical Systems, 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), 453-496. doi: 10.2307/2118564. [2] L. A. Caffarelli and V. Oliker, Weak solutions of one inverse problem in geometric optics, J. Math. Sci., 154 (2008), 37-46. doi: 10.1007/s10958-008-9152-x. [3] 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. [4] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, $2^{nd}$ edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [5] P. Guan and X.-J. Wang, On a Monge-Ampère equation arising in geometric optics, J. Diff. Geom., 48 (1998), 205-223. [6] C. Gutiérrez and F. Tournier, The parallel refractor, Developments in Mathematics, 28 (2013), 325-334. doi: 10.1007/978-1-4614-4075-8_14. [7] F. Jiang and N. S. Trudinger, On Pogorelov estimates in optimal transportation and geometric optics, Bull. Math. Sci., 4 (2014), 407-431. doi: 10.1007/s13373-014-0055-5. [8] A. Karakhanyan, Existence and regularity of the reflector surfaces in $\mathbbR^{n+1}$, Arch. Ration. Mech. Anal., 213 (2014), 833-885. doi: 10.1007/s00205-014-0743-z. [9] A. Karakhanyan and X.-J. Wang, On the reflector shape design, J. Diff. Geom., 84 (2010), 561-610. [10] S. Kochengin and V. Oliker, Determination of reflector surfaces from near-field scattering data, Inverse Problems, 13 (1997), 363-373. doi: 10.1088/0266-5611/13/2/011. [11] Y. Li, J. Liu and L. Nguyen, A degree theory for second order nonlinear elliptic operators with nonlinear oblique boundary conditions,, preprint, (): 26. [12] G. M. Lieberman and N. S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations. Trans. Amer. Math. Soc., 295 (1986), 509-546. doi: 10.1090/S0002-9947-1986-0833695-6. [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] J. Liu, Light reflection is nonlinear optimization, Calc. Var. PDE, 46 (2013), 861-878. doi: 10.1007/s00526-012-0506-3. [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), 151-183. doi: 10.1007/s00205-005-0362-9. [16] V. Oliker, Differential equations for design of a freeform single lens with prescribed irradiance properties, Optical Engineering, 53 (2014), 031301, 10pp. doi: 10.1117/1.OE.53.3.031302. [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), 160-183. doi: 10.1016/j.aam.2014.09.009. [18] N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math., 175 (1995), 151-164. doi: 10.1007/BF02393303. [19] N. S. Trudinger, Recent developments in elliptic partial differential equations of Monge-Ampère type, Proc. Int. Cong. Math., Madrid, 3 (2006), 291-301. [20] N. S. Trudinger, On the prescribed Jacobian equation, Gakuto Intl. Series, Math. Sci. Appl., Proc. Intl. Conf. for the 25th Anniversary of Viscosity Solutions, XX (2008), 243-255. [21] N. S. Trudinger, On the local theory of prescribed Jacobian equations, Discrete Contin. Dyn. Syst., 34 (2014), 1663-1681. doi: 10.3934/dcds.2014.34.1663. [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), 143-174. [23] 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. [24] J. Urbas, Mass Transfer Problems, Lecture Notes, Univ. of Bonn, 1998. [25] J. Urbas, Oblique boundary value problems for equations of Monge-Ampère type, Calc. Var. PDE, 7 (1998), 19-39. doi: 10.1007/s005260050097. [26] G. von Nessi, On the second boundary value problem for a class of modified-Hessian equations, Comm. Partial Differential Equations, 35 (2010), 745-785. doi: 10.1080/03605301003632317. [27] X.-J. Wang, On the design of a reflector antenna, Inverse problems, 12 (1996), 351-375. doi: 10.1088/0266-5611/12/3/013. [28] X.-J. Wang, On the design of a reflector antenna II, Calc. Var. PDE, 20 (2004), 329-341. doi: 10.1007/s00526-003-0239-4.

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##### References:
 [1] L. A. Caffarelli, Boundary regularity of maps with convex potentials II, Ann. of Math., 144 (1996), 453-496. doi: 10.2307/2118564. [2] L. A. Caffarelli and V. Oliker, Weak solutions of one inverse problem in geometric optics, J. Math. Sci., 154 (2008), 37-46. doi: 10.1007/s10958-008-9152-x. [3] 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. [4] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, $2^{nd}$ edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [5] P. Guan and X.-J. Wang, On a Monge-Ampère equation arising in geometric optics, J. Diff. Geom., 48 (1998), 205-223. [6] C. Gutiérrez and F. Tournier, The parallel refractor, Developments in Mathematics, 28 (2013), 325-334. doi: 10.1007/978-1-4614-4075-8_14. [7] F. Jiang and N. S. Trudinger, On Pogorelov estimates in optimal transportation and geometric optics, Bull. Math. Sci., 4 (2014), 407-431. doi: 10.1007/s13373-014-0055-5. [8] A. Karakhanyan, Existence and regularity of the reflector surfaces in $\mathbbR^{n+1}$, Arch. Ration. Mech. Anal., 213 (2014), 833-885. doi: 10.1007/s00205-014-0743-z. [9] A. Karakhanyan and X.-J. Wang, On the reflector shape design, J. Diff. Geom., 84 (2010), 561-610. [10] S. Kochengin and V. Oliker, Determination of reflector surfaces from near-field scattering data, Inverse Problems, 13 (1997), 363-373. doi: 10.1088/0266-5611/13/2/011. [11] Y. Li, J. Liu and L. Nguyen, A degree theory for second order nonlinear elliptic operators with nonlinear oblique boundary conditions,, preprint, (): 26. [12] G. M. Lieberman and N. S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations. Trans. Amer. Math. Soc., 295 (1986), 509-546. doi: 10.1090/S0002-9947-1986-0833695-6. [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] J. Liu, Light reflection is nonlinear optimization, Calc. Var. PDE, 46 (2013), 861-878. doi: 10.1007/s00526-012-0506-3. [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), 151-183. doi: 10.1007/s00205-005-0362-9. [16] V. Oliker, Differential equations for design of a freeform single lens with prescribed irradiance properties, Optical Engineering, 53 (2014), 031301, 10pp. doi: 10.1117/1.OE.53.3.031302. [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), 160-183. doi: 10.1016/j.aam.2014.09.009. [18] N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math., 175 (1995), 151-164. doi: 10.1007/BF02393303. [19] N. S. Trudinger, Recent developments in elliptic partial differential equations of Monge-Ampère type, Proc. Int. Cong. Math., Madrid, 3 (2006), 291-301. [20] N. S. Trudinger, On the prescribed Jacobian equation, Gakuto Intl. Series, Math. Sci. Appl., Proc. Intl. Conf. for the 25th Anniversary of Viscosity Solutions, XX (2008), 243-255. [21] N. S. Trudinger, On the local theory of prescribed Jacobian equations, Discrete Contin. Dyn. Syst., 34 (2014), 1663-1681. doi: 10.3934/dcds.2014.34.1663. [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), 143-174. [23] 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. [24] J. Urbas, Mass Transfer Problems, Lecture Notes, Univ. of Bonn, 1998. [25] J. Urbas, Oblique boundary value problems for equations of Monge-Ampère type, Calc. Var. PDE, 7 (1998), 19-39. doi: 10.1007/s005260050097. [26] G. von Nessi, On the second boundary value problem for a class of modified-Hessian equations, Comm. Partial Differential Equations, 35 (2010), 745-785. doi: 10.1080/03605301003632317. [27] X.-J. Wang, On the design of a reflector antenna, Inverse problems, 12 (1996), 351-375. doi: 10.1088/0266-5611/12/3/013. [28] X.-J. Wang, On the design of a reflector antenna II, Calc. Var. PDE, 20 (2004), 329-341. doi: 10.1007/s00526-003-0239-4.
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