Archivio della ricerca di Triestehttps://arts.units.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Fri, 18 Jun 2021 14:00:29 GMT2021-06-18T14:00:29Z1041A prescribed anisotropic mean curvature equation modeling the corneal shape: a paradigm of nonlinear analysishttp://hdl.handle.net/11368/2915510Titolo: A prescribed anisotropic mean curvature equation modeling the corneal shape: a paradigm of nonlinear analysis
Abstract: In this paper we survey, complete and refine some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation egin{equation*} { m -div}left({ abla u}/{sqrt{1 + | abla u|^2}} ight) = -au + {b}/{sqrt{1 + | abla u|^2}}, end{equation*} in a bounded Lipschitz domain $Omega subset RR^N$, 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.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11368/29155102018-01-01T00:00:00ZRadially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shapehttp://hdl.handle.net/11368/2922393Titolo: Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape
Abstract: We prove existence and uniqueness of classical solutions of the anisotropic prescribed mean curvature problem
egin{equation*}
{
m -div}left({
abla u}/{sqrt{1 + |
abla u|^2}}
ight) = -au + {b}/{sqrt{1 + |
abla u|^2}}, ext{ in } B, quad u=0, ext{ 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.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/11368/29223932015-01-01T00:00:00ZQualitative analysis of a curvature equation modelling MEMS with vertical loadshttp://hdl.handle.net/11368/2961811Titolo: Qualitative analysis of a curvature equation modelling MEMS with vertical loads
Abstract: We investigate existence, multiplicity and qualitative properties of the solutions of the Dirichlet problem for a singularly perturbed prescribed mean curvature equation, which appears in the theory of micro-electro-mechanical systems (MEMS) when the effects of capillarity and vertical forces are taken into account.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11368/29618112020-01-01T00:00:00ZRadial solutions of the Dirichlet problem for a class of quasilinear elliptic equations arising in optometryhttp://hdl.handle.net/11368/2932001Titolo: Radial solutions of the Dirichlet problem for a class of quasilinear elliptic equations arising in optometry
Abstract: This paper deals with the quasilinear elliptic problem egin{align*} { m -div} left({ abla u}/{sqrt{1 + | abla u|^2}} ight)+a(x) u &= b(x)/sqrt{1 + | abla u|^2} ext { in } B, ;; u=0 , ext{ on } partial B, end{align*} where $B$ is an open ball in $RR^N$, with $Nge 2$, and $a,b in C^1(overline B) $ are given radially symmetric functions, with $a(x) ge 0$ in $B$. This class of anisotropic prescribed mean curvature equations appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Unlike all previous works published on these subjects, existence and uniqueness of solutions of the above problem are here analyzed in the case where the coefficients $a, b$ are not necessarily constant and no sign condition is assumed on $b$.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11368/29320012019-01-01T00:00:00Z