D. Cassani | Insubria University (original) (raw)
Papers by D. Cassani
We consider in the whole plane the following Hamiltonian coupling of Schrödinger equations −∆u + ... more We consider in the whole plane the following Hamiltonian coupling of Schrödinger equations −∆u + V 0 u = g(v) −∆v + V 0 v = f (u) where V 0 > 0, f, g have critical growth in the sense of Moser. We prove that the (nonempty) set S of ground state solutions is compact in H 1 (R 2) × H 1 (R 2) up to translations. Moreover, for each (u, v) ∈ S, one has that u, v are uniformly bounded in L ∞ (R 2) and uniformly decaying at infinity. Then we prove that actually the ground state is positive and radially symmetric. We apply those results to prove the existence of semiclassical ground states solutions to the singularly perturbed system −ε 2 ∆ϕ + V (x)ϕ = g(ψ) −ε 2 ∆ψ + V (x)ψ = f (ϕ) where V ∈ C(R 2) is a Schrödinger potential bounded away from zero. Namely, as the adimensionalized Planck constant ε → 0, we prove the existence of minimal energy solutions which concentrate around the closest local minima of the potential with some precise asymptotic rate.
We are concerned with the existence of ground states for nonlinear Choquard equations involving a... more We are concerned with the existence of ground states for nonlinear Choquard equations involving a critical nonlinearity in the sense of Hardy-Littlewood-Sobolev. Our result complements previous results by Moroz and Van Schaftingen where the subcritical case was considered. Then, we focus on the existence of semi-classical states and by using a truncation argument approach, we establish the existence and concentration of single peak solutions concentrating around minima of the Schrödinger potential, as the Planck constant goes to zero. The result is robust in the sense that the nonlinearity is not required to satisfy monotonicity conditions nor the Ambrosetti-Rabinowitz condition.
We investigate connections between Hardy's inequality in the whole space R n and embedding inequa... more We investigate connections between Hardy's inequality in the whole space R n and embedding inequalities for Sobolev-Lorentz spaces. In particular, we complete previous results due to [1, Alvino] and [29, Talenti] by establishing optimal embedding inequalities for the Sobolev-Lorentz quasinorm ∇ · p,q also in the range p < q < ∞, which remained essentially open since [1]. Attainability of the best embedding constants is also studied, as well as the limiting case when q = ∞. Here, we surprisingly discover that the Hardy inequality is equivalent to the corresponding Sobolev-Marcinkiewicz embedding inequality. Moreover, the latter turns out to be attained by the so-called " ghost " extremal functions of [6, Brezis-Vazquez], in striking contrast with the Hardy inequality, which is never attained. In this sense, our functional approach seems to be more natural than the classical Sobolev setting, answering a question raised in [6].
We are concerned with the following equation −ε 2 ∆u + V (x)u = f (u), u(x) > 0 in R 2. By a vari... more We are concerned with the following equation −ε 2 ∆u + V (x)u = f (u), u(x) > 0 in R 2. By a variational approach, we construct a solution u ε which concentrates, as ε → 0, around arbitrarily given isolated local minima of the confining potential V : here the nonlinearity f has a quite general Moser's critical growth, as in particular we do not require the monotonicity of f (s)/s nor the Ambrosetti-Rabinowitz condition.
We study the following singularly perturbed nonlocal Schrödinger equation −ε 2 ∆u + V (x)u = ε µ−... more We study the following singularly perturbed nonlocal Schrödinger equation −ε 2 ∆u + V (x)u = ε µ−2 1 |x| µ * F (u) f (u) in R 2 , where V (x) is a continuous real function on R 2 , F (s) is the primitive of f (s), 0 < µ < 2 and ε is a positive parameter. Assuming that the nonlinearity f (s) has critical exponential growth in the sense of Trudinger-Moser, we establish the existence and concentration of solutions by variational methods. Mathematics Subject Classifications (2010): 35J20, 35J60, 35B33
We prove existence of variational solutions for the Hamiltonian coupling of nonlinear Schrödinger... more We prove existence of variational solutions for the Hamiltonian coupling of nonlinear Schrödinger equations in the whole plane, when the nonlinearities exhibit su-percritical growth with respect to the Trudinger-Moser inequality. We discover linking type solutions which have finite energy in a suitable Lorentz-Sobolev space setting.
Combining a priori estimates with penalization techniques and an implicit function argument based... more Combining a priori estimates with penalization techniques and an implicit function argument based on Campanato's near operators theory, we obtain the existence of periodic solutions for a fourth order integro-differential equation modelling actuators in MEMS devices. New insights are also derived for the stationary problem improving previous existence results by removing smallness assumptions on the domain.
Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) h... more Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang [SIAM J. Imaging Sci., 1 (2008), pp. 248–272]. The method in a nutshell consists of a discrete Fourier transform-based alternating minimization algorithm with periodic boundary conditions and in which two fast Fourier transforms (FFTs) are required per iteration. In this paper, we propose an alternating minimization algorithm for the continuous version of the total variation image deblurring problem. We establish convergence of the proposed continuous alternating minimization algorithm. The continuous setting is very useful to have a unifying representation of the algorithm, independently of the discrete approximation of the deconvolution problem, in particular concerning the strategies for dealing with boundary artifacts. Indeed, an accurate restoration of blurred and noisy images requires a proper treatment of the boundary. A discrete version of our continuous alternating minimization algorithm is obtained following two different strategies: the imposition of appropriate boundary conditions and the enlargement of the domain. The first is computationally useful in the case of a symmetric blur, while the second can be efficiently applied for a nonsymmetric blur. Numerical tests show that our algorithm generates higher quality images in comparable running times with respect to the Fast Total Variation deconvolution algorithm.
We first investigate concentration and vanishing phenomena concerning Moser type inequalities in ... more We first investigate concentration and vanishing phenomena concerning Moser type inequalities in the whole plane which involve complete and reduced Sobolev norms. In particular we show that the critical Ruf inequality is equivalent to an improved version of the subcritical Adachi-Tanaka inequality which we prove to be attained. Then, we consider the limiting space D 1,2 (R 2), completion of smooth compactly supported functions with respect to the Dirichlet norm ∇ ⋅ 2 , and we prove an optimal Lorentz-Zygmund type inequality with explicit extremals and from which can be derived classical inequalities in H 1 (R 2) such as the Adachi-Tanaka inequality and a version of Ruf's inequality.
We establish existence and regularity results for a time dependent fourth order integro-different... more We establish existence and regularity results for a time dependent fourth order integro-differential equation with a possibly singular nonlinearity which has applications in designing MicroElectroMechanicalSystems. The key ingredient in our approach, besides basic theory of hyperbolic equations in Hilbert spaces, exploit Near Operators Theory introduced by Campanato.
It is well known that classical Sobolev's embeddings may be improved within the framework of Lore... more It is well known that classical Sobolev's embeddings may be improved within the framework of Lorentz spaces. However, the value of the best possible constants in the corresponding inequalities is known in just one case by the work of A. Alvino. Here, we determine optimal constants for the embedding of the space D 1,p (R n), 1 < p < n, into the whole Lorentz space scale L p * ,q (R n), p ≤ q ≤ ∞; including the case of Alvino q = p of which we give a new proof. We finally exhibit extremal functions for the embedding inequalities by solving related elliptic problems.
We study the so-called limiting Sobolev cases for embeddings of the spaces W 1,n 0 (Ω), where Ω ⊂... more We study the so-called limiting Sobolev cases for embeddings of the spaces W 1,n 0 (Ω), where Ω ⊂ R n is a bounded domain. Differently from J. Moser, we consider optimal embeddings into Zygmund spaces: we derive related Euler-Lagrange equations, and show that Moser's concentrating sequences are the solutions of these equations and thus realize the best constants of the related embedding inequalities. Furthermore, we exhibit a group invariance, and show that Moser's sequence is generated by this group invariance and that the solutions of the limiting equation are unique up to this invariance. Finally, we derive a related Pohozaev identity, and use it to prove that equations related to perturbed optimal embeddings do not have solutions.
—We consider fourth order nonlinear problems which describe electrostatic actuation in MicroElect... more —We consider fourth order nonlinear problems which describe electrostatic actuation in MicroElectroMechan-icalSystems (MEMS) both in the stationary case and in the evolution case; we prove existence, uniqueness and regularity theorems by exploiting the Near Operators Theory.
We survey old and new results about the so-called limiting Sobolev case for the embedding of the ... more We survey old and new results about the so-called limiting Sobolev case for the embedding of the space W 1,n 0 (Ω) into suitable spaces of functions having exponential summability. In particular, we discuss a new notion of criticality with respect to attainability of the best constant in the related embedding inequalities and the connection with existence and nonexistence of solutions to boundary value problems, in which Moser's functions are cast in a new framework. Then, we prove a new version of Moser's inequality in Zygmund spaces with respect to the full Sobolev norm and without boundary conditions.
We prove existence of solutions for a nonlocal equation arising from the mathematical modeling of... more We prove existence of solutions for a nonlocal equation arising from the mathematical modeling of MicroElectroMechanicalSystems. The existence result, obtained within a suitable Implicit Function Theorem framework, is established under rather general boundary conditions and for bounded domains whose diameter is fairly small.
We prove a family of Hardy-Rellich inequalities with optimal constants and additional boundary te... more We prove a family of Hardy-Rellich inequalities with optimal constants and additional boundary terms. These inequalities are used to study the behavior of extremal solutions to biharmonic Gelfand-type equations under Steklov boundary conditions.
In this paper we prove the existence and qualitative properties of positive bound state solutions... more In this paper we prove the existence and qualitative properties of positive bound state solutions for a class of quasilinear Schrödinger equations in dimension N ≥ 3: we investigate the case of unbounded potentials which can not be covered in a standard function space setting. This leads us to use a nonstandard penalization technique, in a borderline Orlicz space framework, which enable us to build up a one parameter family of classical solutions which have finite energy and exhibit, as the parameter goes to zero, a concentrating behavior around some point which is localized.
We study optimal embeddings for the space of functions whose Laplacian ∆u belongs to L 1 (Ω), whe... more We study optimal embeddings for the space of functions whose Laplacian ∆u belongs to L 1 (Ω), where Ω ⊂ R N is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W 2,1 (Ω) in which the whole set of second order derivatives is considered. In particular, in the limiting Sobolev case, when N = 2, we establish a sharp embedding inequality into the Zygmund space L exp (Ω). On one hand, this result enables us to improve the Brezis–Merle [13] regularity estimate for the Dirichlet problem ∆u = f (x) ∈ L 1 (Ω), u = 0 on ∂Ω
This paper deals with direct and inverse evolution problems which come up in studying Micro-Elect... more This paper deals with direct and inverse evolution problems which come up in studying Micro-Electro-Mechanical-Systems: here we consider a nonlinear and nonlocal MEMS model. The inverse problem consists of recovering a time varying Coulomb potential by exploiting some accessible measurements, which depend on the dynamic displacement of the system. Local existence, uniqueness and continuous dependence results are proved for both direct and inverse problems.
We consider in the whole plane the following Hamiltonian coupling of Schrödinger equations −∆u + ... more We consider in the whole plane the following Hamiltonian coupling of Schrödinger equations −∆u + V 0 u = g(v) −∆v + V 0 v = f (u) where V 0 > 0, f, g have critical growth in the sense of Moser. We prove that the (nonempty) set S of ground state solutions is compact in H 1 (R 2) × H 1 (R 2) up to translations. Moreover, for each (u, v) ∈ S, one has that u, v are uniformly bounded in L ∞ (R 2) and uniformly decaying at infinity. Then we prove that actually the ground state is positive and radially symmetric. We apply those results to prove the existence of semiclassical ground states solutions to the singularly perturbed system −ε 2 ∆ϕ + V (x)ϕ = g(ψ) −ε 2 ∆ψ + V (x)ψ = f (ϕ) where V ∈ C(R 2) is a Schrödinger potential bounded away from zero. Namely, as the adimensionalized Planck constant ε → 0, we prove the existence of minimal energy solutions which concentrate around the closest local minima of the potential with some precise asymptotic rate.
We are concerned with the existence of ground states for nonlinear Choquard equations involving a... more We are concerned with the existence of ground states for nonlinear Choquard equations involving a critical nonlinearity in the sense of Hardy-Littlewood-Sobolev. Our result complements previous results by Moroz and Van Schaftingen where the subcritical case was considered. Then, we focus on the existence of semi-classical states and by using a truncation argument approach, we establish the existence and concentration of single peak solutions concentrating around minima of the Schrödinger potential, as the Planck constant goes to zero. The result is robust in the sense that the nonlinearity is not required to satisfy monotonicity conditions nor the Ambrosetti-Rabinowitz condition.
We investigate connections between Hardy's inequality in the whole space R n and embedding inequa... more We investigate connections between Hardy's inequality in the whole space R n and embedding inequalities for Sobolev-Lorentz spaces. In particular, we complete previous results due to [1, Alvino] and [29, Talenti] by establishing optimal embedding inequalities for the Sobolev-Lorentz quasinorm ∇ · p,q also in the range p < q < ∞, which remained essentially open since [1]. Attainability of the best embedding constants is also studied, as well as the limiting case when q = ∞. Here, we surprisingly discover that the Hardy inequality is equivalent to the corresponding Sobolev-Marcinkiewicz embedding inequality. Moreover, the latter turns out to be attained by the so-called " ghost " extremal functions of [6, Brezis-Vazquez], in striking contrast with the Hardy inequality, which is never attained. In this sense, our functional approach seems to be more natural than the classical Sobolev setting, answering a question raised in [6].
We are concerned with the following equation −ε 2 ∆u + V (x)u = f (u), u(x) > 0 in R 2. By a vari... more We are concerned with the following equation −ε 2 ∆u + V (x)u = f (u), u(x) > 0 in R 2. By a variational approach, we construct a solution u ε which concentrates, as ε → 0, around arbitrarily given isolated local minima of the confining potential V : here the nonlinearity f has a quite general Moser's critical growth, as in particular we do not require the monotonicity of f (s)/s nor the Ambrosetti-Rabinowitz condition.
We study the following singularly perturbed nonlocal Schrödinger equation −ε 2 ∆u + V (x)u = ε µ−... more We study the following singularly perturbed nonlocal Schrödinger equation −ε 2 ∆u + V (x)u = ε µ−2 1 |x| µ * F (u) f (u) in R 2 , where V (x) is a continuous real function on R 2 , F (s) is the primitive of f (s), 0 < µ < 2 and ε is a positive parameter. Assuming that the nonlinearity f (s) has critical exponential growth in the sense of Trudinger-Moser, we establish the existence and concentration of solutions by variational methods. Mathematics Subject Classifications (2010): 35J20, 35J60, 35B33
We prove existence of variational solutions for the Hamiltonian coupling of nonlinear Schrödinger... more We prove existence of variational solutions for the Hamiltonian coupling of nonlinear Schrödinger equations in the whole plane, when the nonlinearities exhibit su-percritical growth with respect to the Trudinger-Moser inequality. We discover linking type solutions which have finite energy in a suitable Lorentz-Sobolev space setting.
Combining a priori estimates with penalization techniques and an implicit function argument based... more Combining a priori estimates with penalization techniques and an implicit function argument based on Campanato's near operators theory, we obtain the existence of periodic solutions for a fourth order integro-differential equation modelling actuators in MEMS devices. New insights are also derived for the stationary problem improving previous existence results by removing smallness assumptions on the domain.
Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) h... more Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang [SIAM J. Imaging Sci., 1 (2008), pp. 248–272]. The method in a nutshell consists of a discrete Fourier transform-based alternating minimization algorithm with periodic boundary conditions and in which two fast Fourier transforms (FFTs) are required per iteration. In this paper, we propose an alternating minimization algorithm for the continuous version of the total variation image deblurring problem. We establish convergence of the proposed continuous alternating minimization algorithm. The continuous setting is very useful to have a unifying representation of the algorithm, independently of the discrete approximation of the deconvolution problem, in particular concerning the strategies for dealing with boundary artifacts. Indeed, an accurate restoration of blurred and noisy images requires a proper treatment of the boundary. A discrete version of our continuous alternating minimization algorithm is obtained following two different strategies: the imposition of appropriate boundary conditions and the enlargement of the domain. The first is computationally useful in the case of a symmetric blur, while the second can be efficiently applied for a nonsymmetric blur. Numerical tests show that our algorithm generates higher quality images in comparable running times with respect to the Fast Total Variation deconvolution algorithm.
We first investigate concentration and vanishing phenomena concerning Moser type inequalities in ... more We first investigate concentration and vanishing phenomena concerning Moser type inequalities in the whole plane which involve complete and reduced Sobolev norms. In particular we show that the critical Ruf inequality is equivalent to an improved version of the subcritical Adachi-Tanaka inequality which we prove to be attained. Then, we consider the limiting space D 1,2 (R 2), completion of smooth compactly supported functions with respect to the Dirichlet norm ∇ ⋅ 2 , and we prove an optimal Lorentz-Zygmund type inequality with explicit extremals and from which can be derived classical inequalities in H 1 (R 2) such as the Adachi-Tanaka inequality and a version of Ruf's inequality.
We establish existence and regularity results for a time dependent fourth order integro-different... more We establish existence and regularity results for a time dependent fourth order integro-differential equation with a possibly singular nonlinearity which has applications in designing MicroElectroMechanicalSystems. The key ingredient in our approach, besides basic theory of hyperbolic equations in Hilbert spaces, exploit Near Operators Theory introduced by Campanato.
It is well known that classical Sobolev's embeddings may be improved within the framework of Lore... more It is well known that classical Sobolev's embeddings may be improved within the framework of Lorentz spaces. However, the value of the best possible constants in the corresponding inequalities is known in just one case by the work of A. Alvino. Here, we determine optimal constants for the embedding of the space D 1,p (R n), 1 < p < n, into the whole Lorentz space scale L p * ,q (R n), p ≤ q ≤ ∞; including the case of Alvino q = p of which we give a new proof. We finally exhibit extremal functions for the embedding inequalities by solving related elliptic problems.
We study the so-called limiting Sobolev cases for embeddings of the spaces W 1,n 0 (Ω), where Ω ⊂... more We study the so-called limiting Sobolev cases for embeddings of the spaces W 1,n 0 (Ω), where Ω ⊂ R n is a bounded domain. Differently from J. Moser, we consider optimal embeddings into Zygmund spaces: we derive related Euler-Lagrange equations, and show that Moser's concentrating sequences are the solutions of these equations and thus realize the best constants of the related embedding inequalities. Furthermore, we exhibit a group invariance, and show that Moser's sequence is generated by this group invariance and that the solutions of the limiting equation are unique up to this invariance. Finally, we derive a related Pohozaev identity, and use it to prove that equations related to perturbed optimal embeddings do not have solutions.
—We consider fourth order nonlinear problems which describe electrostatic actuation in MicroElect... more —We consider fourth order nonlinear problems which describe electrostatic actuation in MicroElectroMechan-icalSystems (MEMS) both in the stationary case and in the evolution case; we prove existence, uniqueness and regularity theorems by exploiting the Near Operators Theory.
We survey old and new results about the so-called limiting Sobolev case for the embedding of the ... more We survey old and new results about the so-called limiting Sobolev case for the embedding of the space W 1,n 0 (Ω) into suitable spaces of functions having exponential summability. In particular, we discuss a new notion of criticality with respect to attainability of the best constant in the related embedding inequalities and the connection with existence and nonexistence of solutions to boundary value problems, in which Moser's functions are cast in a new framework. Then, we prove a new version of Moser's inequality in Zygmund spaces with respect to the full Sobolev norm and without boundary conditions.
We prove existence of solutions for a nonlocal equation arising from the mathematical modeling of... more We prove existence of solutions for a nonlocal equation arising from the mathematical modeling of MicroElectroMechanicalSystems. The existence result, obtained within a suitable Implicit Function Theorem framework, is established under rather general boundary conditions and for bounded domains whose diameter is fairly small.
We prove a family of Hardy-Rellich inequalities with optimal constants and additional boundary te... more We prove a family of Hardy-Rellich inequalities with optimal constants and additional boundary terms. These inequalities are used to study the behavior of extremal solutions to biharmonic Gelfand-type equations under Steklov boundary conditions.
In this paper we prove the existence and qualitative properties of positive bound state solutions... more In this paper we prove the existence and qualitative properties of positive bound state solutions for a class of quasilinear Schrödinger equations in dimension N ≥ 3: we investigate the case of unbounded potentials which can not be covered in a standard function space setting. This leads us to use a nonstandard penalization technique, in a borderline Orlicz space framework, which enable us to build up a one parameter family of classical solutions which have finite energy and exhibit, as the parameter goes to zero, a concentrating behavior around some point which is localized.
We study optimal embeddings for the space of functions whose Laplacian ∆u belongs to L 1 (Ω), whe... more We study optimal embeddings for the space of functions whose Laplacian ∆u belongs to L 1 (Ω), where Ω ⊂ R N is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W 2,1 (Ω) in which the whole set of second order derivatives is considered. In particular, in the limiting Sobolev case, when N = 2, we establish a sharp embedding inequality into the Zygmund space L exp (Ω). On one hand, this result enables us to improve the Brezis–Merle [13] regularity estimate for the Dirichlet problem ∆u = f (x) ∈ L 1 (Ω), u = 0 on ∂Ω
This paper deals with direct and inverse evolution problems which come up in studying Micro-Elect... more This paper deals with direct and inverse evolution problems which come up in studying Micro-Electro-Mechanical-Systems: here we consider a nonlinear and nonlocal MEMS model. The inverse problem consists of recovering a time varying Coulomb potential by exploiting some accessible measurements, which depend on the dynamic displacement of the system. Local existence, uniqueness and continuous dependence results are proved for both direct and inverse problems.