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

Chiara Corsato; Colette De Coster; Franco Obersnel; Pierpaolo Omari; Alessandro Soranzo

doi:10.3934/dcdss.2018013

Article Contents

2018, Volume 11, Issue 2: 213-256. Doi: 10.3934/dcdss.2018013

This issue Previous Article Quasilinear elliptic equations with measures and multi-valued lower order terms Next Article Positive subharmonic solutions to superlinear ODEs with indefinite weight

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
* Corresponding author: Pierpaolo Omari

Received: December 2016

Revised: April 2017

Published: April 2018

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".

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Abstract

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.

Keywords:

Prescribed anisotropic mean curvature equation,
positive solution,
Dirichlet boundary condition,
generalized solution,
classical solution,
singular solution,
existence,
uniqueness,
regularity,
boundary behaviour,
bounded variation function,
implicit function theorem,
topological degree,
variational method,
lower and upper solutions.

Mathematics Subject Classification: Primary: 35J93, 35J25, 35J62, 35J67; Secondary: 35B09, 35B51, 35J20.

Citation:

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Figure 1. Graph of a generalized solution on an arbitrary domain.

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Figure 2. Graph of a singular solution on a thick spherical shell

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Figure 3. Classical solutions emanating from the trivial line: $\|\nabla u (a, b)\|_\infty$ is plotted, in applicates, versus $a$, in abscissas, and $b$, in ordinates

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Figure 4. Profile of the upper solution

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Figure 5. Profile of the lower solution

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