PROC
On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space
Chiara Corsato Franco Obersnel Pierpaolo Omari Sabrina Rivetti
We develop a lower and upper solution method for the Dirichlet problem associated with the prescribed mean curvature equation in Minkowski space \begin{equation*} \begin{cases} -{\rm div}\Big( \nabla u /\sqrt{1 - |\nabla u|^2}\Big)= f(x,u) & \hbox{ in } \Omega, \\ u=0& \hbox{ on } \partial \Omega. \end{cases} \end{equation*} Here $\Omega$ is a bounded regular domain in $\mathbb {R}^N$ and the function $f$ satisfies the Carathéodory conditions. The obtained results display various peculiarities due to the special features of the involved differential operator.
keywords: lower and upper solutions Mean curvature partial differential equation quasilinear Dirichlet condition existence elliptic Minkowski space multiplicity.
DCDS-S
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
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
PROC
Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape
Chiara Corsato Colette De Coster Pierpaolo Omari
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: positive solution Dirichlet boundary condition Anisotropic prescribed mean curvature equation uniqueness radially symmetric solution existence shooting method.

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