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

2015 ◽  
Author(s):  
Pierpaolo Omari ◽  
Colette De Coster ◽  
Chiara Corsato
Keyword(s):  
Mean Curvature ◽  
Equation Modeling ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Radially Symmetric Solutions ◽  
Corneal Shape ◽  
Anisotropic Mean Curvature

scholarly journals An Efficient Semi-Analytical Solution of a One-Dimensional Curvature Equation that Describes the Human Corneal Shape

2019 ◽  
Vol 24 (1) ◽  
pp. 8 ◽  
Author(s):  
Marwan Abukhaled ◽  
Suheil Khuri
Keyword(s):  
Fixed Point ◽  
Analytical Solution ◽  
Numerical Approach ◽  
Equation Modeling ◽  
Fixed Point Iteration ◽  
One Dimensional ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Corneal Shape ◽  
Anisotropic Mean Curvature

In this paper, a numerical approach is proposed to find a semi analytical solution for a prescribed anisotropic mean curvature equation modeling the human corneal shape. The method is based on an integral operator that is constructed in terms of Green’s function coupled with the implementation of Picard’s or Mann’s fixed point iteration schemes. Using the contraction principle, it will be shown that the method is convergent for both fixed point iteration schemes. Numerical examples will be presented to demonstrate the applicability, efficiency, and high accuracy of the proposed method.

scholarly journals Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1185-1205
Author(s):  
Franco Obersnel ◽  
Pierpaolo Omari
Keyword(s):  
Mean Curvature ◽  
Solution Method ◽  
The Novel ◽  
Open Ball ◽  
Radial Solutions ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Radially Symmetric Solutions ◽  
The Mean ◽  
Curvature Problem

Abstract This paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem: \left\{\begin{array}{ll}-\text{div}\left(\frac{\nabla v}{\sqrt{1+|\nabla v{|}^{2}}}\right)=f(x,v,\nabla v)& \text{in}\hspace{.5em}\text{Ω},\\ {a}_{0}v+{a}_{1}\tfrac{\partial v}{\partial \nu }=0& \text{on}\hspace{.5em}\partial \text{Ω},\end{array}\right. with \text{Ω} an open ball in {{\mathbb{R}}}^{N} , in the presence of one or more couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition. The novel assumptions introduced on the function f allow us to complement or improve several results in the literature.

Gradient estimates of mean curvature equation with Neumann boundary condition in domains of Riemannian manifold

2017 ◽  
Vol 28 (08) ◽  
pp. 1750065
Author(s):  
Jinju Xu ◽  
Dekai Zhang
Keyword(s):  
Riemannian Manifold ◽  
Mean Curvature ◽  
Neumann Boundary Condition ◽  
Existence Result ◽  
Neumann Boundary ◽  
Gradient Estimates ◽  
The Maximum Principle ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Prescribed Mean Curvature Equation

We study the prescribed mean curvature equation with Neumann boundary conditions in domains of Riemannian manifold. The main goal is to establish the gradient estimates for solutions by the maximum principle. As a consequence, we obtain an existence result.

A generalization of the asymptotic behavior of Palais-Smale sequences on a manifold with boundary

2018 ◽  
Vol 29 (10) ◽  
pp. 1850069
Author(s):  
Hong Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Mean Curvature ◽  
Compact Manifold ◽  
Boundary Value ◽  
Fundamental Solutions ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Manifold With Boundary ◽  
Prescribed Mean Curvature Equation

In this paper, we study the asymptotic behavior of Palais-Smale sequences associated with the prescribed mean curvature equation on a compact manifold with boundary. We prove that every such sequence converges to a solution of the associated equation plus finitely many “bubbles” obtained by rescaling fundamental solutions of the corresponding Euclidean boundary value problem.

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