American Institute of Mathematical Sciences

Journals

PROC
We produce a detailed proof of a result of C.V. Co ffman 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 $u = 0$.
keywords: positive solution Prescribed mean curvature equation variational method Dirichlet problem Nehari method.
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.
keywords: Dirichlet problem boundary behaviour. regularity Quasilinear ordinary differential equation bounded variation solution capillarity equation existence classical solution
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.
keywords: prescribed mean curvature equation bounded variation function Neumann boundary condition Capillary surface variational methods
PROC
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
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