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Benedetta Pellacci

Publications and source records attributed to Benedetta Pellacci.

At least 19 recordsLinked to original sources
arXiv2026

Optimization of the total tumor population under Gompertz growth

We study optimal control problems for a stationary reaction--diffusion model describing the spatial distribution of a tumor cell population with Gompertz growth. The control $m(x)$ represents a treatment term acting as a density-dependent removal rate and it is subject to $L^{1}-L^{\infty}$ constraints. When the intrinsic growth rate is constant, the uniform distribution of the treatment is shown to be the unique minimizer. For the maximization problem, we prove that every optimal control is of bang-bang type. In addition, we show that in the one dimensional case and for sufficiently large diffusion rates, the positivity set of optimal controls is an interval sticking to one of the extrema of the domain. Finally, numerical simulations complement the theoretical analysis and explore regimes that are not fully covered by the results proved in the paper. The computations confirm the bang-bang structure of maximizers, and illustrate how the shape of optimal controls and the associated states are affected by spatial heterogeneity in the growth rate, localized admissible treatment regions, and the diffusion coefficient. Moreover, they reveal a monotone dependence of the optimized total population on the diffusion coefficient: this is a new phenomenon with respect to the logistic setting.

math.AP↗
arXiv2026

Sign-changing solutions to the Yamabe problem on a spherical cap

Spherical caps play a crucial role in establishing a criterion for the existence of solutions to the Yamabe problem on a compact Riemannian manifold with boundary, similar to the role played by the standard sphere in the problem on a closed Riemannian manifold. This problem is expressed in terms of a nonlinear boundary-value problem, where both the nonlinearity and the boundary condition are critical in the Sobolev sense. This work focuses on the existence of multiple solutions to the Yamabe problem on spherical caps. We show that if the spherical cap is contained in a hemisphere of the standard $n$-sphere and $n = 5$ or $n \geq 7$, the Yamabe problem has infinitely many sign-changing solutions. Our approach takes advantage of symmetries and is based on a careful analysis of the loss of compactness of the variational problem.

math.AP↗
arXiv2025

Sign-changing solutions to the Yamabe problem on manifolds with boundary

Let $(M, g)$ be a compact Riemannian manifold with boundary. The Yamabe problem concerning the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary is equivalent, in analytic terms, to finding a positive solution to a nonlinear boundary-value problem with critical growth. While the existence of positive solutions to this problem is by now well understood, the existence of sign-changing (nodal) solutions remains largely open. In this work we establish the existence of least-energy sign-changing solutions when the manifold is positive and the mean curvature of the boundary is a non-negative constant. More precisely, we prove that if $n\ge7$ and $M$ has a nonumbilic boundary point, then the problem admits least-energy nodal solutions. Our approach is variational and relies on the analysis of suitable conformal invariants and sharp energy estimates.

math.DG↗
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 $\Omega\subset \mathbb{R}^{N}$, $N\ge2$, 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 $u$, $P\in \partial \Omega$, tends to a point of maximal mean curvature of $\partial \Omega$. Furthermore, we show that $D$ is the intersection with $\Omega$ 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↗
arXiv2023

Partially concentrating standing waves for weakly coupled Schr\"odinger systems

We study the existence of standing waves for the following weakly coupled system of two Schr\"odinger equations in $\mathbb{R}^N$, $N=2,3$, \[ \begin{cases} i \hslash \partial_{t}\psi_{1}=-\frac{\hslash^2}{2m_{1}}\Delta \psi_{1}+ {V_1}(x)\psi_{1}-\mu_{1}|\psi_{1}|^{2}\psi_{1}-\beta|\psi_{2}|^{2}\psi_{1} & \\ i \hslash \partial_{t}\psi_{2}=-\frac{\hslash^2}{2m_{2}}\Delta \psi_{2}+ {V_2}(x)\psi_{2}-\mu_{2}|\psi_{2}|^{2}\psi_{2}-\beta|\psi_{1}|^{2}\psi_{2},& \end{cases} \] 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

Spectral optimization for weighted anisotropic problems with Robin conditions

We study a weighted eigenvalue problem with anisotropic diffusion in bounded Lipschitz domains $\Omega\subset \mathbb{R}^{N} $, $N\ge1$, under Robin boundary conditions, proving the existence of two positive eigenvalues $\lambda^{\pm}$ respectively associated with a positive and a negative eigenfunction. Next, we analyze the minimization of $\lambda^{\pm}$ with respect to the sign-changing weight, showing that the optimal eigenvalues $\Lambda^{\pm}$ are equal and the optimal weights are of bang-bang type, namely piece-wise constant functions, each one taking only two values. As a consequence, the problem is equivalent to the minimization with respect to the subsets of $\Omega$ satisfying a volume constraint. Then, we completely solve the optimization problem in one dimension, in the case of homogeneous Dirichlet or Neumann conditions, showing new phenomena induced by the presence of the anisotropic diffusion. The optmization problem for $\lambda^{+}$ naturally arises in the study of the optimal spatial arrangement of resources for a species to survive in a heterogeneous habitat.

math.AP↗
arXiv2022

An upper bound for the least energy of a sign-changing solution to a zero mass problem

We give an upper bound for the least energy of a sign-changing solution to the the nonlinear scalar field equation $$-\Delta u = f(u), \qquad u\in D^{1,2}(\mathbb{R}^{N}),$$ where $N\geq5$ and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that $12c_0$ if $N=5,6$ and smaller than $10c_0$ if $N\geq 7$, where $c_0$ is the ground state energy.

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 $\Omega\subset \mathbb{R}^{N}$, 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 $\Omega$ and $D$ is connected. Furthermore, the boundary of $D$ intersects the boundary of the box $\Omega$, and more precisely, ${\mathcal H}^{N-1}(\partial D \cap \partial \Omega)\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

Symmetric positive solutions to nonlinear Choquard equations with potentials

Existence results for a class of Choquard equations with potentials are established. The potential has a limit at infinity and it is taken invariant under the action of a closed subgroup of linear isometries of $\mathbb{R}^N$. As a consequence, the positive solution found will be invariant under the same action. Power nonlinearities with exponent greater or equal than two or less than two will be handled. Our results include the physical case.

math.AP↗
arXiv2021

Asymptotic behavior of positive solutions of semilinear elliptic problems with increasing powers

We prove existence results of two solutions of the problem \[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{ in $\Omega$}, \\ \quad u>0 &\text{ in $\Omega$}, \\ \quad u=0 & \text{ on $\partial \Omega$}, \end{cases} \] where $L(v)=-{\rm div}(M(x)\nabla v)$ is a linear operator, $p\in (2,2^{*}]$ and $\lambda$ and $ m$ sufficiently large. Then their asymptotical limit as $m\to +\infty$ is investigated showing different behaviors.

math.AP↗
arXiv2019

Normalized concentrating solutions to nonlinear elliptic problems

We prove the existence of solutions $(\lambda, v)\in \mathbb{R}\times H^{1}(\Omega)$ of the elliptic problem \[ \begin{cases} -\Delta v+(V(x)+\lambda) v =v^{p}\ &\text{ in $ \Omega, $} \ v>0,\qquad \int_\Omega v^2\,dx =\rho. \end{cases} \] Any $v$ solving such problem (for some $\lambda$) is called a normalized solution, where the normalization is settled in $L^2(\Omega)$. Here $\Omega$ 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 Schr\"odinger equation, or that of quadratic ergodic Mean Field Games systems. We prove the existence of solutions concentrating at suitable points of $\Omega$ as the prescribed mass $\rho$ 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

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~$ct^\beta$ for any given~$c>0$ and~$\beta\in\left(0,\frac12\right)$.

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 $\Omega$; 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 $\Omega$. 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 $\Omega$ is a planar polygon, providing quantitative estimates of the optimal level convergence, as well as of the involved eigenvalues.

math.AP↗
arXiv2018

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 $\Omega\subset\mathbb{R}^N$, 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 $\Omega$, the optimal shapes for this second problem are indeed spherical; moreover, for general $\Omega$, we show that small volume spectral drops are asymptotically spherical, with center at points of $\partial\Omega$ having large mean curvature.

math.AP↗
arXiv2018

Positive multipeak solutions to a zero mass problem in exterior domains

We establish the existence of positive multipeak solutions to the nonlinear scalar field equation with zero mass $$-\Delta u = f(u), \qquad u\in D_0^{1,2}(\Omega_R),$$ where $\Omega_R:=\{x \in \mathbb{R}^N:|u|>R\}$ with $R>0$, $N\geq4$, and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. We show that the number of positive multipeak solutions becomes arbitrarily large as $R \to \infty$.

math.AP↗
arXiv2017

Oscillating solutions for nonlinear Helmholtz Equations

Existence results for radially symmetric oscillating solutions for a class of nonlinear autonomous Helmholtz equations are given and their exact asymptotic behavior at infinity is established. Some generalizations to nonautonomous radial equations as well as existence results for nonradial solutions are found. Our theorems prove the existence of standing waves solutions of nonlinear Klein-Gordon or Schrödinger equations with large frequencies.

math.AP↗