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### Open Access Journals

PROC

We produce a detailed proof of a result of C.V. Coffman and W.K.
Ziemer [1] on the existence of positive solutions of the Dirichlet problem for
the prescribed mean curvature equation

-div$(\nablau/\sqrt(1+|\nablau|^2)=\lambdaf(x,u)$ in $\Omega,$ $u=0$ on $\partial\Omega$
assuming that $f$ has a superlinear behaviour at $u = 0$.

DCDS

We discuss existence and regularity of bounded variation solutions of the Dirichlet problem for the one-dimensional capillarity-type equation
\begin{equation*}
\Big( u'/{ \sqrt{1+{u'}^2}}\Big)'
= f(t,u) \quad \hbox{ in } {]-r,r[},
\qquad
u(-r)=a, \, u(r) = b.
\end{equation*}
We prove interior regularity of solutions and we obtain a precise description of their boundary behaviour. This is achieved by a direct and elementary approach that exploits the properties of the zero set of the right-hand side $f$ of the equation.

PROC

We discuss existence and multiplicity of bounded variation solutions
of the non-homogeneous Neumann problem for the prescribed mean curvature
equation

-div$(\nabla u/\sqrt(1+|\nablau|^2))=g(x,u)+h$ in $\Omega$

-$\nablau*v/\sqrt(1+|\nablau|^2)=k$ on $\partial\Omega$
where $g(x, s)$ is periodic with respect to $s$. Our approach is variational and
makes use of non-smooth critical point theory in the space of bounded variation
functions.

-$\nablau*v/\sqrt(1+|\nablau|^2)=k$ on $\partial\Omega$

PROC

On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space

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.

DCDS-S

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

, with

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.

$Ω \subset \mathbb{R}^N$ |

$a,b>0$ |

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

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.

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