Abstract
We study a functional modelling the progressive lens design, which is a combination of Willmore functional and total Gauss curvature. First, we prove the existence for the minimizers of this class of functionals among the class of revolution surfaces rotated by the curves y = f(x) about the x-axis. Then, choosing such a minimiser as background surfaces to approximate the functional by a quadratic functional, we prove the existence and uniqueness of the solution to the Euler-Lagrange equation for the quadratic functionals. Our results not only provide a strictly mathematical proof for numerical methods, but also give a more reasonable and more extensive choice for the background surfaces.
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References
Bauer M, Kuwert E. Existence of minimizing the Willmore surfaces of prescribed genes. Int Math Res Not IMRN, 2003, 2003: 553–576
Bergner M, Dall’Acqua A, Fröhlich S. Symmetry Willmore surfaces of revolution satisfying natural boundary conditions. Calc Var Partial Diffrential Equations, 2010, 39: 361–378
Canham P B. The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J Theoret Biol, 1970, 26: 61–76
Chen J, Li Y. Radially symmetric solutions to the graphic Willmore surface equation. J Geom Anal, 2017, 27: 671–681
Dall’Acqua A, Fröhlich S, Grunau H C, Schieweck F. Symmetry Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data. Adv Calc Var, 2011, 4: 1–81
Eichmann S, Koeller A. Symmetry for Willmore surfaces of revolution. J Geom Anal, 2017, 27: 618–642
Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. New York: Springer-Verlag, 1983
Helfrich W. Elastic properties of lipid bilayers: Theory and possible experiments. Z Naturalforsch Teil C, 1973, 28: 693–703
Jiang W, Bao W, Tang Q, Wang H. A variational-difference numerical method for designing progressive-addition lenses. Comput -Aided Des, 2014, 48: 17–27
Kuwert E, Schätzle R. The Willmore functional. In: Mingione G, ed. Topics in Modern Regularity Theory. CRM Series, Vol 13. Pisa: Ed Norm, 2012, 1–115
Li Y. Some remarks on Willmore surfaces embedded in ℝ3. J Geom Anal, 2016, 26: 2411–2424
Loos J, Greiner G, Seidel H P. A variational approach to progressive lens design. Comput -Aided Des, 1998, 30: 595–602
Marques F C, Neves A. The Willmore conjecture. Jahresber Dtsch Math -Ver, 2014, 116: 201–222
Schäzle R. The Willmore boundary problem. Calc Var Partial Differential Equations, 2010, 37: 275–302
Simon L. Existence of surfaces minimizing the Willmore functional. Comm Anal Geom, 1993, 1: 281–326
Wang J, Gulliver R, Santosa F. Analysis of a variational approach to progressive lens design. SIAM J Appl Math, 2003, 64: 277–296
Wang J, Santosa F. A numerical methods for progressive lens design. Math Models Methods Appl Sci, 2004, 14: 619–640
Willmore T J. Note on embedded surfaces. An Ştiinţ Univ Al I Cuza Iaşsi Mat, 1965, 11: 493–496
Acknowledgements
This paper started in January of 2011 when the first author (Huaiyu Jian) was visiting the National University of Singapore. He would like to express his sincere thanks to Professor Weizhu Bao and Professor Wei Jiang for introducing the problem to him and discussing the problem with him. This work was supported by the National Natural Science Foundation of China (Grant No. 11771237).
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Jian, H., Zeng, H. Existence and uniqueness for variational problem from progressive lens design. Front. Math. China 15, 491–505 (2020). https://doi.org/10.1007/s11464-020-0845-x
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DOI: https://doi.org/10.1007/s11464-020-0845-x
Keywords
- Variational problem
- Willmore surfaces of revolution
- fourth-order elliptic partial differential equation
- Dirichlet boundary value problem
- existence and uniqueness