arXiv subjects

Gianmaria Verzini

Publications and source records attributed to Gianmaria Verzini.

At least 19 recordsLinked to original sources

arXiv2026

The nonlinear Schrödinger equation on products of $\mathbb{R}^N$ and compact metric graphs

We study the stationary focusing nonlinear Schrödinger equation on the product $\mathbb{R}^N \times \mathcal{G}$ of the Euclidean space with a compact metric graph, in the mass-constrained variational setting. Such a product is a hybrid structure of a new type: all its faces are $(N+1)$-dimensional and are glued along interfaces of codimension one, so that the energy space is a genuine Sobolev space, while both the metric and the topology of the graph enter the variational problem. We first develop the functional framework, giving two equivalent descriptions of $H^1(\mathbb{R}^N \times \mathcal{G})$, introducing partial rearrangements in each of the two variables together with the corresponding Pólya--Szegő inequalities, and proving Gagliardo--Nirenberg inequalities in a localized form, with a comparison of the optimal constants with those of $\mathbb{R}^{N+1}$ and of the half-space. We then study the mass-constrained problem. Ground states exist for every mass when $2<p<2_*:=2+4/(N+1)$. At the critical exponent $p=2_*$, they exist below a graph-dependent threshold lying between one half of the Euclidean critical mass and the full Euclidean critical mass. The latter value is attained when the graph admits a cycle covering, whereas the threshold is exactly halved in the presence of a terminal edge. In both cases, the threshold is sharp. For $2_*<p<2+4/N$ global minimizers do not exist, but we prove the existence of local minimizers below a further mass threshold, for which we give an explicit lower bound. Finally, we describe the dimensional crossover: below a critical mass the minimizers do not depend on the graph variable, and we characterize the threshold below which the semi-trivial solution is a local minimizer in terms of the first nonzero eigenvalue of the Kirchhoff Laplacian on $\mathcal{G}$.

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arXiv2026

On the emergence of dead cores in elliptic systems with sublinear competitive interactions

In this paper, we study semilinear elliptic systems with sublinear coupling terms, showing that, under competition-type interactions, solutions typically have dead cores, i.e., they vanish on certain open subsets of the domain. We apply our results to a large class of solutions treated in the literature, for instance to ground states and least energy sign-changing solutions, under Dirichlet boundary conditions or in the whole space. The proofs are based on a general result stating that subsolutions to a certain sublinear equation have dead cores, and on uniform Hölder bounds.

math.AP↗

arXiv2026

Selection of the angular speed of rotating waves in segregated reaction-diffusion systems with asymmetric competition

We investigate the existence of segregated rotating waves, arising in the singular limit of competition-diffusion systems of the type \[ \partial_t u_i -\partial_{xx} u_i = f(u_i)-βu_i \sum_{j \neq i} a_{ij} u_j,\qquad x\in\mathbb{S}^1,\ t>0, 1\le i,j\le k, \] as $β\to+\infty$. Here $k\ge3$, the reaction $f$ is of Fisher-KPP (logistic) type, and the competition coefficients $a_{ij}>0$ are not necessarily symmetric. Assuming that, for every $i$, \[ \dfrac{a_{i+1,i}}{a_{i,i+1}}=λ>0, \] we provide a complete characterization of the rotating waves enjoying an equivariant structure, where each density is a suitable rotation of any other one: such waves exist if and only if $λ$ belongs to an explicit range, in which case the angular velocity $ω=ω(λ)$ is uniquely prescribed, as is the rotating profile. In particular, stationary solutions (with $ω=0$) exist only in the symmetric case $λ=1$. This marks a strong difference with the same problem with either Dirichlet or Neumann boundary conditions, where it is known that no periodic in time solution exists, also in the asymmetric case, sheding more light on some conjectures and open problems concerning the long time behavior of competition-diffusion systems.

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arXiv2025

Shape optimization of a small favorable region in a periodically fragmented environment

We consider a shape optimization problem for the persistence threshold of a biological species dispersing in a periodically fragmented environment, the unknown shape corresponding to the portion of the habitat which is favorable to the population. Analytically, this translates in the minimization of a weighted eigenvalue of the periodic Laplacian, with respect to a bang-bang indefinite weight. For such problem, we exploit some recent results obtained in the framework of Dirichlet or Neumann boundary conditions, to provide a full description of the singularly perturbed regime in which the volume of the favorable zone vanishes. First, we show that the optimal favorable zone shrinks to a connected, convex, nearly spherical set, in $C^{1,1}$ sense. Secondly, we show that the spherical asymmetry of the optimal favorable zone decays exponentially, with respect to a negative power of its volume, in the $C^{1,α}$ sense, for every $α<1$.

math.AP↗

arXiv2025

Miminization of the first eigenvalue of the Dirichlet Laplacian with a small volume obstacle

We consider the well-known shape optimization problem with spectral cost: minimizing the first eigenvalue of the Dirichlet Laplacian among all subdomains $Ω$ having prescribed volume and contained in a fixed box $D$; equivalently, we look for the best way to remove a compact set (obstacle) $K\subset\overline{D}$ of Lebesgue measure $|K|=\varepsilon$, $0<\varepsilon<|D|$, in order to minimize the first Dirichlet eigenvalue of the set $Ω= D \setminus K$. In the small volume regime $\varepsilon\to0$, we prove that the optimal obstacles accumulate, in a suitable sense, to points of $\partial D$ where $|\nabla ϕ_0|$ is minimal, where $ϕ_0$ denotes the first eigenfunction of the Dirichlet Laplacian on $D$. Moreover, we provide a fairly detailed description of the convergence of the optimal eigenvalues, eigenfunctions and free boundaries. Our results are based on sharp estimates of the optimal eigenvalues, in terms of a suitable notion of relative capacity.

math.AP↗

arXiv2025

Energy local minimizers for the nonlinear Schrödinger equation on product spaces

We investigate the existence of local minimizers with prescribed $L^2$-norm for the energy functional associated to the mass-supercritical nonlinear Schrödinger equation on the product space $\mathbb{R}^N \times M^k$, where $(M^k,g)$ is a compact Riemannian manifold, thus complementing the study of the mass-subcritical case performed by Terracini, Tzvetkov and Visciglia in [\emph{Anal. PDE} 2014, arXiv:1205.0342]. First we prove that, for small $L^2$-mass, the problem admits local minimizers. Next, we show that when the $L^2$-norm is sufficiently small, the local minimizers are constants along $M^k$, and they coincide with those of the corresponding problem on $\mathbb{R}^N$. Finally, under certain conditions, we show that the local minimizers obtained above are nontrivial along $M^k$. The latter situation occurs, for instance, for every $M^k$ of dimension $k\ge 2$, with the choice of an appropriate metric $\hat g$, and in $\mathbb{R}\times\mathbb{S}^k$, $k\ge 3$, where $\mathbb{S}^k$ is endowed with the standard round metric.

math.AP↗

arXiv2025

Normalized solutions for the nonlinear Schrödinger equation with potential: the purely Sobolev critical case

We study the existence and multiplicity of positive solutions in $H^1(\mathbb{R}^N)$, $N\ge3$, with prescribed $L^2$-norm, for the (stationary) nonlinear Schrödinger equation with Sobolev critical power nonlinearity. It is well known that, in the free case, the associated energy functional has a mountain pass geometry on the $L^2$-sphere. This boils down, in higher dimensions, to the existence of a mountain pass solution which is (a suitable scaling of) the Aubin-Talenti function. In this paper, we consider the same problem, in presence of a weakly attractive, possibly irregular, potential, wondering (i) whether a local minimum solution appears, thus providing an orbitally stable family of solitons, and (ii) if the existence of a mountain-pass solution persists. We provide positive answers, depending on suitable assumptions on the potential and on the mass value. Moreover, by the Hopf-Cole transform, we give some applications of our results to the existence of multiple solutions to ergodic Mean Field Games systems with potential and quadratic Hamiltonian.

math.AP↗

arXiv2024

Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem

We complete the study concerning the minimization of the positive principal eigenvalue associated with a weighted Neumann problem settled in a bounded regular domain $Ω\subset \mathbb{R}^{N}$, $N\ge2$, for the weight varying in a suitable class of sign-changing bounded functions. Denoting with $u$ the optimal eigenfunction and with $D$ its super-level set, corresponding to the positivity set of the optimal weight, we prove that, as the measure of $D$ tends to zero, the unique maximum point of $u$, $P\in \partial Ω$, tends to a point of maximal mean curvature of $\partial Ω$. Furthermore, we show that $D$ is the intersection with $Ω$ of a $C^{1,1}$ nearly spherical set, and we provide a quantitative estimate of the spherical asymmetry, which decays like a power of the measure of $D$. These results provide, in the small volume regime, a fully detailed answer to some long-standing questions in this framework.

math.AP↗

arXiv2024

Normalized solutions for Sobolev critical Schrödinger equations on bounded domains

We study the existence and multiplicity of positive solutions with prescribed $L^2$-norm for the Sobolev critical Schrödinger equation on a bounded domain $Ω\subset\mathbb{R}^N$, $N\ge3$: \[ -ΔU = λU + U^{2^{*}-1},\qquad U\in H^1_0(Ω),\qquad \int_ΩU^2\,dx = ρ^{2}, \] where $2^*=\frac{2N}{N-2}$. First, we consider a general bounded domain $Ω$ in dimension $N\ge3$, with a restriction, only in dimension $N=3$, involving its inradius and first Dirichlet eigenvalue. In this general case we show the existence of a mountain pass solution on the $L^2$-sphere, for $ρ$ belonging to a subset of positive measure of the interval $(0,ρ^{**})$, for a suitable threshold $ρ^{**}>0$. Next, assuming that $Ω$ is star-shaped, we extend the previous result to all values $ρ\in(0,ρ^{**})$. With respect to that of local minimizers, already known in the literature, the existence of mountain pass solutions in the Sobolev critical case is much more elusive. In particular, our proofs are based on the sharp analysis of the bounded Palais-Smale sequences, provided by a nonstandard adaptation of the Struwe monotonicity trick, that we develop.

math.AP↗

arXiv2024

Partially concentrating standing waves for weakly coupled Schrödinger systems

We study the existence of standing waves for the following weakly coupled system of two Schrödinger equations in $\mathbb{R}^N$, $N=2,3$, \[ \begin{cases} i \hslash \partial_{t}ψ_{1}=-\frac{\hslash^2}{2m_{1}}Δψ_{1}+ {V_1}(x)ψ_{1}-μ_{1}|ψ_{1}|^{2}ψ_{1}-β|ψ_{2}|^{2}ψ_{1} & \\ i \hslash \partial_{t}ψ_{2}=-\frac{\hslash^2}{2m_{2}}Δψ_{2}+ {V_2}(x)ψ_{2}-μ_{2}|ψ_{2}|^{2}ψ_{2}-β|ψ_{1}|^{2}ψ_{2},& \end{cases} \] where $V_1$ and $V_2$ are radial potentials bounded from below. We address the case $m_{1}\sim \hslash^2\to0$, $m_2$ constant, and prove the existence of a standing wave solution with both nontrivial components satisfying a prescribed asymptotic profile. In particular, the second component of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature.

math.AP↗

arXiv2023

Asymptotic properties of an optimal principal Dirichlet eigenvalue arising in population dynamics

We consider a shape optimization problem related to the persistence threshold for a biological species, the unknown shape corresponding to the zone of the habitat which is favorable to the population. Analytically, this translates in the minimization of a weighted eigenvalue of the Dirichlet Laplacian, with respect to a bang-bang indefinite weight. For such problem, we provide a full description of the singularly perturbed regime in which the volume of the favorable zone vanishes, with particular attention to the interplay between its location and shape. First, we show that the optimal favorable zone shrinks to a connected, nearly spherical set, in $C^{1,1}$ sense, which aims at maximizing its distance from the lethal boundary. Secondly, we show that the spherical asymmetry of the optimal favorable zone decays exponentially, with respect to a negative power of its volume, in the $C^{1,α}$ sense, for every $α<1$. This latter property is based on sharp quantitative asymmetry estimates for the optimization of a weighted eigenvalue problem on the full space, of independent interest.

math.AP↗

arXiv2023

Ergodic Mean Field Games: existence of local minimizers up to the Sobolev critical case

We investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case.

math.AP↗

arXiv2022

Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditions

We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain $Ω\subset \mathbb{R}^N$, where the bang-bang weight equals a positive constant $\overline{m}$ on a ball $B\subsetΩ$ and a negative constant $-\underline{m}$ on $Ω\setminus B$. The corresponding positive principal eigenvalue provides a threshold to detect persistence/extinction of a species whose evolution is described by the heterogeneous Fisher-KPP equation in population dynamics. In particular, we study the minimization of such eigenvalue with respect to the position of $B$ in $Ω$. We provide sharp asymptotic expansions of the optimal eigenpair in the singularly perturbed regime in which the volume of $B$ vanishes. We deduce that, up to subsequences, the optimal ball concentrates at a point maximizing the distance from $\partialΩ$.

math.AP↗

arXiv2022

Local Hölder and maximal regularity of solutions of elliptic equations with superquadratic gradient terms

We study the local Hölder regularity of strong solutions $u$ of second-order uniformly elliptic equations having a gradient term with superquadratic growth $γ> 2$, and right-hand side in a Lebesgue space $L^q$. When $q > N\frac{γ-1}γ$ ($N$ is the dimension of the Euclidean space), we obtain the optimal Hölder continuity exponent $α_q > \frac{γ-2}{γ-1}$. This allows us to prove some new results of maximal regularity type, which consist in estimating the Hessian matrix of $u$ in $L^q$. Our methods are based on blow-up techniques and a Liouville theorem.

math.AP↗

arXiv2022

Rotating Spirals in segregated reaction-diffusion systems

We give a complete characterization of the boundary traces $φ_i$ ($i=1,\dots,K$) supporting spiraling waves, rotating with a given angular speed $ω$, which appear as singular limits of competition-diffusion systems of the type \[ \frac{\partial}{\partial t} u_i -Δu_i = μu_i -βu_i \sum_{j \neq i} a_{ij} u_j \text{ in } Ω\times\mathbb{R}^+, \qquad u_i = φ_i \text{ on $\partialΩ\times\mathbb{R}^+$}, \qquad u_i(\mathbf{x},0) = u_{i,0}(\mathbf{x}) \text{ for $\mathbf{x} \in Ω$} \] as $β\to +\infty$. Here $Ω$ is a rotationally invariant planar set and $a_{ij}>0$ for every $i$ and $j$. We tackle also the homogeneous Dirichlet and Neumann boundary conditions, as well as entire solutions in the plane. As a byproduct of our analysis we detect explicit families of eternal, entire solutions of the pure heat equation, parameterized by $ω\in\mathbb{R}$, which reduce to homogeneous harmonic polynomials for $ω=0$.

math.AP↗

arXiv2021

Singular analysis of the optimizers of the principal eigenvalue in indefinite weighted Neumann problems

We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain $Ω\subset \mathbb{R}^{N}$, within a suitable class of sign-changing weights. Denoting with $u$ the optimal eigenfunction and with $D$ its super-level set associated to the optimal weight, we perform the analysis of the singular limit of the optimal eigenvalue as the measure of $D $ tends to zero. We show that, when the measure of $D$ is sufficiently small, $u $ has a unique local maximum point lying on the boundary of $Ω$ and $D$ is connected. Furthermore, the boundary of $D$ intersects the boundary of the box $Ω$, and more precisely, ${\mathcal H}^{N-1}(\partial D \cap \partial Ω)\ge C|D|^{(N-1)/N} $ for some universal constant $C>0$. Though widely expected, these properties are still unknown if the measure of $D$ is arbitrary.

math.AP↗

arXiv2021

Normalized solutions to mass supercritical Schrodinger equations with negative potential

We study the existence of positive solutions with prescribed $L^2$-norm for the Schrödinger equation \[ -Δu-V(x)u+λu=|u|^{p-2}u\qquadλ\in \mathbb{R},\quad u\in H^1(\mathbb{R}^N), \] where $V\ge 0$, $N\ge 1$ and $p\in\left(2+\frac 4 N,2^*\right)$, $2^*:=\frac{2N}{N-2}$ if $N\ge 3$ and $2^*:=+\infty$ if $N=1,2$. We treat two cases. Firstly, under an explicit smallness assumption on $V$ and no condition on the mass, we prove the existence of a mountain pass solution at positive energy level, and we exclude the existence of solutions with negative energy. Secondly, requiring that the mass is smaller than some explicit bound, depending on $V$, and that $V$ is not too small in a suitable sense, we find two solutions: a local minimizer with negative energy, and a mountain pass solution with positive energy. Moreover, a nonexistence result is proved.

math.AP↗

arXiv2021

Multiplicity of solutions on a Nehari set in an invariant cone

For $1<p<2$ and $q$ large, we prove the existence of two positive, nonconstant, radial and radially nondreacreasing solutions of the supercritical equation \[-Δ_p u+u^{p-1}=u^{q-1}\] under Neumann boundary conditions, in the unit ball of $\mathbb R^N$. We use a variational approach in an invariant cone. We distinguish the two solutions upon their energy: one is a ground state inside a Nehari-type subset of the cone, the other is obtained via a mountain pass argument inside the Nehari set. As a byproduct of our proofs, we detect the limit profile of the low energy solution as $q\to\infty$ and show that the constant solution 1 is a local minimum on the Nehari set.

math.AP↗