Open Access Research

A one-dimensional prescribed curvature equation modeling the corneal shape

Isabel Coelho12, Chiara Corsato3 and Pierpaolo Omari3*

Author Affiliations

1 Area Departamental de Matemática, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro 1, Lisboa, 1950-062, Portugal

2 Département de Mathématique, Université Libre de Bruxelles, CP 214 Boulevard du Triomphe, Bruxelles, 1052, Belgium

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

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Boundary Value Problems 2014, 2014:127  doi:10.1186/1687-2770-2014-127

Published: 20 May 2014

Abstract

We prove existence, uniqueness, and stability of solutions of the prescribed curvature problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M1">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M3">View MathML</a>, for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/127/mathml/M5">View MathML</a>. We also develop a linear monotone iterative scheme for approximating the solution. This equation has been proposed as a model of the corneal shape in the recent paper (Okrasiński and Płociniczak in Nonlinear Anal., Real World Appl. 13:1498-1505, 2012), where a simplified version obtained by partial linearization has been investigated.

Keywords:
mean curvature equation; mixed boundary condition; positive solution; existence; uniqueness; linear stability; order stability; Lyapunov stability; lower and upper solutions; monotone approximation; topological degree