Pierpaolo Omari - Academia.edu (original) (raw)

Papers by Pierpaolo Omari

Research paper thumbnail of Qualitative analysis of a curvature equation modelling MEMS with vertical loads

Nonlinear Analysis: Real World Applications, 2020

Abstract We investigate existence, multiplicity and qualitative properties of the solutions of th... more Abstract We investigate existence, multiplicity and qualitative properties of the solutions of the Dirichlet problem for the singularly perturbed prescribed mean curvature equation − ( 1 − b u ) div ( ∇ u 1 + | ∇ u | 2 ) = a ( u − R ) 2 + b 1 + | ∇ u | 2 , in Ω , u = 0 , on ∂ Ω , where a , b , R are given constants and Ω is a bounded regular domain in R N . This model appears in the theory of micro-electro-mechanical systems (MEMS) when the effects of capillarity and vertical forces are taken into account.

Research paper thumbnail of On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space

Conference Publications, 2013

We develop a lower and upper solution method for the Dirichlet problem associated with the prescr... more 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\OmegaOmega is a bounded regular domain in mathbbRN\mathbb {R}^NmathbbRN and the function fff satisfies the Caratheodory conditions. The obtained results display various peculiarities due to the special features of the involved differential operator.

Research paper thumbnail of Characterizing the formation of singularities in a superlinear indefinite problem related to the mean curvature operator

Journal of Differential Equations, 2020

Abstract The aim of this paper is characterizing the development of singularities by the positive... more Abstract The aim of this paper is characterizing the development of singularities by the positive solutions of the quasilinear indefinite Neumann problem − ( u ′ / 1 + ( u ′ ) 2 ) ′ = λ a ( x ) f ( u ) in ( 0 , 1 ) , u ′ ( 0 ) = 0 , u ′ ( 1 ) = 0 , where λ ∈ R is a parameter, a ∈ L ∞ ( 0 , 1 ) changes sign once in ( 0 , 1 ) at the point z ∈ ( 0 , 1 ) , and f ∈ C ( R ) ∩ C 1 [ 0 , + ∞ ) is positive and increasing in ( 0 , + ∞ ) with a potential, ∫ 0 s f ( t ) d t , superlinear at +∞. In this paper, by providing a precise description of the asymptotic profile of the derivatives of the solutions of the problem as λ → 0 + , we can characterize the existence of singular bounded variation solutions of the problem in terms of the integrability of this limiting profile, which is in turn equivalent to the condition ( ∫ x z a ( t ) d t ) − 1 2 ∈ L 1 ( 0 , z ) and ( ∫ x z a ( t ) d t ) − 1 2 ∈ L 1 ( z , 1 ) . No previous result of this nature is known in the context of the theory of superlinear indefinite problems.

Research paper thumbnail of A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis

Discrete & Continuous Dynamical Systems - S, 2018

In this paper we survey, complete and refine some recent results concerning the Dirichlet problem... more In this paper we survey, complete and refine some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation div ✓ ru/ q 1 + |ru| 2 ◆ = au + b/ q 1 + |ru| 2 , in a bounded Lipschitz domain ⌦ ⇢ 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. Contents 1. Introduction 2 2. Small classical solutions on arbitrary domains 11 2.1. Global uniqueness of classical solutions 11 2.2. Local existence of classical solutions 12 2.3. A maximal branch of classical solutions 13

Research paper thumbnail of Bifurcation of positive solutions for a one-dimensional indefinite quasilinear Neumann problem

Nonlinear Analysis, 2017

We study the structure of the set of the positive regular solutions of the onedimensional quasili... more We study the structure of the set of the positive regular solutions of the onedimensional quasilinear Neumann problem involving the curvature operator − u / 1 + (u) 2 = λa(x)f (u), u (0) = 0, u (1) = 0. Here λ ∈ R is a parameter, a ∈ L 1 (0, 1) changes sign, and f ∈ C(R). We focus on the case where the slope of f at 0, f (0), is finite and non-zero, and the potential of f is superlinear at infinity, but also the two limiting cases where f (0) = 0, or f (0) = +∞, are discussed. We investigate, in some special configurations, the possible development of singularities and the corresponding appearance in this problem of bounded variation solutions.

Research paper thumbnail of Existence and localization of solutions of second order elliptic problems using lower and upper solutions in the reversed order

Topological Methods in Nonlinear Analysis, 1996

Research paper thumbnail of Boundary value problems for forced nonlinear equations at resonance

Lecture Notes in Mathematics, 1985

Research paper thumbnail of On a Semilinear Elliptic Neumann Problem with Asymmetric Nonlinearities

Transactions of the American Mathematical Society, 1995

Research paper thumbnail of On the existence of periodic solutions of forced Liénard differential equations

Nonlinear Analysis: Theory, Methods & Applications, 1987

Research paper thumbnail of Periodic solutions of a second order ordinary differential equation: A necessary and sufficient condition for nonresonance

Journal of Differential Equations, 1991

Research paper thumbnail of Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential∗

Communications in Partial Differential Equations, 1996

Let Ω be a bounded domain in IR{sup N}, with N ⥠1, having a smooth boundary âΩ. We denote by ... more Let Ω be a bounded domain in IR{sup N}, with N ⥠1, having a smooth boundary âΩ. We denote by A the quasilinear elliptic second order differential operator defined by Au+div(a({vert_bar}â{sub u}{vert_bar}²)â{sub u}). We suppose that the function a:[O,+ââO, +â] is of class C¹ and satisfies the following ellipticity and growth conditions of Leray-Lions type (cf. e.g. [22]): there

Research paper thumbnail of Multiple Positive Solutions of a One-Dimensional Prescribed Mean Curvature Problem

Communications in Contemporary Mathematics, 2007

We discuss existence, non-existence and multiplicity of positive solutions of the Dirichlet probl... more We discuss existence, non-existence and multiplicity of positive solutions of the Dirichlet problem for the one-dimensional prescribed curvature equation [Formula: see text] in connection with the changes of concavity of the function f. The proofs are based on an upper and lower solution method, we specifically develop for this problem, combined with a careful analysis of the time-map associated with some related autonomous equations.

Research paper thumbnail of A note on nonlinear oscillations at resonance

Acta Mathematica Sinica, 1987

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

Georgian Mathematical Journal, 2017

We discuss existence, multiplicity, localisation and stability properties of solutions of the Dir... more We discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz–Minkowski spaceThe obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann–Lemaître–Robertson–Walker, as well as Schwarzschild–Reissner–Nordström, spacetimes.

Research paper thumbnail of Giovanni Torelli, un matematico appassionato: il profilo scientifico, in breve

Research paper thumbnail of Multiple bounded variation solutions of a capillarity problem

Conference Publications 2011

We discuss existence and multiplicity of bounded variation solutions of the non-homogeneous Neuma... more We discuss existence and multiplicity of bounded variation solutions of the non-homogeneous Neumann problem for the prescribed mean curvature equation    −div ∇u/ 1 + |∇u| 2 = g(x, u) + h in Ω, −∇u • ν/ 1 + |∇u| 2 = κ on ∂Ω, 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.

Research paper thumbnail of Positive solutions of superlinear indefinite prescribed mean curvature problems

This paper analyzes the superlinear indefinite prescribed mean curvature problem −div ∇u/1 + |∇u|... more This paper analyzes the superlinear indefinite prescribed mean curvature problem −div ∇u/1 + |∇u|2 = λa(x)h(u)in Ω,u = 0on ∂Ω, where Ω is a bounded domain in ℝN with a regular boundary ∂Ω, h ∈ C0(ℝ...

Research paper thumbnail of Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation

Open Mathematics

This paper focuses on the existence and the multiplicity of classical radially symmetric solution... more This paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem: \left\{\begin{array}{ll}-\text{div}\left(\frac{\nabla v}{\sqrt{1+|\nabla v{|}^{2}}}\right)=f(x,v,\nabla v)& \text{in}\hspace{.5em}\text{Ω},\\ {a}_{0}v+{a}_{1}\tfrac{\partial v}{\partial \nu }=0& \text{on}\hspace{.5em}\partial \text{Ω},\end{array}\right. with \text{Ω} an open ball in {{\mathbb{R}}}^{N} , in the presence of one or more couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition. The novel assumptions introduced on the function f allow us to complement or improve several results in the literature.

Research paper thumbnail of On the periodic Ambrosetti–Prodi problem for a class of ODEs with nonlinearities indefinite in sign

Applied Mathematics Letters

Research paper thumbnail of Regular Versus Singular Solutions in a Quasilinear Indefinite Problem with an Asymptotically Linear Potential

Advanced Nonlinear Studies

The aim of this paper is analyzing the positive solutions of the quasilinear problem-\bigl{(}u^{\... more The aim of this paper is analyzing the positive solutions of the quasilinear problem-\bigl{(}u^{\prime}/\sqrt{1+(u^{\prime})^{2}}\big{)}^{\prime}=\lambda a(x)f(u)% \quad\text{in }(0,1),\qquad u^{\prime}(0)=0,\quad u^{\prime}(1)=0,where {\lambda\in\mathbb{R}} is a parameter, {a\in L^{\infty}(0,1)} changes sign once in {(0,1)} and satisfies {\int_{0}^{1}a(x)\,dx<0}, and {f\in\mathcal{C}^{1}(\mathbb{R})} is positive and increasing in {(0,+\infty)} with a potential, {F(s)=\int_{0}^{s}f(t)\,dt}, quadratic at zero and linear at {+\infty}. The main result of this paper establishes that this problem possesses a component of positive bounded variation solutions, {\mathscr{C}_{\lambda_{0}}^{+}}, bifurcating from {(\lambda,0)} at some {\lambda_{0}>0} and from {(\lambda,\infty)} at some {\lambda_{\infty}>0}. It also establishes that {\mathscr{C}_{\lambda_{0}}^{+}} consists of regular solutions if and only if\int_{0}^{z}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty\quad\text%...

Research paper thumbnail of Qualitative analysis of a curvature equation modelling MEMS with vertical loads

Nonlinear Analysis: Real World Applications, 2020

Abstract We investigate existence, multiplicity and qualitative properties of the solutions of th... more Abstract We investigate existence, multiplicity and qualitative properties of the solutions of the Dirichlet problem for the singularly perturbed prescribed mean curvature equation − ( 1 − b u ) div ( ∇ u 1 + | ∇ u | 2 ) = a ( u − R ) 2 + b 1 + | ∇ u | 2 , in Ω , u = 0 , on ∂ Ω , where a , b , R are given constants and Ω is a bounded regular domain in R N . This model appears in the theory of micro-electro-mechanical systems (MEMS) when the effects of capillarity and vertical forces are taken into account.

Research paper thumbnail of On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space

Conference Publications, 2013

We develop a lower and upper solution method for the Dirichlet problem associated with the prescr... more 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\OmegaOmega is a bounded regular domain in mathbbRN\mathbb {R}^NmathbbRN and the function fff satisfies the Caratheodory conditions. The obtained results display various peculiarities due to the special features of the involved differential operator.

Research paper thumbnail of Characterizing the formation of singularities in a superlinear indefinite problem related to the mean curvature operator

Journal of Differential Equations, 2020

Abstract The aim of this paper is characterizing the development of singularities by the positive... more Abstract The aim of this paper is characterizing the development of singularities by the positive solutions of the quasilinear indefinite Neumann problem − ( u ′ / 1 + ( u ′ ) 2 ) ′ = λ a ( x ) f ( u ) in ( 0 , 1 ) , u ′ ( 0 ) = 0 , u ′ ( 1 ) = 0 , where λ ∈ R is a parameter, a ∈ L ∞ ( 0 , 1 ) changes sign once in ( 0 , 1 ) at the point z ∈ ( 0 , 1 ) , and f ∈ C ( R ) ∩ C 1 [ 0 , + ∞ ) is positive and increasing in ( 0 , + ∞ ) with a potential, ∫ 0 s f ( t ) d t , superlinear at +∞. In this paper, by providing a precise description of the asymptotic profile of the derivatives of the solutions of the problem as λ → 0 + , we can characterize the existence of singular bounded variation solutions of the problem in terms of the integrability of this limiting profile, which is in turn equivalent to the condition ( ∫ x z a ( t ) d t ) − 1 2 ∈ L 1 ( 0 , z ) and ( ∫ x z a ( t ) d t ) − 1 2 ∈ L 1 ( z , 1 ) . No previous result of this nature is known in the context of the theory of superlinear indefinite problems.

Research paper thumbnail of A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis

Discrete & Continuous Dynamical Systems - S, 2018

In this paper we survey, complete and refine some recent results concerning the Dirichlet problem... more In this paper we survey, complete and refine some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation div ✓ ru/ q 1 + |ru| 2 ◆ = au + b/ q 1 + |ru| 2 , in a bounded Lipschitz domain ⌦ ⇢ 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. Contents 1. Introduction 2 2. Small classical solutions on arbitrary domains 11 2.1. Global uniqueness of classical solutions 11 2.2. Local existence of classical solutions 12 2.3. A maximal branch of classical solutions 13

Research paper thumbnail of Bifurcation of positive solutions for a one-dimensional indefinite quasilinear Neumann problem

Nonlinear Analysis, 2017

We study the structure of the set of the positive regular solutions of the onedimensional quasili... more We study the structure of the set of the positive regular solutions of the onedimensional quasilinear Neumann problem involving the curvature operator − u / 1 + (u) 2 = λa(x)f (u), u (0) = 0, u (1) = 0. Here λ ∈ R is a parameter, a ∈ L 1 (0, 1) changes sign, and f ∈ C(R). We focus on the case where the slope of f at 0, f (0), is finite and non-zero, and the potential of f is superlinear at infinity, but also the two limiting cases where f (0) = 0, or f (0) = +∞, are discussed. We investigate, in some special configurations, the possible development of singularities and the corresponding appearance in this problem of bounded variation solutions.

Research paper thumbnail of Existence and localization of solutions of second order elliptic problems using lower and upper solutions in the reversed order

Topological Methods in Nonlinear Analysis, 1996

Research paper thumbnail of Boundary value problems for forced nonlinear equations at resonance

Lecture Notes in Mathematics, 1985

Research paper thumbnail of On a Semilinear Elliptic Neumann Problem with Asymmetric Nonlinearities

Transactions of the American Mathematical Society, 1995

Research paper thumbnail of On the existence of periodic solutions of forced Liénard differential equations

Nonlinear Analysis: Theory, Methods & Applications, 1987

Research paper thumbnail of Periodic solutions of a second order ordinary differential equation: A necessary and sufficient condition for nonresonance

Journal of Differential Equations, 1991

Research paper thumbnail of Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential∗

Communications in Partial Differential Equations, 1996

Let Ω be a bounded domain in IR{sup N}, with N ⥠1, having a smooth boundary âΩ. We denote by ... more Let Ω be a bounded domain in IR{sup N}, with N ⥠1, having a smooth boundary âΩ. We denote by A the quasilinear elliptic second order differential operator defined by Au+div(a({vert_bar}â{sub u}{vert_bar}²)â{sub u}). We suppose that the function a:[O,+ââO, +â] is of class C¹ and satisfies the following ellipticity and growth conditions of Leray-Lions type (cf. e.g. [22]): there

Research paper thumbnail of Multiple Positive Solutions of a One-Dimensional Prescribed Mean Curvature Problem

Communications in Contemporary Mathematics, 2007

We discuss existence, non-existence and multiplicity of positive solutions of the Dirichlet probl... more We discuss existence, non-existence and multiplicity of positive solutions of the Dirichlet problem for the one-dimensional prescribed curvature equation [Formula: see text] in connection with the changes of concavity of the function f. The proofs are based on an upper and lower solution method, we specifically develop for this problem, combined with a careful analysis of the time-map associated with some related autonomous equations.

Research paper thumbnail of A note on nonlinear oscillations at resonance

Acta Mathematica Sinica, 1987

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Research paper thumbnail of The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

Georgian Mathematical Journal, 2017

We discuss existence, multiplicity, localisation and stability properties of solutions of the Dir... more We discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz–Minkowski spaceThe obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann–Lemaître–Robertson–Walker, as well as Schwarzschild–Reissner–Nordström, spacetimes.

Research paper thumbnail of Giovanni Torelli, un matematico appassionato: il profilo scientifico, in breve

Research paper thumbnail of Multiple bounded variation solutions of a capillarity problem

Conference Publications 2011

We discuss existence and multiplicity of bounded variation solutions of the non-homogeneous Neuma... more We discuss existence and multiplicity of bounded variation solutions of the non-homogeneous Neumann problem for the prescribed mean curvature equation    −div ∇u/ 1 + |∇u| 2 = g(x, u) + h in Ω, −∇u • ν/ 1 + |∇u| 2 = κ on ∂Ω, 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.

Research paper thumbnail of Positive solutions of superlinear indefinite prescribed mean curvature problems

This paper analyzes the superlinear indefinite prescribed mean curvature problem −div ∇u/1 + |∇u|... more This paper analyzes the superlinear indefinite prescribed mean curvature problem −div ∇u/1 + |∇u|2 = λa(x)h(u)in Ω,u = 0on ∂Ω, where Ω is a bounded domain in ℝN with a regular boundary ∂Ω, h ∈ C0(ℝ...

Research paper thumbnail of Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation

Open Mathematics

This paper focuses on the existence and the multiplicity of classical radially symmetric solution... more This paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem: \left\{\begin{array}{ll}-\text{div}\left(\frac{\nabla v}{\sqrt{1+|\nabla v{|}^{2}}}\right)=f(x,v,\nabla v)& \text{in}\hspace{.5em}\text{Ω},\\ {a}_{0}v+{a}_{1}\tfrac{\partial v}{\partial \nu }=0& \text{on}\hspace{.5em}\partial \text{Ω},\end{array}\right. with \text{Ω} an open ball in {{\mathbb{R}}}^{N} , in the presence of one or more couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition. The novel assumptions introduced on the function f allow us to complement or improve several results in the literature.

Research paper thumbnail of On the periodic Ambrosetti–Prodi problem for a class of ODEs with nonlinearities indefinite in sign

Applied Mathematics Letters

Research paper thumbnail of Regular Versus Singular Solutions in a Quasilinear Indefinite Problem with an Asymptotically Linear Potential

Advanced Nonlinear Studies

The aim of this paper is analyzing the positive solutions of the quasilinear problem-\bigl{(}u^{\... more The aim of this paper is analyzing the positive solutions of the quasilinear problem-\bigl{(}u^{\prime}/\sqrt{1+(u^{\prime})^{2}}\big{)}^{\prime}=\lambda a(x)f(u)% \quad\text{in }(0,1),\qquad u^{\prime}(0)=0,\quad u^{\prime}(1)=0,where {\lambda\in\mathbb{R}} is a parameter, {a\in L^{\infty}(0,1)} changes sign once in {(0,1)} and satisfies {\int_{0}^{1}a(x)\,dx<0}, and {f\in\mathcal{C}^{1}(\mathbb{R})} is positive and increasing in {(0,+\infty)} with a potential, {F(s)=\int_{0}^{s}f(t)\,dt}, quadratic at zero and linear at {+\infty}. The main result of this paper establishes that this problem possesses a component of positive bounded variation solutions, {\mathscr{C}_{\lambda_{0}}^{+}}, bifurcating from {(\lambda,0)} at some {\lambda_{0}>0} and from {(\lambda,\infty)} at some {\lambda_{\infty}>0}. It also establishes that {\mathscr{C}_{\lambda_{0}}^{+}} consists of regular solutions if and only if\int_{0}^{z}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty\quad\text%...