arXiv subjects

Caterina Sportelli

Publications and source records attributed to Caterina Sportelli.

At least 19 recordsLinked to original sources

arXiv2026

A free boundary analysis of tumor invasion driven by angiogenesis

We discuss a free boundary model for tumor invasion that describes a cloud of cells that diffuse and, at the same time, are drifted along the vector field of the chemotactic direction. The model captures the evolution of a solid tumor, including the process of angiogenesis, which consists in the formation of new blood vessels that supply the tumor with oxygen and other nutrients, thereby promoting its spread and growth. We prove that, once formed, the tumor survives through time, maintaining strictly positive thickness. An explicit expression in terms of the initial data is derived. Moreover, we distinguish two regimes depending on the ratio $κ$ between the spreading of tumor cells and the growth of the tumor mass. If $κ$ is sufficiently large, then the tumor grows exponentially in time and invades the entire host tissue. In contrast, if $κ$ is small enough, then either the tumor remains bounded in size over time or may experience a fast contraction.

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arXiv2026

Homoclinic solutions for nonlocal equations and applications to the theory of atom dislocation

We establish the existence of homoclinic solutions for suitable systems of nonlocal equations whose forcing term is of gradient type. The elliptic operator under consideration is the fractional Laplacian and the potentials that we take into account are of two types: the first one is a spatially homogeneous function with a strict local maximum at the origin, the second one is a spatially inhomogeneous potential satisfying the Ambrosetti-Rabinowitz condition coupled to a quadratic term with spatially dependent growth at infinity. The existence of these special solutions has interesting consequences for the theory of atomic edge dislocations in crystals according to the Peierls-Nabarro model and its generalization to fractional equations. Specifically, for the first type of potentials, the results obtained give the existence of a crystal configuration with atoms located at both extrema in an unstable rest position, up to an arbitrarily small modification of the structural potential and a "pinch" of a particle at any given position. For the second type of potentials, the results obtained also entail the existence of a crystal configuration reaching an equilibrium at infinity, up to an arbitrarily small superquadratic perturbation of the classical Peierls-Nabarro potential.

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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].

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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.

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arXiv2025

Nonlocal operators in divergence form and existence theory for integrable data

We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to $L^1(Ω)$ and to be suitably dominated. We also prove that the solution that we find converges, as $s\nearrow 1$, to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in $L^1(Ω)$ and therefore the usual regularity theory cannot be leveraged to our benefit in this framework. The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as $s\nearrow 1$, every classical operator in divergence form.

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arXiv2025

The Neumann condition for the superposition of fractional Laplacians

We present a new functional setting for Neumann conditions related to the superposition of (possibly infinitely many) fractional Laplace operators. We will introduce some bespoke functional framework and present minimization properties, existence and uniqueness results, asymptotic formulas, spectral analyses, rigidity results, integration by parts formulas, superpositions of fractional perimeters, as well as a study of the associated heat equation.

math.AP↗

arXiv2025

Logistic diffusion equations governed by the superposition of operators of mixed fractional order

We discuss the existence of stationary solutions for logistic diffusion equations of Fisher-Kolmogoroff-Petrovski-Piskunov type driven by the superposition of fractional operators in a bounded region with "hostile" environmental conditions, modeled by homogeneous external Dirichlet data. We provide a range of results on the existence and nonexistence of solutions tied to the spectral properties of the ambient space, corresponding to either survival or extinction of the population. We also discuss how the possible presence of nonlocal phenomena of concentration and diffusion affect the endurance or disappearance of the population. In particular, we give examples in which both classical and anomalous diffusion leads to the extinction of the species, while the presence of an arbitrarily small concentration pattern enables survival.

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arXiv2024

Optimal embedding results for fractional Sobolev spaces

This paper deals with the fractional Sobolev spaces $W^{s, p}(Ω)$, with $s\in (0, 1]$ and $p\in[1,+\infty]$. Here, we use the interpolation results in [4] to provide suitable conditions on the exponents $s$ and $p$ so that the spaces $W^{s, p}(Ω)$ realize a continuous embedding when either $Ω=\mathbb R^N$ or $Ω$ is any open and bounded domain with Lipschitz boundary. Our results enhance the classical continuous embedding and, when $Ω$ is any open bounded domain with Lipschitz boundary, we also improve the classical compact embeddings. All the results stated here are proved to be optimal. Also, our strategy does not require the use of Besov or other interpolation spaces.

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arXiv2024

A general theory for the $(s, p)$-superposition of nonlinear fractional operators

We consider the continuous superposition of operators of the form \[ \iint_{[0, 1]\times (1, N)} (-Δ)_p^s \,u\,dμ(s,p), \] where $μ$ denotes a signed measure over the set $[0, 1]\times (1, N)$, joined to a nonlinearity satisfying a proper subcritical growth. The novelty of the paper relies in the fact that, differently from the existing literature, the superposition occurs in both $s$ and $p$. Here we introduce a new framework which is so broad to include, for example, the scenarios of the finite sum of different (in both $s$ and $p$) Laplacians, or of a fractional $p$-Laplacian plus a $p$-Laplacian, or even combinations involving some fractional Laplacians with the "wrong" sign. The development of this new setting comes with two applications, which are related to the Weierstrass Theorem and a Mountain Pass technique. The results obtained contribute to the existing literature with several specific cases of interest which are entirely new.

math.AP↗

arXiv2024

Some nonlinear problems for the superposition of fractional operators with Neumann boundary conditions

We discuss the existence theory of a nonlinear problem of nonlocal type subject to Neumann boundary conditions. Differently from the existing literature, the elliptic operator under consideration is obtained as a superposition of operators of mixed order. The setting that we introduce is very general and comprises, for instance, the sum of two fractional Laplacians, or of a fractional Laplacian and a Laplacian, as particular cases (the situation in which there are infinitely many operators, and even a continuous distribution of operators, can be considered as well). New bits of functional analysis are introduced to deal with this problem. An eigenvalue analysis divides the existence theory into two streams, one related to a Mountain Pass method, the other to a Linking technique.

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arXiv2023

An existence theory for nonlinear superposition operators of mixed fractional order

We establish the existence of multiple solutions for a nonlinear problem of critical type. The problem considered is fractional in nature, since it is obtained by the superposition of $(s,p)$-fractional Laplacians of different orders. The results obtained are new even in the case of the sum of two different fractional $p$-Laplacians, or the sum of a fractional $p$-Laplacian and a classical $p$-Laplacian, but our framework is general enough to address also the sum of finitely, or even infinitely many, operators. In fact, we can also consider the superposition of a continuum of operators, modulated by a general signed measure on the fractional exponents. When this measure is not positive, the contributions of the individual operators to the whole superposition operator is allowed to change sign. In this situation, our structural assumption is that the positive measure on the higher fractional exponents dominates the rest of the signed measure.

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arXiv2023

An existence theory for superposition operators of mixed order subject to jumping nonlinearities

We consider a superposition operator of the form $$ \int_{[0, 1]} (-Δ)^s u\, dμ(s),$$ for a signed measure $μ$ on the interval of fractional exponents $[0,1]$, joined to a nonlinearity whose term of homogeneity equal to one is "jumping", i.e. it may present different coefficients in front of the negative and positive parts. The signed measure is supposed to possess a positive contribution coming from the higher exponents that overcomes its negative contribution (if any). The problem taken into account is also of "critical" type, though in this case the critical exponent needs to be carefully selected in terms of the signed measure $μ$. Not only the operator and the nonlinearity considered here are very general, but our results are new even in special cases of interest and include known results as particular subcases. The possibility of considering operators "with the wrong sign" is also a complete novelty in this setting.

math.AP↗

arXiv2023

On existence and multiplicity of solutions for generalized (p, q)-Laplacian equations on unbounded domains

This paper deals with the existence and multiplicity of solutions for the generalized $(p, q)$-Laplacian equation \begin{align*} &-{\text{ div}}(A(x, u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x, u)|\nabla u|^p -{\text{ div}}(B(x, u)|\nabla u|^{q-2}\nabla u) \\ &\quad\qquad+\frac1q B_t(x, u)|\nabla u|^q + V(x)|u|^{p-2} u+ W(x)|u|^{q-2} u= g(x, u)\quad\qquad\mbox{ in } \mathbb{R}^N, \end{align*} where $1<q\le p< N$, $A, B:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ are suitable $C^1$ Carathéodory functions with $A_t(x, u)=\frac{\partial A}{\partial t}(x, u), B_t(x, u)=\frac{\partial B}{\partial t}(x, u)$, $V, W:\mathbb{R}^N\to\mathbb{R}$ are proper ``weight functions" and $g:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ is a Carathéodory map. Notwithstanding the occurrence of some coefficients which rely upon the solution itself makes the use of variational techniques more challenging, under suitable assumptions on the involved functions, we are able to exploit the variational nature of our problem. In particular, the existence of a nontrivial solution is derived via a generalized version of the Ambrosetti-Rabinowitz Mountain Pass Theorem, based on a weaker version of the classical Cerami-Palais-Smale condition. Finally, the multiplicity result, which is thoroughly new also even in the simpler case $q=p$, is gained under symmetry assumptions and a sharp decomposition of the ambient space.

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arXiv2023

Nonlocal critical growth elliptic problems with jumping nonlinearities

In this paper we study a nonlocal critical growth elliptic problem driven by the fractional Laplacian in presence of jumping nonlinearities. In the main results of the paper we prove the existence of a nontrivial solution for the problem under consideration, using variational and topological methods and applying a new linking theorems recently got by Perera and Sportelli in [10]. The existence results provided in this paper can be seen as the nonlocal counterpart of the ones obtained in [10] in the context of the Laplacian equations. In the nonlocal framework the arguments used in the classical setting have to be refined. Indeed the presence of the fractional Laplacian operator gives rise to some additional difficulties, that we are able to overcome proving new regularity results for weak solutions of nonlocal problems, which are of independent interest.

math.AP↗

arXiv2023

Theoretical aspects in penalty hyperparameters optimization

Learning processes are useful methodologies able to improve knowledge of real phenomena. These are often dependent on hyperparameters, variables set before the training process and regulating the learning procedure. Hyperparameters optimization problem is an open issue in learning approaches since it can strongly affect any real data analysis. They are usually selected using Grid-Search or Cross Validation techniques. No automatic tuning procedure exists especially if we focus on an unsupervised learning scenario. This study aims to assess some theoretical considerations for tuning penalty hyperparameters in optimization problems. It considers a bi-level formulation tuning problem in an unsupervised context, by using Gradient-based methods. Suitable conditions for the existence of a minimizer in an infinite-dimensional Hilbert space are outlined, together with some theoretical results, applicable in all those situations when it is unnecessary or not possible obtaining an exact minimizer. An iterative algorithmic strategy is considered, equipped with a stopping criterion via Ekeland's variational principle.

math.OC↗

arXiv2022

New linking theorems with applications to critical growth elliptic problems with jumping nonlinearities

We study critical growth elliptic problems with jumping nonlinearities. Standard linking arguments based on decompositions of $H^1_0(Ω)$ into eigenspaces of $- Δ$ cannot be used to obtain nontrivial solutions to such problems. We show that the associated variational functional admits certain linking structures based on splittings of $H^1_0(Ω)$ into nonlinear submanifolds. In order to capture these linking geometries, we prove several generalizations of the classical linking theorem of Rabinowitz that are not based on linear subspaces. We then use these new linking theorems to obtain nontrivial solutions of our problems. Our abstract results are of independent interest and can be used to obtain nontrivial solutions of other types of problems with jumping nonlinearities as well.

math.AP↗

arXiv2022

A multiplicity result for critical elliptic problems involving differences of local and nonlocal operators

We study some critical elliptic problems involving the difference of two nonlocal operators, or the difference of a local operator and a nonlocal operator. The main result is the existence of two nontrivial weak solutions, one with negative energy and the other with positive energy, for all sufficiently small values of a parameter. The proof is based on an abstract result recently obtained in [20].

math.AP↗

arXiv2022

Multiple solutions for coupled gradient-type quasilinear elliptic systems with supercritical growth

In this paper we consider the following coupled gradient-type quasilinear elliptic system \begin{equation*} \left\{ \begin{array}{ll} - {\rm div} ( a(x, u, \nabla u) ) + A_t (x, u, \nabla u) = G_u(x, u, v) &\hbox{ in $Ω$,}\\[10pt] - {\rm div} ( b(x, v, \nabla v) ) + B_t(x, v, \nabla v) = G_v\left(x, u, v\right) &\hbox{ in $Ω$,}\\[10pt] u = v = 0 &\hbox{ on $\partialΩ$,} \end{array} \right. \end{equation*} where $Ω$ is an open bounded domain in $\mathbb{R}^N$, $N\ge 2$. We suppose that some $\mathcal{C}^{1}$-Carathéodory functions $A, B:Ω\times\mathbb{R}\times\mathbb{R}^N\rightarrow\mathbb{R}$ exist such that $a(x,t,ξ) = \nabla_ξ A(x,t,ξ)$, $A_t(x,t,ξ) = \frac{\partial A}{\partial t} (x,t,ξ)$, $b(x,t,ξ) = \nabla_ξ B(x,t,ξ)$, $B_t(x,t,ξ) =\frac{\partial B}{\partial t}(x,t,ξ)$, and that $G_u(x, u, v)$, $G_v(x, u, v)$ are the partial derivatives of a $\mathcal{C}^{1}$-Carathéodory nonlinearity $G:Ω\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$. Roughly speaking, we assume that $A(x,t,ξ)$ grows at least as $(1+|t|^{s_1p_1})|ξ|^{p_1}$, $p_1 > 1$, $s_1 \ge 0$, while $B(x,t,ξ)$ grows as $(1+|t|^{s_2p_2})|ξ|^{p_2}$, $p_2 > 1$, $s_2 \ge 0$, and that $G(x, u, v)$ can also have a supercritical growth related to $s_1$ and $s_2$. Since the coefficients depend on the solution and its gradient themselves, the study of the interaction of two different norms in a suitable Banach space is needed. In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones.

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