arXiv subjects

Chiara Trifone

Publications and source records attributed to Chiara Trifone.

6 recordsLinked to original sources

arXiv2026

Some counterexamples concerning a model of eradication of invasive species

In this note we study the regularity properties of the \emph{Effort Function $E(β)$}. This can be defined as the minimal $L^1$-norm of the control $\tilde α(x) \geq 0$ so that the PDE $$U_t = ΔU + f(U) - \tilde α(t,x) U, \quad U \in [0,1], %\ \tilde α\geq 0,$$ admits a traveling profile $U(x - βt)$ connecting $0$ to $1$. The above PDE models the evolution of an invasive species, where the control $\tilde α(t,x)$ describes the percentage amount of species removal. In the literature some conditions are assumed on the properties of $f(U)$ and (consequently) on the effort function $E(β)$, the main ones being that the traveling profile has a simple structure and that the effort function can be used as an anisotropic perimeter to generated a cost function over sets of finite perimeter. This note gives some examples showing that in the interesting cases, i.e. when the PDE has a non trivial dynamics, these assumptions are in general not satisfied. We also show that $E(β)$ is in general only $C^1$.

math.AP↗

arXiv2026

Invariant manifolds and transversality for a traveling-wave ODE in a controlled reaction-diffusion model

We study the parameter dependence and transversality properties of trajectories of the controlled ODE $$ U_x = P, \qquad P_x = - f(U) - βP + \tilde α(x)U, $$ arising from traveling-wave solutions of a bistable reaction-diffusion equation with control. Optimal profiles minimizing the $L^1$-norm of the control $\tilde α$ are determined by the unstable and stable manifolds $Γ_u$, $Γ_s$, the critical curve $$ P^*(U) = \sqrt{Uf(U)}, $$ and the associated function $$ F(U) = \frac{dP^*}{dU} + \frac{f(U)}{P^*(U)}. $$ We establish generic transversality properties for these objects with respect to perturbations of the speed $β$ and the nonlinear source $f$, together with quantitative estimates on the first- and second-order dependence of the invariant manifolds on these parameters. We also analyze the loss of regularity at the critical speed $β^{**}$, where the unstable manifold approaches the strongly stable manifold of the equilibrium $(U^*,0)$. These results provide the ODE ingredients used in the analysis of the structure and regularity of optimal profiles in paper [S. Bianchini and C. Trifone. Traveling profiles and control cost for a PDE describing the evolution of invasive species. 2026. arXiv: 2606.23979 [math.OC]].

math.AP↗

arXiv2026

Traveling profiles and control cost for a PDE describing the evolution of invasive species

We develop a detailed analysis of optimal traveling waves $U(t,x) = U(x - βt)$ for a model of invasive-species control proposed in [Bressan, Chiri, and Salehi, Math. Models Methods Appl. Sci., 2022] : the relative density $U \in [0,1]$ of the invasive species satisfies the following reaction-diffusion equation with a positive control \begin{equation} \label{Equa:PDE_abstract} U_t = U_{xx} + f(U) - \tilde α(t,x) U, \quad U \in [0,1], \ \tilde α\geq 0. \end{equation} The control $\tilde α(t,x)$ represents the fraction of the population removed at $(t,x)$: the minimal control effort $E(β,f)$ required to sustain a traveling invasion front with prescribed speed $β$ is defined as the minimal $L^1$-norm of $\tilde α$ for a traveling wave solution $U(x-βt)$ to the PDE. In order to study large scale dynamics $(t,x) \mapsto (εt,εx)$, a fundamental role is played by the structure of traveling waves and the convexity and regularity properties of $E$. The main results of this paper are the following: 1)In the phase plane $(U,P=U_x)$, there exists a unique optimal profile $P_β(U)$ minimizing the effort; 2) It satisfies explicit first-order conditions, which are both necessary and sufficient; 3) The associated control is acting on an open subset of the set $\{U : P_β(U) = \sqrt{U f(U)}\}$, in particular it is uniformly integrable, and it depends smoothly on $(β,f)$ on a dense open set; 4)The effort function $E(β,f)$ is only $C^1$ w.r.t. $β$ and Lipschitz w.r.t. $f$ in the $C^2$-topology, and is asymptocally linear for $β\to \infty$; 5)$β\mapsto E(β,f)$ is in general neither convex nor subadditive.

math.OC↗

arXiv2026

$Γ$-convergence of convolution-type functionals for free discontinuity problems

We prove compactness with respect to $Γ$-convergence for a general class of non-local energies modelled after the ones considered in [Gobbino, CPAM (1998)]. We give an integral representation result for the limits, which are free discontinuity functionals defined on the space of generalised special functions of bounded variation. We then characterise the bulk and surface energy densities of the obtained limits by means of minimisation problems on small cubes for the approximating energies.

math.AP↗

arXiv2025

Singular perturbations models in phase transitions for anisotropic higher-order materials

We discuss a model for phase transitions in which a double-well potential is singularly perturbed by possibly several terms involving different, arbitrarily high orders of derivation. We study by $Γ$-convergence the asymptotic behaviour as $\varepsilon\to 0$ of the functionals \begin{equation*} F_\varepsilon(u):=\int_Ω\Bigl[\frac{1}{\varepsilon}W(u)+\sum_{\ell=1}^{k}q_\ell\varepsilon^{2\ell-1}|\nabla^{(\ell)}u|_\ell^2\Bigr]\,dx, \qquad u\in H^k(Ω), \end{equation*} for fixed $k>1$ integer, addressing also to the case in which the coefficients $q_1,...,q_{k-1}$ are negative and $|\cdot|_\ell$ is any norm on the space of symmetric $\ell$-tensors for each $\ell\in\{1,...,k\}$. The negativity of the coefficients leads to the lack of a priori bounds on the functionals; such issue is overcome by proving a nonlinear interpolation inequality. With this inequality at our disposal, a compactness result is achieved by resorting to the recent paper [10]. A further difficulty is the presence of general tensor norms which carry anisotropies, making standard slicing arguments not suitable. We prove that the $Γ$-limit is finite only on sharp interfaces and that it equals an anisotropic perimeter, with a surface energy density described by a cell formula.

math.AP↗

arXiv2024

Homogenization of non-local energies on disconnected sets

We consider the problem of the homogenization of non-local quadratic energies defined on $δ$-periodic disconnected sets defined by a double integral, depending on a kernel concentrated at scale $\varepsilon$. For kernels with unbounded support we show that we may have three regimes: (i) $\varepsilon<\!<δ$, for which the $Γ$-limit even in the strong topology of $L^2$ is $0$; (ii) $\frac\varepsilonδ\toκ$, in which the energies are coercive with respect to a convergence of interpolated functions, and the limit is governed by a non-local homogenization formula parameterized by $κ$; (iii) $δ<\!<\varepsilon$, for which the $Γ$-limit is computed with respect to a coarse-grained convergence and exhibits a separation-of-scales effect; namely, it is the same as the one obtained by formally first letting $δ\to 0$ (which turns out to be a pointwise weak limit, thanks to an iterated use of Jensen's inequality), and then, noting that the outcome is a nonlocal energy studied by Bourgain, Brezis and Mironescu, letting $\varepsilon\to0$. A slightly more complex description is necessary for case (ii) if the kernel is compactly supported.

math.AP↗