Archivio della ricerca di Triestehttps://arts.units.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Tue, 21 Sep 2021 17:32:27 GMT2021-09-21T17:32:27Z1051On the lower and upper solution method for the prescribed mean curvature equation in Minkowski spacehttp://hdl.handle.net/11368/2966173.1Titolo: On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space
Abstract: We develop a lower and upper solution method for the Dirichlet problem associated with the prescribed mean curvature equation in Minkowski space {-div(∇u/√1-|∇u|2=f(x,u) in Ω u=0 on ∂Ω Here Ω is a bounded regular domain in ℝNand the function f satisfies the Carathéodory conditions. The obtained results display various peculiarities due to the special features of the involved differential operator.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/11368/2966173.12013-01-01T00:00:00ZPositive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski spacehttp://hdl.handle.net/11368/2589620Titolo: Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space
Abstract: We prove the existence of multiple positive solutions of the Dirichlet problem for 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, \nabla u)
& \hbox{ in } \Omega,
\\
u=0& \hbox{ on } \partial \Omega.
\end{cases}
\end{equation*}
Here $\Omega$ is a bounded regular domain in $\RR^N$
and the function $f=f(x,s,\xi)$ is
either sublinear, or superlinear, or sub-superlinear near $s=0$.
The proof combines topological and variational methods.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/11368/25896202013-01-01T00:00:00ZAsymmetric Poincaré inequalities and solvability of capillarity problemshttp://hdl.handle.net/11368/2767321Titolo: Asymmetric Poincaré inequalities and solvability of capillarity problems
Abstract: In this paper we first establish an asymmetric version of the Poincaré inequality in the space of bounded variation functions, then, basically relying on this result, we discuss the existence, the non- existence and the multiplicity of bounded variation solutions of a class of capillarity problems with asymmetric perturbations.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/11368/27673212014-01-01T00:00:00ZExistence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutionshttp://hdl.handle.net/11368/2504937Titolo: Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutions
Abstract: We develop a lower and upper solutions method for the periodic problem associated with the capillarity equation
\begin{equation*}
-\Big( u'/{ \sqrt{1+{u'}^2}}\Big)'
= f(t,u)
\end{equation*}
in the space of bounded variation functions. We get the existence of periodic
solutions both in the case where the lower solution
$\alpha$ and the upper solution
$\beta$ satisfy $\alpha \le \beta$, and in the case where
$\alpha \not\le \beta$.
In the former case we also prove regularity and order stability of solutions.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11368/25049372012-01-01T00:00:00ZPOSITIVE RADIAL SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE MINKOWSKI-CURVATURE EQUATION IN A BALLhttp://hdl.handle.net/11368/2921638.7Titolo: POSITIVE RADIAL SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE MINKOWSKI-CURVATURE EQUATION IN A BALL
Abstract: We study the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation
$$
\begin{cases}
\displaystyle
-\text{div}\bigg(\frac{\nabla v}{\sqrt{1-|\nabla v|^2}}\bigg)=f(|x|,v) & \text{in } B_R,\\v=0 & \text{on } \partial B_R,
\end{cases}
$$
where $B_R$ is a ball in $\mathbb{R}^N$ ($N\ge 2$). According to the behaviour of $f=f(r,s)$ near $s=0$, we prove the existence of either one, two or three positive solutions. All results are obtained by reduction to an equivalent non-singular one-dimensional problem, to which variational methods can be applied in a standard way.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/11368/2921638.72014-01-01T00:00:00Z