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

Chiara Corsato; Colette De  Coster; Pierpaolo Omari

doi:10.3934/proc.2015.0297

Article Contents

2015, Volume 2015: 297-303. Doi: 10.3934/proc.2015.0297 open access

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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

Received: August 2014

Revised: January 2015

Published: October 2015

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Abstract

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:

Anisotropic prescribed mean curvature equation,
Dirichlet boundary condition,
radially symmetric solution,
positive solution,
existence,
uniqueness,
shooting method.

Mathematics Subject Classification: Primary: 35J93, 35J25; Secondary: 35B07, 35B09, 35A09, 35A02, 35A24, 34A12.

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