arXiv subjects

Gianmaria Verzini

Publications and source records attributed to Gianmaria Verzini.

At least 37 records · Page 2Linked to original sources

arXiv2020

Normalized solutions of mass supercritical Schrödinger equations with potential

This paper is concerned with the existence of normalized solutions of the nonlinear Schrödinger equation \[ -Δu+V(x)u+λu = |u|^{p-2}u \qquad\text{in $\mathbb{R}^N$} \] in the mass supercritical and Sobolev subcritical case $2+\frac{4}{N}<p<2^*$. We prove the existence of a solution $(u,λ)\in H^1(\mathbb{R}^N)\times\mathbb{R}^+$ with prescribed $L^2$-norm $\|u\|_2=ρ$ under various conditions on the potential $V:\mathbb{R}^N\to\mathbb{R}$, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.

math.AP↗

arXiv2020

Local minimizers in absence of ground states for the critical NLS energy on metric graphs

We consider the mass-critical nonlinear Schrödinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in arXiv:1605.07666, where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved.

math.AP↗

arXiv2020

Regularized variational principles for the perturbed Kepler problem

The goal of the paper is to develop a method that will combine the use of variational techniques with regularization methods in order to study existence and multiplicity results for the periodic and the Dirichlet problem associated to the perturbed Kepler system \[ \ddot x = -\frac{x}{|x|^3} + p(t), \quad x \in \mathbb{R}^d, \] where $d\geq 1$, and $p:\mathbb{R}\to\mathbb{R}^d$ is smooth and $T$-periodic, $T>0$. The existence of critical points for the action functional associated to the problem is proved via a non-local change of variables inspired by Levi-Civita and Kustaanheimo-Stiefel techniques. As an application we will prove that the perturbed Kepler problem has infinitely many generalized $T$-periodic solutions for $d=2$ and $d=3$, without any symmetry assumptions on $p$.

math.CA↗

arXiv2019

Spiraling asymptotic profiles of competition-diffusion systems

This paper describes the structure of the nodal set of segregation profiles arising in the singular limit of planar, stationary, reaction-diffusion systems with strongly competitive interactions of Lotka-Volterra type, when the matrix of the inter-specific competition coefficients is asymmetric and the competition parameter tends to infinity. Unlike the symmetric case, when it is known that the nodal set consists in a locally finite collection of curves meeting with equal angles at a locally finite number of singular points, the asymmetric case shows the emergence of spiraling nodal curves, still meeting at locally isolated points with finite vanishing order.

math.AP↗

arXiv2019

Normalized concentrating solutions to nonlinear elliptic problems

We prove the existence of solutions $(λ, v)\in \mathbb{R}\times H^{1}(Ω)$ of the elliptic problem \[ \begin{cases} -Δv+(V(x)+λ) v =v^{p}\ &\text{ in $ Ω, $} \ v>0,\qquad \int_Ωv^2\,dx =ρ. \end{cases} \] Any $v$ solving such problem (for some $λ$) is called a normalized solution, where the normalization is settled in $L^2(Ω)$. Here $Ω$ is either the whole space $\mathbb R^N$ or a bounded smooth domain of $\mathbb R^N$, in which case we assume $V\equiv0$ and homogeneous Dirichlet or Neumann boundary conditions. Moreover, $1 1$ if $N=1,2$. Normalized solutions appear in different contexts, such as the study of the Nonlinear Schrödinger equation, or that of quadratic ergodic Mean Field Games systems. We prove the existence of solutions concentrating at suitable points of $Ω$ as the prescribed mass $ρ$ is either small (when $p<1+\frac 4N$) or large (when $p>1+\frac 4N$) or it approaches some critical threshold (when $p=1+\frac 4N$).

math.AP↗

arXiv2019

Asymptotic spherical shapes in some spectral optimization problems

We study the optimization of the positive principal eigenvalue of an indefinite weighted problem, associated with the Neumann Laplacian in a box $Ω\subset\mathbb{R}^N$, which arises in the investigation of the survival threshold in population dynamics. When trying to minimize such eigenvalue with respect to the weight, one is lead to consider a shape optimization problem, which is known to admit no spherical optimal shapes (despite some previously stated conjectures). We investigate whether spherical shapes can be recovered in some singular perturbation limit. More precisely we show that, whenever the negative part of the weight diverges, the above shape optimization problem approaches in the limit the so called spectral drop problem, which involves the minimization of the first eigenvalue of the mixed Dirichlet-Neumann Laplacian. We prove that, for suitable choices of the box $Ω$, the optimal shapes for this second problem are indeed spherical; moreover, for general $Ω$, we show that small volume spectral drops are asymptotically spherical, with center at points of $\partialΩ$ having large mean curvature.

math.AP↗

arXiv2019

Time-fractional equations with reaction terms: fundamental solutions and asymptotics

We analyze the fundamental solution of a time-fractional problem, establishing existence and uniqueness in an appropriate functional space. We also focus on the one-dimensional spatial setting in the case in which the time-fractional exponent is equal to, or larger than, $\frac12$. In this situation, we prove that the speed of invasion of the fundamental solution is at least `almost of square root type', namely it is larger than~$ct^β$ for any given~$c>0$ and~$β\in\left(0,\frac12\right)$.

math.AP↗

arXiv2019

Quantitative analysis of a singularly perturbed shape optimization problem in a polygon

We carry on our study of the connection between two shape optimization problems with spectral cost. On the one hand, we consider the optimal design problem for the survival threshold of a population living in a heterogenous habitat $Ω$; this problem arises when searching for the optimal shape and location of a shelter zone in order to prevent extinction of the species. On the other hand, we deal with the spectral drop problem, which consists in minimizing a mixed Dirichlet-Neumann eigenvalue in a box $Ω$. In a previous paper arXiv:1811.01623 we proved that the latter one can be obtained as a singular perturbation of the former, when the region outside the refuge is more and more hostile. In this paper we sharpen our analysis in case $Ω$ is a planar polygon, providing quantitative estimates of the optimal level convergence, as well as of the involved eigenvalues.

math.AP↗

arXiv2018

Best dispersal strategies in spatially heterogeneous environments: optimization of the principal eigenvalue for indefinite fractional Neumann problems

We study the positive principal eigenvalue of a weighted problem associated with the Neumann spectral fractional Laplacian. This analysis is related to the investigation of the survival threshold in population dynamics. Our main result concerns the optimization of such threshold with respect to the fractional order $s\in(0,1]$, the case $s=1$ corresponding to the standard Neumann Laplacian: when the habitat is not too fragmented, the principal positive eigenvalue can not have local minima for $0<s<1$. As a consequence, the best strategy for survival is either following the diffusion with $s=1$ (i.e. Brownian diffusion), or with the lowest possible $s$ (i.e. diffusion allowing long jumps), depending on the size of the domain. In addition, we show that analogous results hold for the standard fractional Laplacian in $\mathbb{R}^N$, in periodic environments.

math.AP↗

arXiv2018

Normalized solutions for Nonlinear Schrödinger systems on bounded domains

We analyze $L^2$-normalized solutions of nonlinear Schrödinger systems of Gross-Pitaevskii type, on bounded domains, with homogeneous Dirichlet boundary conditions. We provide sufficient conditions for the existence of orbitally stable standing waves. Such waves correspond to global minimizers of the associated energy in the $L^2$-subcritical and critical cases, and to local ones in the $L^2$-supercritical case. Notably, our study includes also the Sobolev-critical case.

math.AP↗

arXiv2017

On the honeycomb conjecture for a class of minimal convex partitions

We prove that the planar hexagonal honeycomb is asymptotically optimal for a large class of optimal partition problems, in which the cells are assumed to be convex, and the criterion is to minimize either the sum or the maximum among the energies of the cells, the cost being a shape functional $F$ which satisfies a few assumptions. They are: monotonicity under inclusions; homogeneity under dilations; a Faber-Krahn inequality for convex hexagons; a convexity-type inequality for the map which associates with every $n \in \mathbb{N}$ the minimizers of $F$ among convex $n$-gons with given area. In particular, our result allows to obtain the honeycomb conjecture for the Cheeger constant and for the logarithmic capacity (still assuming the cells to be convex). Moreover we show that, in order to get the conjecture also for the first Dirichlet eigenvalue of the Laplacian, it is sufficient to establish some facts about the behaviour of $λ_1$ among convex pentagons, hexagons, and heptagons with prescribed area.

math.OC↗

arXiv2016

Normalized bound states for the nonlinear Schrodinger equation in bounded domains

Given $ρ>0$, we study the elliptic problem \[ \text{find } (U,λ)\in H^1_0(Ω)\times \mathbb{R} \text{ such that } \begin{cases} -ΔU+λU=|U|^{p-1}U \int_Ω U^2\, dx=ρ, \end{cases} \] where $Ω\subset\mathbb{R}^N$ is a bounded domain and $p>1$ is Sobolev-subcritical, searching for conditions (about $ρ$, $N$ and $p$) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when $p$ is $L^2$-subcritical, i.e. $1 0$. In the $L^2$-critical and supercritical case, i.e. when $1+4/N \leq p < 2^*-1$, we show that, for any $k\in\mathbb{N}$, the problem admits solutions having Morse index bounded above by $k$ only if $ρ$ is sufficiently small. Next we provide existence results for certain ranges of $ρ$, which can be estimated in terms of the Dirichlet eigenvalues of $-Δ$ in $H^1_0(Ω)$, extending to general domains and to changing sign solutions some results obtained in [Noris, Tavares, Verzini, Analysis & PDE, 2014] for positive solutions in the ball.

math.AP↗

arXiv2015

Bifurcation and segregation in quadratic two-populations Mean Field Games systems

We search for non-constant normalized solutions to the semilinear elliptic system \[ \begin{cases} - νΔv_i + g_i(v_j^2) v_i = λ_i v_i,\quad v_i>0 & \text{in $Ω$} \\ \partial_n v_i = 0 & \text{on $\partial Ω$}\\ \int_Ωv_i^2\,dx = 1, & 1\leq i,j\leq 2, \quad j\neq i, \end{cases} \] where $ν>0$, $Ω\subset \mathbb{R}^N$ is smooth and bounded, the functions $g_i$ are positive and increasing, and both the functions $v_i$ and the parameters $λ_i$ are unknown. This system is obtained, via the Hopf-Cole transformation, from a two-populations ergodic Mean Field Games system, which describes Nash equilibria in differential games with identical players. In these models, each population consists of a very large number of indistinguishable rational agents, aiming at minimizing some long-time average criterion. Firstly, we discuss existence of nontrivial solutions, using variational methods when $g_i(s)=s$, and bifurcation ones in the general case; secondly, for selected families of nontrivial solutions, we address the appearing of segregation in the vanishing viscosity limit, i.e. \[ \int_Ω v_1 v_2 \to 0 \qquad \text{as }ν\to0. \]

math.AP↗

arXiv2014

Positive solutions with a complex behavior for superlinear indefinite ODEs on the real line

We show the existence of infinitely many positive solutions, defined on the real line, for the nonlinear scalar ODE \[ \ddot u + (a^+(t) - μa^-(t)) u^3 = 0, \] where $a$ is a periodic, sign-changing function, and the parameter $μ>0$ is large. Such solutions are characterized by the fact of being either small or large in each interval of positivity of $a$. In this way, we find periodic solutions, having minimal period arbitrarily large, and bounded non-periodic solutions, exhibiting a complex behavior. The proof is variational, exploiting suitable natural constraints of Nehari type.

math.CA↗

arXiv2014

Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials

For the cubic Schrödinger system with trapping potentials in $\mathbb{R}^N$, $N\leq3$, or in bounded domains, we investigate the existence and the orbital stability of standing waves having components with prescribed $L^2$-mass. We provide a variational characterization of such solutions, which gives information on the stability through of a condition of Grillakis-Shatah-Strauss type. As an application, we show existence of conditionally orbitally stable solitary waves when: a) the masses are small, for almost every scattering lengths, and b) in the defocusing, weakly interacting case, for any masses.

math.AP↗

arXiv2013

Strong competition versus fractional diffusion: the case of Lotka-Volterra interaction

We consider a system of differential equations with nonlinear Steklov boundary conditions, related to the fractional problem $$(-Δ)^s u_i = f_i(x,u_i) - βu_i^p \sum_{j\neq i} a_{ij} u_j^p,$$ where $i = i,\dots, k$, $s\in(0,1)$, $p>0$, $a_{ij}>0$ and $β>0$. When $k=2$ we develop a quasi-optimal regularity theory in $C^{0,α}$, uniformly w.r.t. $β$, for every $α< α_{\mathrm opt}=min(1,2s)$; moreover we show that the traces of the limiting profiles as $β\to+\infty$ are Lipschitz continuous and segregated. Such results are extended to the case of $k\geq3$ densities, with some restrictions on $s$, $p$ and $a_{ij}$.

math.AP↗

arXiv2013

Existence and orbital stability of the ground states with prescribed mass for the L^2-critical and supercritical NLS on bounded domains

We study solutions of a semilinear elliptic equation with prescribed mass and Dirichlet homogeneous boundary conditions in the unitary ball. Such problem arises in the search of solitary wave solutions for nonlinear Schrödinger equations (NLS) with Sobolev subcritical power nonlinearity on bounded domains. Necessary and sufficient conditions are provided for the existence of such solutions. Moreover, we show that standing waves associated to least energy solutions are always orbitally stable when the nonlinearity is L^2-critical and subcritical, while they are almost always stable in the L^2-supercritical regime. The proofs are obtained in connection with the study of a variational problem with two constraints, of independent interest: to maximize the L^{p+1}-norm among functions having prescribed L^2 and H^1_0-norm.

math.AP↗

arXiv2013

Bounded solutions for a forced bounded oscillator without friction

Under the validity of a Landesman-Lazer type condition, we prove the existence of solutions bounded on the real line, together with their first derivatives, for some second order nonlinear differential equation of the form $\ddot u + g(u) = p(t)$, where the reaction term $g$ is bounded. The proof is variational, and relies on a dual version of the Nehari method for the existence of oscillating solutions to superlinear equations.

math.CA↗