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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 |
References:
[1] |
L. A. Caffarelli, Boundary regularity of maps with convex potentials II,, Ann. of Math., 144 (1996), 453.
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.
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, 8 (1991), 443.
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, (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.
|
[6] |
C. Gutiérrez and F. Tournier, The parallel refractor,, Developments in Mathematics, 28 (2013), 325.
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.
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.
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.
|
[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. |
[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. |
[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. |
[14] |
J. Liu, Light reflection is nonlinear optimization,, Calc. Var. PDE, 46 (2013), 861.
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.
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).
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.
doi: 10.1016/j.aam.2014.09.009. |
[18] |
N. S. Trudinger, On the Dirichlet problem for Hessian equations,, Acta Math., 175 (1995), 151.
doi: 10.1007/BF02393303. |
[19] |
N. S. Trudinger, Recent developments in elliptic partial differential equations of Monge-Ampère type,, Proc. Int. Cong. Math., 3 (2006), 291.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[27] |
X.-J. Wang, On the design of a reflector antenna,, Inverse problems, 12 (1996), 351.
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.
doi: 10.1007/s00526-003-0239-4. |
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. |
[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. |
[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. |
[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. |
[5] |
P. Guan and X.-J. Wang, On a Monge-Ampère equation arising in geometric optics,, J. Diff. Geom., 48 (1998), 205.
|
[6] |
C. Gutiérrez and F. Tournier, The parallel refractor,, Developments in Mathematics, 28 (2013), 325.
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.
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.
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.
|
[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. |
[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. |
[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. |
[14] |
J. Liu, Light reflection is nonlinear optimization,, Calc. Var. PDE, 46 (2013), 861.
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.
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).
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.
doi: 10.1016/j.aam.2014.09.009. |
[18] |
N. S. Trudinger, On the Dirichlet problem for Hessian equations,, Acta Math., 175 (1995), 151.
doi: 10.1007/BF02393303. |
[19] |
N. S. Trudinger, Recent developments in elliptic partial differential equations of Monge-Ampère type,, Proc. Int. Cong. Math., 3 (2006), 291.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[27] |
X.-J. Wang, On the design of a reflector antenna,, Inverse problems, 12 (1996), 351.
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.
doi: 10.1007/s00526-003-0239-4. |
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