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

Chiara Corsato (1) , Colette De Coster (2) , Franco Obersnel (1) , Pierpaolo Omari (1) , Alessandro Soranzo (1)

1 Università degli studi di Trieste = University of Trieste
2 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015

In this paper we survey, complete and refine some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation (Equation Presented), in a bounded Lipschitz domain Ω ⊂ RN, 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.

Mots clés: 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.

Domaines: Mathématiques [math]

Chiara Corsato, Colette De Coster, Franco Obersnel, Pierpaolo Omari, Alessandro Soranzo. A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis. Discrete and Continuous Dynamical Systems - Series S, 2018, 11 (2), pp.213-256. ⟨10.3934/dcdss.2018013⟩. ⟨hal-03135736⟩