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

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

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.
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), 243 (2009), 213.   Google Scholar

[2]

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

[3]

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

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M. Bergner, On the Dirichlet problem for the prescribed mean curvature equation over general domains,, Differential Geom. Appl. 27 (2009), 27 (2009), 335.   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), 243 (2007), 208.   Google Scholar

[6]

I. Coelho, C. Corsato, P. Omari, A one-dimensional prescribed curvature equation modeling the corneal shape,, Bound. Value Probl. 2014, 2014 (2014).  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), (2015).   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

T. Marquardt, Remark on the anisotropic prescribed mean curvature equation on arbitrary domains,, Math. Z. 264 (2010), 264 (2010), 507.   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), 33 (2013), 305.   Google Scholar

[13]

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

[14]

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

[15]

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

[16]

Ł. Płociniczak, G.W.Griffits, W.E.Schiesser, ODE/PDE analysis of corneal curvature,, Comput. Biol. Med. 53 (2014), 53 (2014), 30.  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), 23 (2015), 443.   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), 414 (2014), 461.   Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

M. Bergner, On the Dirichlet problem for the prescribed mean curvature equation over general domains,, Differential Geom. Appl. 27 (2009), 27 (2009), 335.   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), 243 (2007), 208.   Google Scholar

[6]

I. Coelho, C. Corsato, P. Omari, A one-dimensional prescribed curvature equation modeling the corneal shape,, Bound. Value Probl. 2014, 2014 (2014).  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), (2015).   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

T. Marquardt, Remark on the anisotropic prescribed mean curvature equation on arbitrary domains,, Math. Z. 264 (2010), 264 (2010), 507.   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), 33 (2013), 305.   Google Scholar

[13]

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

[14]

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

[15]

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

[16]

Ł. Płociniczak, G.W.Griffits, W.E.Schiesser, ODE/PDE analysis of corneal curvature,, Comput. Biol. Med. 53 (2014), 53 (2014), 30.  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), 23 (2015), 443.   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), 414 (2014), 461.   Google Scholar

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