arXiv subjects

Daniele Cassani

Publications and source records attributed to Daniele Cassani.

25 records · Page 2Linked to original sources

arXiv2017

Equivalent and attained version of Hardy's inequality in $\mathbb{R}^n$

We investigate connections between Hardy's inequality in the whole space $\mathbb{R}^n$ and embedding inequalities for Sobolev-Lorentz spaces. In particular, we complete previous results due to [A. Alvino, Sulla diseguaglianza di Sobolev in spazi di Lorentz, (1977)] and [G. Talenti, An inequality between $u^*$ and $|{\rm{grad}} u^*|$, (1992)] by establishing optimal embedding inequalities for the Sobolev-Lorentz quasinorm $\|\nabla\,\cdot\,\|_{p,q}$ also in the range $p < q<\infty$, which remained essentially open since the work of Alvino. Attainability of the best embedding constants is also studied, as well as the limiting case when $q=\infty$. 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 [Brezis-Vázquez, Blow-up solutions of some nonlinear elliptic problems, (1977)], 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 by Brezis and Vázquez.

math.FA↗

arXiv2017

Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent

For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation, $ -Δu+V_{μ,ν} u=(I_α\ast |u|^{\frac{N+α}{N}}){|u|}^{\fracα{N}-1}u$, in $\mathbb{R}^N$ where $N\ge 3$, $V_{μ, ν} : \mathbb{R}^N \to \mathbb{R}$ is an external potential defined for $μ,ν\in\mathbb{R}$ and $x \in \mathbb{R}^N$ by $V_{μ, ν} (x)=1-μ/(ν^2 + |x|^2)$ and $I_α: \mathbb{R}^N \to 0$ is the Riesz potential for $α\in (0,N)$, we exhibit two thresholds $μ_ν,μ^ν>0$ such that the equation admits a positive ground state solution if and only if $μ_ν<μ<μ^ν$ and no ground state solution exists for $μ<μ_ν$. Moreover, if $μ>\max\left\{μ^ν,\frac{N^2(N-2)}{4(N+1)}\right\}$, then equation still admits a sign changing ground state solution provided $N\ge4$ or in dimension $N=3$ if in addition $\frac{3}{2}<α<3$ and ker$(-Δ+ V_{μ, ν}) = \{0\}$, namely in the non-resonant case.

math.AP↗

arXiv2016

Ground states and semiclassical states of nonlinear Choquard equations involving Hardy-Littlewood-Sobolev critical growth

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.

math.AP↗

arXiv2016

A priori estimates and positivity for semiclassical ground states for systems of critical Schroedinger equations in dimension two

We consider in the whole plane the Hamiltonian coupling of semilinear Schroedinger equations which have critical growth in the sense of Moser. We prove that the (nonempty) set S of ground state solutions is compact up to translations. Moreover, ground states are uniformly bounded 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 when the external potential stays bounded away from zero. Namely, as the adimensionalized Planck constant vanishes, we prove the existence of minimal energy solutions which concentrate around the closest local minima of the potential with some precise asymptotic rate.

math.AP↗

arXiv2016

Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in $\R^2$

We study the following singularly perturbed nonlocal Schrödinger equation $$ -\vr^2Δu +V(x)u =\vr^{μ-2}\Big[\frac{1}{|x|^μ}\ast F(u)\Big]f(u) \quad \mbox{in} \quad \R^2, $$ where $V(x)$ is a continuous real function on $\R^2$, $F(s)$ is the primitive of $f(s)$, $0<μ<2$ and $\vr$ 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.

math.AP↗

arXiv2014

Equivalent Moser type inequalities in R2 and the zero mass case

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 $\mathcal{D}^{1,2}(\mathbb{R}^2)$, completion of smooth compactly supported functions with respect to the Dirichlet norm $\|\nabla\cdot\|_2$, and we prove an optimal Lorentz-Zygmund type inequality with explicit extremals and from which can be derived classical inequalities in $H^1(\mathbb{R}^2)$ such as the Adachi-Tanaka inequality and a version of Ruf's inequality.

math.FA↗

arXiv2013

A Fast Alternating Minimization Algorithm for Total Variation Deblurring Without Boundary Artifacts

Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang [{\em 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 one is computationally useful in the case of a symmetric blur, while the second one 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.

math.FA↗