# American Institute of Mathematical Sciences

April  2018, 11(2): 213-256. doi: 10.3934/dcdss.2018013

## A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis

 1 Università degli Studi di Trieste, Dipartimento di Scienze Economiche, Aziendali, Matematiche e Statistiche, Piazzale Europa 1,34127 Trieste, Italy 2 Univ. Valenciennes, EA 4015 -LAMAV -FR CNRS 2956, F-59313 Valenciennes, France 3 Università degli Studi di Trieste, Dipartimento di Matematica e Geoscienze-Sezione di Matematica e Informatica, Via A. Valerio 12/1,34127 Trieste, Italy

* Corresponding author: Pierpaolo Omari

Received  December 2016 Revised  April 2017 Published  January 2018

Fund Project: This paper was written under the auspices of INdAM-GNAMPA. The third and the fourth named authors have also been supported by the University of Trieste, in the frame of the 2015 FRA project "Differential Equations: Qualitative and Computational Theory"

In this paper we survey, complete and refine some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation
 $\begin{equation*}{\rm{ -div}}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}},\end{equation*}$
in a bounded Lipschitz domain
 $Ω \subset \mathbb{R}^N$
, with
 $a,b>0$
parameters. This equation appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Here we show how various techniques of nonlinear functional analysis can successfully be applied to derive a complete picture of the solvability patterns of the problem.
Citation: Chiara Corsato, Colette De Coster, Franco Obersnel, Pierpaolo Omari, Alessandro Soranzo. A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 213-256. doi: 10.3934/dcdss.2018013
##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems Clarendon Press, Oxford, 2000. Google Scholar [2] G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 293-318. doi: 10.1007/BF01781073. Google Scholar [3] M. Athanassenas and J. Clutterbuck, A capillarity problem for compressible liquids, Pacific J. Math., 243 (2009), 213-232. doi: 10.2140/pjm.2009.243.213. Google Scholar [4] M. Athanassenas and R. Finn, Compressible fluids in a capillary tube, Pacific J. Math., 224 (2006), 201-229. doi: 10.2140/pjm.2006.224.201. Google Scholar [5] M. Bergner, The Dirichlet problem for graphs of prescribed anisotropic mean curvature in $\mathbb{R}^{n+1}$, Analysis (Munich), 28 (2008), 149-166. doi: 10.1524/anly.2008.0906. Google Scholar [6] M. Bergner, On the Dirichlet problem for the prescribed mean curvature equation over general domains, Differential Geom. Appl., 27 (2009), 335-343. doi: 10.1016/j.difgeo.2009.03.002. Google Scholar [7] D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical solutions of a prescribed curvature equation, J. Differential Equations, 243 (2007), 208-237. doi: 10.1016/j.jde.2007.05.031. Google Scholar [8] D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical positive solutions of a prescribed curvature equation with singularities, Rend. Istit. Mat. Univ. Trieste, 39 (2007), 63-85. Google Scholar [9] I. Coelho, C. Corsato and P. Omari, A one-dimensional prescribed curvature equation modeling the corneal shape Bound. Value Probl. , 2014 (2014), p127. doi: 10.1186/1687-2770-2014-127. Google Scholar [10] C. Corsato, C. De Coster and P. Omari, Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape, Discrete Contin. Dyn. Syst.(Suppl.), 2015 (2015), 297-303. doi: 10.3934/proc.2015.0297. Google Scholar [11] C. Corsato, C. De Coster and P. Omari, The Dirichlet problem for a prescribed anisotropic mean curvature equation: existence, uniqueness and regularity of solutions, J. Differential Equations, 260 (2016), 4572-4618. doi: 10.1016/j.jde.2015.11.024. Google Scholar [12] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations Chapman & Hall/CRC, Boca Raton, 2011. doi: 10.1201/b10802. Google Scholar [13] I. Ekeland and R. Temam, Convex Analysis and Variational Problems Society for Industrial and Applied Mathematics, Philadelphia, 1999. doi: 10.1137/1.9781611971088. Google Scholar [14] L. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions CRC Press, Boca Raton, 1992. Google Scholar [15] D. G. de Figueiredo, Lectures on the Ekeland Variational Principle with Applications and Detours Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 81 Springer, Berlin, 1989. Google Scholar [16] R. Finn, On the equations of capillarity, J. Math. Fluid Mech., 3 (2001), 139-151. doi: 10.1007/PL00000966. Google Scholar [17] R. Finn, Capillarity problems for compressible fluids, Mem. Differential Equations Math. Phys., 33 (2004), 47-55. Google Scholar [18] R. Finn and G. Luli, On the capillary problem for compressible fluids, J. Math. Fluid Mech., 9 (2007), 87-103. doi: 10.1007/s00021-005-0203-5. Google Scholar [19] E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, Rend. Sem. Mat. Univ. Padova, 27 (1957), 284-305. Google Scholar [20] C. Gerhardt, Existence and regularity of capillary surfaces, Boll. Un. Mat. Ital. (4), 10 (1974), 317-335. Google Scholar [21] C. Gerhardt, Existence, regularity, and boundary behavior of generalized surfaces of prescribed mean curvature, Math. Z., 139 (1974), 173-198. doi: 10.1007/BF01418314. Google Scholar [22] C. Gerhardt, On the regularity of solutions to variational problems in $BV(Ω)$, Math. Z., 149 (1976), 281-286. doi: 10.1007/BF01175590. Google Scholar [23] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order Springer, New York, 2001. Google Scholar [24] E. Giusti, On the equation of surfaces of prescribed mean curvature. Existence and uniqueness without boundary conditions, Invent. Math., 46 (1978), 111-137. doi: 10.1007/BF01393250. Google Scholar [25] E. Giusti, Generalized solutions for the mean curvature equation, Pacific J. Math., 88 (1980), 297-321. doi: 10.2140/pjm.1980.88.297. Google Scholar [26] E. Giusti, Minimal Surfaces and Functions of Bounded Variations Birkhäuser, Basel, 1984. doi: 10.1007/978-1-4684-9486-0. Google Scholar [27] M. Goebel, On Fréchet-differentiability of Nemytskij operators acting in Hölder spaces, Glasgow Math. J., 33 (1991), 1-5. doi: 10.1017/S0017089500007965. Google Scholar [28] K. Hayasida and Y. Ikeda, Prescribed mean curvature equations under the transformation with non-orthogonal curvilinear coordinates, Nonlinear Anal., 67 (2007), 1-25. doi: 10.1016/j.na.2006.07.016. Google Scholar [29] K. Hayasida and M. Nakatani, On the Dirichlet problem of prescribed mean curvature equations without H-convexity condition, Nagoya Math. J., 157 (2000), 177-209. doi: 10.1017/S0027763000007248. Google Scholar [30] R. Huff and J. McCuan, Minimal graphs with discontinuous boundary values, J. Aust. Math. Soc., 86 (2009), 75-95. doi: 10.1017/S1446788708000335. Google Scholar [31] H. Jenkins and J. Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math., 229 (1968), 170-187. doi: 10.1515/crll.1968.229.170. Google Scholar [32] G. A. Ladyzhenskaya and N. N. Ural'tseva, Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations, Comm. Pur. Appl. Math., 23 (1970), 677-703. doi: 10.1002/cpa.3160230409. Google Scholar [33] A. Lichnewsky, Principe du maximum local et solutions généralisées de problémes du type hypersurfaces minimales, Bull. Soc. Math. France, 102 (1974), 417-433. Google Scholar [34] A. Lichnewsky, Sur le comportement au bord des solutions généralisées du probléme non paramétrique des surfaces minimales, J. Math. Pures Appl., 53 (1974), 397-425. Google Scholar [35] A. Lichnewsky, Solutions généralisées du probléme des surfaces minimales pour des données au bord non bornées, J. Math. Pures Appl., 57 (1978), 231-253. Google Scholar [36] A. Lichnewsky and R. Temam, Pseudosolutions of the time-dependent minimal surface problem, J. Differential Equation, 30 (1978), 340-364. doi: 10.1016/0022-0396(78)90005-0. Google Scholar [37] J. López-Gómez, P. Omari and S. Rivetti, Positive solutions of one-dimensional indefinite capillarity-type problems: A variational approach, J. Differential Equations, 262 (2017), 2335-2392. doi: 10.1016/j.jde.2016.10.046. Google Scholar [38] J. López-Gómez, P. Omari and S. Rivetti, Bifurcation of positive solutions for a one-dimensional indefinite quasilinear Neumann problem, Nonlinear Anal., 155 (2017), 1-51. doi: 10.1016/j.na.2017.01.007. Google Scholar [39] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems Birkhäuser, Basel, 1995. Google Scholar [40] T. Marquardt, Remark on the anisotropic prescribed mean curvature equation on arbitrary domains, Math. Z., 264 (2010), 507-511. doi: 10.1007/s00209-009-0476-0. Google Scholar [41] M. Miranda, Superfici minime illimitate, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1977), 313-322. Google Scholar [42] M. Miranda, Maximum principles and minimal surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 667-681. Google Scholar [43] C. B. Morrey Jr. , Multiple Integrals in the Calculus of Variations Springer, New York, 1966. Google Scholar [44] J. Nečas, Direct Methods in the Theory of Elliptic Equations Springer, New York, 2012. doi: 10.1007/978-3-642-10455-8. Google Scholar [45] F. Obersnel and P. Omari, Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation, Discrete Contin. Dyn. Syst., 33 (2013), 305-320. doi: 10.3934/dcds.2013.33.305. Google Scholar [46] W. Okrasiński and Ł. Płociniczak, A nonlinear mathematical model of the corneal shape, Nonlinear Anal. Real World Appl., 13 (2012), 1498-1505. doi: 10.1016/j.nonrwa.2011.11.014. Google Scholar [47] W. Okrasiński and Ł. Płociniczak, Bessel function model of corneal topography, Appl. Math. Comput., 223 (2013), 436-443. doi: 10.1016/j.amc.2013.07.097. Google Scholar [48] W. Okrasiński and Ł. Płociniczak, Regularization of an ill-posed problem in corneal topography, Inverse Probl. Sci. Eng., 21 (2013), 1090-1097. doi: 10.1080/17415977.2012.753443. Google Scholar [49] H. Pan and R. Xing, Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations. Ⅱ, Nonlinear Anal., 74 (2011), 3751-3768. doi: 10.1016/j.na.2011.03.020. Google Scholar [50] Ł. Płociniczak, G. W. Griffiths and W. E. Schiesser, ODE/PDE analysis of corneal curvature, Computers in Biology and Medicine, 53 (2014), 30-41. Google Scholar [51] Ł. Płociniczak and W. Okrasiński, Nonlinear parameter identification in a corneal geometry model, Inverse Probl. Sci. Eng., 23 (2015), 443-456. doi: 10.1080/17415977.2014.922074. Google Scholar [52] Ł. Płociniczak, W. Okrasiński, J. J. Nieto and O. Domínguez, On a nonlinear boundary value problem modeling corneal shape, J. Math. Anal. Appl., 414 (2014), 461-471. doi: 10.1016/j.jmaa.2014.01.010. Google Scholar [53] P. H. Rabinowitz, A global theorem for nonlinear eigenvalue problems and applications, in Contributions to nonlinear functional analysis (eds. E. H. Zarantonello), Academic Press, (1971), 11-36 Google Scholar [54] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Phil. Trans. R. Soc. Lond. A, 264 (1969), 413-496. doi: 10.1098/rsta.1969.0033. Google Scholar [55] R. Temam, Solutions généralisées de certaines équations du type hypersurfaces minima, Arch. Rational Mech. Anal., 44 (1971/72), 121-156. doi: 10.1007/BF00281813. Google Scholar

show all references

##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems Clarendon Press, Oxford, 2000. Google Scholar [2] G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 293-318. doi: 10.1007/BF01781073. Google Scholar [3] M. Athanassenas and J. Clutterbuck, A capillarity problem for compressible liquids, Pacific J. Math., 243 (2009), 213-232. doi: 10.2140/pjm.2009.243.213. Google Scholar [4] M. Athanassenas and R. Finn, Compressible fluids in a capillary tube, Pacific J. Math., 224 (2006), 201-229. doi: 10.2140/pjm.2006.224.201. Google Scholar [5] M. Bergner, The Dirichlet problem for graphs of prescribed anisotropic mean curvature in $\mathbb{R}^{n+1}$, Analysis (Munich), 28 (2008), 149-166. doi: 10.1524/anly.2008.0906. Google Scholar [6] M. Bergner, On the Dirichlet problem for the prescribed mean curvature equation over general domains, Differential Geom. Appl., 27 (2009), 335-343. doi: 10.1016/j.difgeo.2009.03.002. Google Scholar [7] D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical solutions of a prescribed curvature equation, J. Differential Equations, 243 (2007), 208-237. doi: 10.1016/j.jde.2007.05.031. Google Scholar [8] D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical positive solutions of a prescribed curvature equation with singularities, Rend. Istit. Mat. Univ. Trieste, 39 (2007), 63-85. Google Scholar [9] I. Coelho, C. Corsato and P. Omari, A one-dimensional prescribed curvature equation modeling the corneal shape Bound. Value Probl. , 2014 (2014), p127. doi: 10.1186/1687-2770-2014-127. Google Scholar [10] C. Corsato, C. De Coster and P. Omari, Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape, Discrete Contin. Dyn. Syst.(Suppl.), 2015 (2015), 297-303. doi: 10.3934/proc.2015.0297. Google Scholar [11] C. Corsato, C. De Coster and P. Omari, The Dirichlet problem for a prescribed anisotropic mean curvature equation: existence, uniqueness and regularity of solutions, J. Differential Equations, 260 (2016), 4572-4618. doi: 10.1016/j.jde.2015.11.024. Google Scholar [12] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations Chapman & Hall/CRC, Boca Raton, 2011. doi: 10.1201/b10802. Google Scholar [13] I. Ekeland and R. Temam, Convex Analysis and Variational Problems Society for Industrial and Applied Mathematics, Philadelphia, 1999. doi: 10.1137/1.9781611971088. Google Scholar [14] L. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions CRC Press, Boca Raton, 1992. Google Scholar [15] D. G. de Figueiredo, Lectures on the Ekeland Variational Principle with Applications and Detours Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 81 Springer, Berlin, 1989. Google Scholar [16] R. Finn, On the equations of capillarity, J. Math. Fluid Mech., 3 (2001), 139-151. doi: 10.1007/PL00000966. Google Scholar [17] R. Finn, Capillarity problems for compressible fluids, Mem. Differential Equations Math. Phys., 33 (2004), 47-55. Google Scholar [18] R. Finn and G. Luli, On the capillary problem for compressible fluids, J. Math. Fluid Mech., 9 (2007), 87-103. doi: 10.1007/s00021-005-0203-5. Google Scholar [19] E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, Rend. Sem. Mat. Univ. Padova, 27 (1957), 284-305. Google Scholar [20] C. Gerhardt, Existence and regularity of capillary surfaces, Boll. Un. Mat. Ital. (4), 10 (1974), 317-335. Google Scholar [21] C. Gerhardt, Existence, regularity, and boundary behavior of generalized surfaces of prescribed mean curvature, Math. Z., 139 (1974), 173-198. doi: 10.1007/BF01418314. Google Scholar [22] C. Gerhardt, On the regularity of solutions to variational problems in $BV(Ω)$, Math. Z., 149 (1976), 281-286. doi: 10.1007/BF01175590. Google Scholar [23] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order Springer, New York, 2001. Google Scholar [24] E. Giusti, On the equation of surfaces of prescribed mean curvature. Existence and uniqueness without boundary conditions, Invent. Math., 46 (1978), 111-137. doi: 10.1007/BF01393250. Google Scholar [25] E. Giusti, Generalized solutions for the mean curvature equation, Pacific J. Math., 88 (1980), 297-321. doi: 10.2140/pjm.1980.88.297. Google Scholar [26] E. Giusti, Minimal Surfaces and Functions of Bounded Variations Birkhäuser, Basel, 1984. doi: 10.1007/978-1-4684-9486-0. Google Scholar [27] M. Goebel, On Fréchet-differentiability of Nemytskij operators acting in Hölder spaces, Glasgow Math. J., 33 (1991), 1-5. doi: 10.1017/S0017089500007965. Google Scholar [28] K. Hayasida and Y. Ikeda, Prescribed mean curvature equations under the transformation with non-orthogonal curvilinear coordinates, Nonlinear Anal., 67 (2007), 1-25. doi: 10.1016/j.na.2006.07.016. Google Scholar [29] K. Hayasida and M. Nakatani, On the Dirichlet problem of prescribed mean curvature equations without H-convexity condition, Nagoya Math. J., 157 (2000), 177-209. doi: 10.1017/S0027763000007248. Google Scholar [30] R. Huff and J. McCuan, Minimal graphs with discontinuous boundary values, J. Aust. Math. Soc., 86 (2009), 75-95. doi: 10.1017/S1446788708000335. Google Scholar [31] H. Jenkins and J. Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math., 229 (1968), 170-187. doi: 10.1515/crll.1968.229.170. Google Scholar [32] G. A. Ladyzhenskaya and N. N. Ural'tseva, Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations, Comm. Pur. Appl. Math., 23 (1970), 677-703. doi: 10.1002/cpa.3160230409. Google Scholar [33] A. Lichnewsky, Principe du maximum local et solutions généralisées de problémes du type hypersurfaces minimales, Bull. Soc. Math. France, 102 (1974), 417-433. Google Scholar [34] A. Lichnewsky, Sur le comportement au bord des solutions généralisées du probléme non paramétrique des surfaces minimales, J. Math. Pures Appl., 53 (1974), 397-425. Google Scholar [35] A. Lichnewsky, Solutions généralisées du probléme des surfaces minimales pour des données au bord non bornées, J. Math. Pures Appl., 57 (1978), 231-253. Google Scholar [36] A. Lichnewsky and R. Temam, Pseudosolutions of the time-dependent minimal surface problem, J. Differential Equation, 30 (1978), 340-364. doi: 10.1016/0022-0396(78)90005-0. Google Scholar [37] J. López-Gómez, P. Omari and S. Rivetti, Positive solutions of one-dimensional indefinite capillarity-type problems: A variational approach, J. Differential Equations, 262 (2017), 2335-2392. doi: 10.1016/j.jde.2016.10.046. Google Scholar [38] J. López-Gómez, P. Omari and S. Rivetti, Bifurcation of positive solutions for a one-dimensional indefinite quasilinear Neumann problem, Nonlinear Anal., 155 (2017), 1-51. doi: 10.1016/j.na.2017.01.007. Google Scholar [39] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems Birkhäuser, Basel, 1995. Google Scholar [40] T. Marquardt, Remark on the anisotropic prescribed mean curvature equation on arbitrary domains, Math. Z., 264 (2010), 507-511. doi: 10.1007/s00209-009-0476-0. Google Scholar [41] M. Miranda, Superfici minime illimitate, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1977), 313-322. Google Scholar [42] M. Miranda, Maximum principles and minimal surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 667-681. Google Scholar [43] C. B. Morrey Jr. , Multiple Integrals in the Calculus of Variations Springer, New York, 1966. Google Scholar [44] J. Nečas, Direct Methods in the Theory of Elliptic Equations Springer, New York, 2012. doi: 10.1007/978-3-642-10455-8. Google Scholar [45] F. Obersnel and P. Omari, Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation, Discrete Contin. Dyn. Syst., 33 (2013), 305-320. doi: 10.3934/dcds.2013.33.305. Google Scholar [46] W. Okrasiński and Ł. Płociniczak, A nonlinear mathematical model of the corneal shape, Nonlinear Anal. Real World Appl., 13 (2012), 1498-1505. doi: 10.1016/j.nonrwa.2011.11.014. Google Scholar [47] W. Okrasiński and Ł. Płociniczak, Bessel function model of corneal topography, Appl. Math. Comput., 223 (2013), 436-443. doi: 10.1016/j.amc.2013.07.097. Google Scholar [48] W. Okrasiński and Ł. Płociniczak, Regularization of an ill-posed problem in corneal topography, Inverse Probl. Sci. Eng., 21 (2013), 1090-1097. doi: 10.1080/17415977.2012.753443. Google Scholar [49] H. Pan and R. Xing, Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations. Ⅱ, Nonlinear Anal., 74 (2011), 3751-3768. doi: 10.1016/j.na.2011.03.020. Google Scholar [50] Ł. Płociniczak, G. W. Griffiths and W. E. Schiesser, ODE/PDE analysis of corneal curvature, Computers in Biology and Medicine, 53 (2014), 30-41. Google Scholar [51] Ł. Płociniczak and W. Okrasiński, Nonlinear parameter identification in a corneal geometry model, Inverse Probl. Sci. Eng., 23 (2015), 443-456. doi: 10.1080/17415977.2014.922074. Google Scholar [52] Ł. Płociniczak, W. Okrasiński, J. J. Nieto and O. Domínguez, On a nonlinear boundary value problem modeling corneal shape, J. Math. Anal. Appl., 414 (2014), 461-471. doi: 10.1016/j.jmaa.2014.01.010. Google Scholar [53] P. H. Rabinowitz, A global theorem for nonlinear eigenvalue problems and applications, in Contributions to nonlinear functional analysis (eds. E. H. Zarantonello), Academic Press, (1971), 11-36 Google Scholar [54] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Phil. Trans. R. Soc. Lond. A, 264 (1969), 413-496. doi: 10.1098/rsta.1969.0033. Google Scholar [55] R. Temam, Solutions généralisées de certaines équations du type hypersurfaces minima, Arch. Rational Mech. Anal., 44 (1971/72), 121-156. doi: 10.1007/BF00281813. Google Scholar
Graph of a generalized solution on an arbitrary domain.
Graph of a singular solution on a thick spherical shell
Classical solutions emanating from the trivial line: $\|\nabla u (a, b)\|_\infty$ is plotted, in applicates, versus $a$, in abscissas, and $b$, in ordinates
Profile of the upper solution
Profile of the lower solution
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