American Institute of Mathematical Sciences

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2015, 2015(special): 297-303. doi: 10.3934/proc.2015.0297

Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape

Chiara Corsato 1, , Colette De Coster 2, and  Pierpaolo Omari 1,

1. 

Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via A. Valerio 12/1, 34127 Trieste
2. 

Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, F-59313 Valenciennes Cedex 9, France

Received  August 2014 Revised  January 2015 Published  November 2015

We prove existence and uniqueness of classical solutions of the anisotropic prescribed mean curvature problem \begin{equation*} {\rm -div}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}}, \ \text{ in } B, \quad u=0, \ \text{ on } \partial B, \end{equation*} where $a,b>0$ are given parameters and $B$ is a ball in ${\mathbb R}^N$. The solution we find is positive, radially symmetric, radially decreasing and concave. This equation has been proposed as a model of the corneal shape in the recent papers [13,14,15,18,17], where however a linearized version of the equation has been investigated.
Keywords: positive solution, Dirichlet boundary condition, Anisotropic prescribed mean curvature equation, uniqueness, radially symmetric solution, existence, shooting method..
Mathematics Subject Classification: Primary: 35J93, 35J25; Secondary: 35B07, 35B09, 35A09, 35A02, 35A24, 34A1.
Citation: Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297
References:
[1]

M. Athanassenas, J. Clutterbuck, A capillarity problem for compressible liquids, Pacific J. Math. 243 (2009), 213-232.  Google Scholar

[2]

M. Athanassenas, R. Finn, Compressible fluids in a capillary tube, Pacific J. Math. 224 (2004), 201-229.  Google Scholar

[3]

M. Bergner, The Dirichlet problem for graphs of prescribed anisotropic mean curvature in $\mathbb R^{n+1}$, Analysis (Munich) 28 (2008), 149-166.  Google Scholar

[4]

M. Bergner, On the Dirichlet problem for the prescribed mean curvature equation over general domains, Differential Geom. Appl. 27 (2009), 335-343.  Google Scholar

[5]

D. Bonheure, P. Habets, F. Obersnel, P. Omari, Classical and non-classical solutions of a prescribed curvature equation, J. Differential Equations 243 (2007), 208-237.  Google Scholar

[6]

I. Coelho, C. Corsato, P. Omari, A one-dimensional prescribed curvature equation modeling the corneal shape, Bound. Value Probl. 2014, 2014:127, 19 pp.. doi: 10.1186/1687-2770-2014-127.  Google Scholar

[7]

C. Corsato, C. De Coster, P. Omari, The Dirichlet problem for a prescribed anisotropic mean curvature equation: existence, uniqueness and regularity of solutions, preprint (2015), 39 pp., doi: 10.13140/2.1.3837.7766. Available from: https://www.researchgate.net/publication/272826705. Google Scholar

[8]

R. Finn, On the equations of capillarity, J. Math. Fluid Mech. 3 (2001), 139-151.  Google Scholar

[9]

R. Finn, Capillarity problems for compressible fluids, Mem. Differential Equations Math. Phys. 33 (2004), 47-55.  Google Scholar

[10]

R. Finn, G. Luli, On the capillary problem for compressible fluids, J. Math. Fluid Mech. 9 (2007), 87-103.  Google Scholar

[11]

T. Marquardt, Remark on the anisotropic prescribed mean curvature equation on arbitrary domains, Math. Z. 264 (2010), 507-511.  Google Scholar

[12]

F. Obersnel, 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.  Google Scholar

[13]

W. Okrasiński, L. Pl ociniczak, A nonlinear mathematical model of the corneal shape, Nonlinear Anal. Real World Appl. 13 (2012), 1498-1505.  Google Scholar

[14]

W. Okrasiński, L. Pl ociniczak, Bessel function model of corneal topography, Appl. Math. Comput. 223 (2013), 436-443.  Google Scholar

[15]

W. Okrasiński, Ł. Płociniczak, Regularization of an ill-posed problem in corneal topography, Inverse Probl. Sci. Eng. 21 (2013), 1090-1097.  Google Scholar

[16]

Ł. Płociniczak, G.W.Griffits, W.E.Schiesser, ODE/PDE analysis of corneal curvature, Comput. Biol. Med. 53 (2014), 30-41. doi: 10.1016/j.compbiomed.2014.07.003.  Google Scholar

[17]

Ł. Płociniczak, W. Okrasiński, Nonlinear parameter identification in a corneal geometry model, Inverse Probl. Sci. Eng. 23 (2015), 443-456.  Google Scholar

[18]

Ł. Płociniczak, W. Okrasiński, J.J. Nieto, O. Domínguez, On a nonlinear boundary value problem modeling corneal shape, J. Math. Anal. Appl. 414 (2014), 461-471.  Google Scholar

show all references

References:
[1]

M. Athanassenas, J. Clutterbuck, A capillarity problem for compressible liquids, Pacific J. Math. 243 (2009), 213-232.  Google Scholar

[2]

M. Athanassenas, R. Finn, Compressible fluids in a capillary tube, Pacific J. Math. 224 (2004), 201-229.  Google Scholar

[3]

M. Bergner, The Dirichlet problem for graphs of prescribed anisotropic mean curvature in $\mathbb R^{n+1}$, Analysis (Munich) 28 (2008), 149-166.  Google Scholar

[4]

M. Bergner, On the Dirichlet problem for the prescribed mean curvature equation over general domains, Differential Geom. Appl. 27 (2009), 335-343.  Google Scholar

[5]

D. Bonheure, P. Habets, F. Obersnel, P. Omari, Classical and non-classical solutions of a prescribed curvature equation, J. Differential Equations 243 (2007), 208-237.  Google Scholar

[6]

I. Coelho, C. Corsato, P. Omari, A one-dimensional prescribed curvature equation modeling the corneal shape, Bound. Value Probl. 2014, 2014:127, 19 pp.. doi: 10.1186/1687-2770-2014-127.  Google Scholar

[7]

C. Corsato, C. De Coster, P. Omari, The Dirichlet problem for a prescribed anisotropic mean curvature equation: existence, uniqueness and regularity of solutions, preprint (2015), 39 pp., doi: 10.13140/2.1.3837.7766. Available from: https://www.researchgate.net/publication/272826705. Google Scholar

[8]

R. Finn, On the equations of capillarity, J. Math. Fluid Mech. 3 (2001), 139-151.  Google Scholar

[9]

R. Finn, Capillarity problems for compressible fluids, Mem. Differential Equations Math. Phys. 33 (2004), 47-55.  Google Scholar

[10]

R. Finn, G. Luli, On the capillary problem for compressible fluids, J. Math. Fluid Mech. 9 (2007), 87-103.  Google Scholar

[11]

T. Marquardt, Remark on the anisotropic prescribed mean curvature equation on arbitrary domains, Math. Z. 264 (2010), 507-511.  Google Scholar

[12]

F. Obersnel, 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.  Google Scholar

[13]

W. Okrasiński, L. Pl ociniczak, A nonlinear mathematical model of the corneal shape, Nonlinear Anal. Real World Appl. 13 (2012), 1498-1505.  Google Scholar

[14]

W. Okrasiński, L. Pl ociniczak, Bessel function model of corneal topography, Appl. Math. Comput. 223 (2013), 436-443.  Google Scholar

[15]

W. Okrasiński, Ł. Płociniczak, Regularization of an ill-posed problem in corneal topography, Inverse Probl. Sci. Eng. 21 (2013), 1090-1097.  Google Scholar

[16]

Ł. Płociniczak, G.W.Griffits, W.E.Schiesser, ODE/PDE analysis of corneal curvature, Comput. Biol. Med. 53 (2014), 30-41. doi: 10.1016/j.compbiomed.2014.07.003.  Google Scholar

[17]

Ł. Płociniczak, W. Okrasiński, Nonlinear parameter identification in a corneal geometry model, Inverse Probl. Sci. Eng. 23 (2015), 443-456.  Google Scholar

[18]

Ł. Płociniczak, W. Okrasiński, J.J. Nieto, O. Domínguez, On a nonlinear boundary value problem modeling corneal shape, J. Math. Anal. Appl. 414 (2014), 461-471.  Google Scholar

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