arXiv subjects

Enrico Valdinoci

Publications and source records attributed to Enrico Valdinoci.

At least 37 records · Page 2Linked to original sources

arXiv2026

Maximum principles and spectral analysis for the superposition of operators of fractional order

We consider a "superposition operator" obtained through the continuous superposition of operators of mixed fractional order, modulated by a signed Borel finite measure defined over the set $[0, 1]$. The relevance of this operator is rooted in the fact that it incorporates special and significant cases of interest, like the mixed operator $-Δ+ (-Δ)^s$, the (possibly) infinite sum of fractional Laplacians and allows to consider operators carrying a "wrong sign". We first outline weak and strong maximum principles for this type of operators. Then, we complete the spectral analysis for the related Dirichlet eigenvalue problem started in [DPLSV25b].

math.AP↗

arXiv2026

Long-range phase coexistence models with degenerate potentials

This survey offers an overview of recent advances in nonlocal phase transition problems, modeled by Ginzburg--Landau type energies of the form \[ \frac{1}{4}\iint_{\R^{2n}\setminus (\R^n \setminus Ω)^2} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy \;+\; \int_ΩW(u(x))\,dx. \] Here,~$W$ is a smooth and possibly \textit{degenerate} double well potential, with a polynomial control on its second derivatives near the wells. The emphasis is on qualitative properties of minimizers and critical points of the energy functional.

math.AP↗

arXiv2026

Stability and Self-Organized Patterns in Coupled Ecohydrological--Fire Dynamics: A Model of Vegetation--Rainfall--Bushfire Interactions

This paper investigates the conditions for the stability and emergence of patterns in a new three-component reaction-diffusion system. The system describes the coexistence and interaction of water reservoirs, vegetation, and bushfire activity in a given ecosystem. We perform a detailed stability analysis to determine the parameter space where an unstable homogeneous equilibrium becomes stable with respect to spatially nonuniform perturbations. We also use diffusion to generate traveling trains in the form of periodic orbits of the linearized system. These orbits are remnants of an unstable equilibrium in the absence of diffusion and arise from a nonsingular eigenvalue crossing of the imaginary axis, while a third eigenvalue remains real and negative, thereby ensuring linear stability for monocromatic waves. These phenomena differ from ``classical'' Turing and Hopf bifurcations, as the model does not involve distinct ``activators'' and ``inhibitors'', and the effects observed are not the byproduct of diffusion with necessarily differing speeds. Also, differently from the classical Turing pattern, the role of diffusion in this context is to stabilize, rather than destabilize, homogeneous equilibria. We also consider the case of plant competition, showing a suitable form of Turing instability for slow-frequency oscillations in a small rainfall regime.

math.AP↗

arXiv2026

Fredholm alternative for a general class of nonlocal operators

We develop a Fredholm alternative for a fractional elliptic operator~$\mathcal{L}$ of mixed order built on the notion of fractional gradient. This operator constitutes the nonlocal extension of the classical second order elliptic operators with measurable coefficients treated by Neil Trudinger in~\cite{trudinger}. We build~$\mathcal{L}$ by weighing the order~$s$ of the fractional gradient over a measure (which can be either continuous, or discrete, or of mixed type). The coefficients of~$\mathcal{L}$ may also depend on~$s$, giving this operator a possibly non-homogeneous structure with variable exponent. These coefficients can also be either unbounded, or discontinuous, or both. A suitable functional analytic framework is introduced and investigated and our main results strongly rely on some custom analysis of appropriate functional spaces.

math.AP↗

arXiv2026

Optimal decay of heteroclinic solutions of the fractional Allen-Cahn equation with a degenerate potential

We refine the asymptotic estimates for minimizers of a class of nonlocal energy functionals of the form \[ \frac{1}{4} \iint_{\R^{2n} \setminus (\R^n \setminus Ω)^2} \snr{u(x) - u(y)}^2 K(x - y) \,dx\,dy + \int_ΩW(u(x)) \,dx, \] as originally studied in~\cite{DPDV}, and we prove the optimality of our improved bounds. Here, $W$ denotes a possibly \emph{degenerate} oscillatory double-well potential, satisfying a polynomial control on its second derivative near the wells. The kernel~$K$ belongs to a broad class of measurable functions and is modeled on the one of the fractional Laplacian.

math.AP↗

arXiv2026

Nonlocal eigenvalue problems and superposition operators

We study the spectral theory of mixed local and nonlocal operators with lower-order terms in the right-hand side of the equation. This kind of problems is motivated by the analysis of superposition operators of mixed order and with the "wrong sign" of the lower-order terms with respect to the classical elliptic theory. Our results include: -convergence to classical cases when the right-hand side of the eigenvalye equations "localizes", recovering the simplicity and sign-definiteness of eigenfunctions in the limit; -a detailed analysis of disconnected domains, showing that, unlike the classical case, any eigenfunction associated with the first eigenvalue must change sign, and that the first eigenvalue of a union of disconnected domains is strictly smaller than that of its individual components; -examples in which the first eigenvalue is either simple or non-simple in disconnected domains; -a regularity theory that underpins these results.

math.AP↗

arXiv2026

Regularity of the trace of nonlocal minimal graphs

We prove that the trace of nonlocal minimal graphs at points of stickiness is of class~$C^{1,γ}$. As a result, we show that boundary continuity implies boundary differentiability for nonlocal minimal graphs.

math.AP↗

arXiv2026

Non-local planelike minimizers and $Γ$-convergence of periodic energies to a local anisotropic perimeter

We investigate a homogenization problem related to a non-local interface energy with a periodic forcing term. We show the existence of planelike minimizers for such energy. Moreover, we prove that, under suitable assumptions on the non-local kernel and the external field, the sequence of rescaled energies $Γ$-converges to a suitable local anisotropic perimeter, where the anisotropy is defined as the limit of the normalized energy of a planelike minimizer in larger and larger cubes (i.e., what is called in jargon "stable norm"). To obtain this, we also establish several auxiliary results, including: the minimality of the level sets of the minimizers, explicit bounds on the oscillations of the minimizers, density estimates for almost minimizers, and non-local perimeter estimates in the large.

math.AP↗

arXiv2025

On non-local almost minimal sets and an application to the non-local Massari's Problem

We consider a fractional Plateau's problem dealing with sets with prescribed non-local mean curvature. This problem can be seen as a non-local counterpart of the classical Massari's Problem. We obtain existence and regularity results, relying on a suitable version of the non-local theory for almost minimal sets. In this framework, the fractional curvature term in the energy functional can be interpreted as a perturbation of the fractional perimeter. In addition, we also discuss stickiness phenomena for non-local almost minimal sets.

math.AP↗

arXiv2025

$Γ$-convergence for higher order nonlocal phase transitions

For every $0 < s <3/4$, we study the asymptotic behavior of the $\varepsilon$-rescaled sum of the $s$-fractional Allen-Cahn energy and the squared $L^2$-norm of its first variation. We prove that the contribution of the first variation vanishes as $\varepsilon \to 0$. This implies the Gamma-convergence of the initial sum to either the classical perimeter or to the $2s$-fractional perimeter, depending on whether $s \ge 1/2$ or not. This contradicts the expectation of finding curvature-dependent terms in the limit, as suggested by the regime $3/4 \le s < 1$, and as known to hold in low dimensions in the local case.

math.AP↗

arXiv2025

A primer on Fourier Series

This is a textbook on Fourier Series, suitable for both undergraduate and graduate courses. The textbook is endowed with exercises, and full solutions are provided at the end of the book.

math.AP↗

arXiv2025

Nonlocal approximation of minimal surfaces: optimal estimates from stability

Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren-Pitts approach. The Allen-Cahn approximation has proved to be a powerful alternative, and Chodosh and Mantoulidis (in Ann. Math. 2020) used it to give a new proof of Yau's conjecture for generic metrics and establish the multiplicity one conjecture. The primary goal of this paper is to set the ground for a new approximation based on nonlocal minimal surfaces. More precisely, we prove that if $\partial E$ is a stable $s$-minimal surface in $B_1\subset \mathbb R^3$ then: - $\partial E\cap B_{1/2}$ enjoys a $C^{2,α}$ estimate that is robust as $s\uparrow 1$ (i.e. uniform in $s$); - the distance between different connected components of~$\partial E\cap B_{1/2}$ must be at least of order~$(1-s)^{\frac 1 2}$ (optimal sheet separation estimate); - interactions between multiple sheets at distances of order $(1-s)^{\frac 1 2}$ are described by the Dávila--del Pino--Wei system. A second important goal of the paper is to establish that hyperplanes are the only stable $s$-minimal hypersurfaces in $\mathbb R^4$, for $s\in(0,1)$ sufficiently close to $1$. This is done by exploiting suitable modifications of the results described above. In this application, it is crucially used that our curvature and separations estimates hold without any assumption on area bounds (in contrast to the analogous estimates for Allen-Cahn).

math.DG↗

arXiv2025

Density estimates for a nonlocal variational model with a degenerate double-well potential via the Sobolev inequality

We provide density estimates for level sets of minimizers of the energy $\frac{1}{2} \int_Ω\int_Ω \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dxdy+\int_Ω\int_{\mathbb{R}^n\setminusΩ} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dxdy+\int_ΩW(u(x))dx$ where $p\in(1,+\infty)$ and $s\in\left(0,\frac{1}{p}\right)$ and $W$ is a double-well potential with polynomial growth $m\in [p,+\infty)$ from the minima. These kinds of potentials are ''degenerate'', since they detach ''slowly'' from the minima, therefore they provide additional difficulties if one wishes to determine the relative sizes of the ''layers'' and the ''pure phases''. To overcome these challenges, we introduce new barriers allowing us to rely on the fractional Sobolev inequality and on a suitable iteration method. The proofs presented here are robust enough to consider the case of quasilinear nonlocal equations driven by the fractional $p$-Laplacian, but our results are new even for the case $p=2$.

math.AP↗

arXiv2025

Elliptic partial differential equations from an elementary viewpoint

These notes are the outcome of some courses taught to undergraduate and graduate students from the University of Western Australia, the Pontif\'ıcia Universidade Católica do Rio de Janeiro, the Indian Institute of Technology Gandhinagar and the Ukrainian Catholic University in 2021 and 2022.

math.AP↗

arXiv2025

On a class of (non)local superposition operators of arbitrary order

In this paper we introduce a very general setting dealing with the superposition of operators of any positive order and provide a systematic study of them. We also provide examples and counterexamples, as well as characterizing properties of the measures and the functional spaces under consideration. Moreover, we present some applications regarding the existence theory for a class of nonlinear problems involving superposition operators of arbitrary (possibly fractional) order.

math.AP↗

arXiv2025

Bushfires and Balance: Proactive versus Reactive Policies in Prescribed Burning

We introduce a new mathematical model to explore the dynamic relationship between prescribed burning and bushfire occurrence, formulated as a system of ordinary differential equations. The model admits a unique steady-state, and its stability is shown to depend critically on the policy framework governing prescribed burning. In particular, reactive policies, where prescribed burning is increased in response to bushfire events, can lead to system instability unless augmented with appropriate feedback control. Conversely, proactive policies that implement moderate, consistent prescribed burning are more effective at stabilizing the system and mitigating the frequency of bushfires. Additionally, higher vegetation regeneration rates contribute positively to equilibrium stability. While this model does not capture the full complexity of fire management, it highlights the potential risks of emotionally driven, reactive responses and underscores the value of preventive, stability-focused strategies. To the best of our knowledge, this is the first mathematical analysis of preventive burning strategies and their long-term impact on bushfire mitigation.

math.OC↗

arXiv2025

The Lévy flight foraging hypothesis: comparison between stationary distributions and anomalous diffusion

We consider a stationary prey in a given region of space and we aim at detecting optimal foraging strategies. On the one hand, when the prey is uniformly distributed, the best possible strategy for the forager is to be stationary and uniformly distributed in the same region. On the other hand, in several biological settings, foragers cannot be completely stationary, therefore we investigate the best seeking strategy for Lévy foragers in terms of the corresponding Lévy exponent. In this case, we show that the best strategy depends on the region size in which the prey is located: large regions exhibit optimal seeking strategies close to Gaussian random walks, while small regions favor Lévy foragers with small fractional exponent. We also consider optimal strategies in view of the Fourier transform of the distribution of a stationary prey. When this distribution is supported in a suitable volume, then the foraging efficiency functional is monotone increasing with respect to the Lévy exponent and accordingly the optimal strategy is given by the Gaussian dispersal. If instead the Fourier transform of the distribution of a stationary prey is supported in the complement of a suitable volume, then the foraging efficiency functional is monotone decreasing with respect to the Lévy exponent and therefore the optimal strategy is given by a null fractional exponent (which in turn corresponds, from a biological standpoint, to a strategy of ``ambush'' type). We will devote a rigorous quantitative analysis also to emphasize some specific differences between the one-dimensional and the higher-dimensional cases.

math.OC↗

arXiv2025

Nonlocal free boundary minimal surfaces

We introduce the nonlocal analogue of the classical free boundary minimal hypersurfaces in an open domain $Ω$ of $\mathbb{R}^n$ as the (boundaries of) critical points of the fractional perimeter $\operatorname{Per}_s(\cdot,\,Ω)$ with respect to inner variations leaving $Ω$ invariant. We deduce the Euler-Lagrange equations and prove a few surprising features, such as the existence of critical points without boundary and a strong volume constraint in $Ω$ for unbounded hypersurfaces. Moreover, we investigate stickiness properties and regularity across the boundary.

math.AP↗