Archivio della ricerca di Triestehttps://arts.units.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Sun, 25 Jul 2021 16:52:43 GMT2021-07-25T16:52:43Z10201A 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/11368/29155102018-01-01T00:00:00ZPOSITIVE 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/11368/2921638.72014-01-01T00:00:00ZRadially 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/11368/29223932015-01-01T00:00:00ZOn the optimal design of participating life insurance contractshttp://hdl.handle.net/11368/2967948.1Titolo: On the optimal design of participating life insurance contracts
Abstract: In this paper we study how policyholders and equityholders contribute to the formation of a life insurance company issuing participating contracts. The structure of these contracts is stylized and features a guaranteed rate of return and a terminal bonus, as in the pioneering model by Briys and de Varenne (1994, 1997). Policyholders aim at maximizing their preferences by choosing the leverage ratio and the guaranteed level, while being subject to regulatory constraints of fair valuation and solvency. We provide conditions under which non trivial contracts exist and analyze their properties.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11368/2967948.12020-01-01T00:00:00ZOn 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/11368/2966173.12013-01-01T00:00:00ZExtending Nearly-Linear Modelshttp://hdl.handle.net/11368/2945506Titolo: Extending Nearly-Linear Models
Abstract: Nearly-Linear Models are a family of neighbourhood models, obtaining lower/upper probabilities from a given probability by a linear affine transformation with barriers. They include a number of known models as special cases, among them the Pari-Mutuel Model, the ε-contamination model, the Total Variation Model and the vacuous lower/upper probabilities. We classified Nearly-Linear models, investigating their consistency properties, in previous work. Here we focus on how to extend those Nearly-Linear Models that are coherent or at least avoid sure loss. We derive formulae for their natural extensions, interpret a specific model as a natural extension itself of a certain class of lower probabilities, and supply a risk measurement interpretation for one of the natural extensions we compute.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11368/29455062019-01-01T00:00:00ZThe Dirichlet problem for a prescribed anisotropic mean curvature equation: existence, uniqueness and regularity of solutionshttp://hdl.handle.net/11368/2848504Titolo: The Dirichlet problem for a prescribed anisotropic mean curvature equation: existence, uniqueness and regularity of solutions
Abstract: We discuss existence, uniqueness, regularity and boundary behaviour of solutions of 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*}
where $a,b>0$ are given parameters and $\Omega$ is a bounded Lipschitz domain in $\RR^N$.
This equation appears in the modeling theory of capillarity phenomena for compressible fluids and in the description of the geometry of the human cornea.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/11368/28485042016-01-01T00:00:00ZInference with Nearly-Linear uncertainty modelshttp://hdl.handle.net/11368/2965326.4Titolo: Inference with Nearly-Linear uncertainty models
Abstract: Several simplified uncertainty models are derived from a given probability of which they are a perturbation. Among these, we introduced in previous work Nearly-Linear (NL) models. They perform a linear affine transformation of with barriers, obtaining a couple of conjugate lower/upper probabilities, and generalise several well known neighbourhood models. We classified NL models, partitioning them into three subfamilies, and established their basic consistency properties in [5]. In this paper we investigate how to extend NL models that avoid sure loss by means of their natural extension, a basic, although operationally not always simple, inferential procedure in Imprecise Probability Theory. We obtain formulae for computing directly the natural extension in a number of cases, supplying a risk measurement interpretation for one of them. The results in the paper also broaden our knowledge of NL models: we characterise when they avoid sure loss, express some of them as linear (or even convex) combinations of simpler models, and explore relationships with interval probabilities.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/11368/2965326.42021-01-01T00:00:00ZQualitative 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/11368/29618112020-01-01T00:00:00ZWeak Dutch Books versus strict consistency with lower previsionshttp://hdl.handle.net/11368/2903912Titolo: Weak Dutch Books versus strict consistency with lower previsions
Abstract: Several consistency notions for lower previsions (coherence, convexity, others) require that the suprema of certain gambles, having the meaning of gains, are non-negative. The limit situation that a gain supremum is zero is termed Weak Dutch Book (WDB). In the literature, the special case of WDBs with precise probabilities has mostly been analysed, and strict coherence has been proposed as a radical alternative. In this paper the focus is on WDBs and generalised strict coherence, termed strict consistency, with imprecise previsions. We discuss properties of lower previsions incurring WDBs and conditions for strict consistency, showing in both cases how they are differentiated by the degree of consistency of the given uncertainty assessment.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11368/29039122017-01-01T00:00:00Z