arXiv subjects

Edoardo Proietti Lippi

Publications and source records attributed to Edoardo Proietti Lippi.

14 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

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.

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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

Global perturbative elliptic problems with critical growth in the fractional setting

Given $s$, $q\in(0,1)$, and a bounded and integrable function $h$ which is strictly positive in an open set, we show that there exist at least two nonnegative solutions $u$ of the critical problem $$(-Δ)^s u=\varepsilon h(x)u^q+u^{2^*_s-1},$$ as long as $\varepsilon>0$ is sufficiently small. Also, if $h$ is nonnegative, these solutions are strictly positive. The case $s=1$ was established in [APP00], which highlighted, in the classical case, the importance of combining perturbative techniques with variational methods: indeed, one of the two solutions branches off perturbatively in $\varepsilon$ from $u=0$, while the second solution is found by means of the Mountain Pass Theorem. The case $s\in\left(0,\frac12\right]$ was already established, with different methods, in [DMV17] (actually, in [DMV17] it was erroneously believed that the method would have carried through all the fractional cases $s\in(0,1)$, so, in a sense, the results presented here correct and complete the ones in [DMV17]).

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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.

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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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arXiv2021

(Non)local logistic equations with Neumann conditions

We consider here a problem of population dynamics modeled on a logistic equation with both classical and nonlocal diffusion, possibly in combination with a pollination term. The environment considered is a niche with zero-flux, according to a new type of Neumann condition. We discuss the situations that are more favorable for the survival of the species, in terms of the first positive eigenvalue. Quite surprisingly, the eigenvalue analysis for the one dimensional case is structurally different than the higher dimensional setting, and it sensibly depends on the nonlocal character of the dispersal. The mathematical framework of this problem takes into consideration the equation $$ -αΔu +β(-Δ)^su =(m-μu)u+τ\;J\star u \qquad{\mbox{in }}\; Ω,$$ where $m$ can change sign. This equation is endowed with a set of Neumann condition that combines the classical normal derivative prescription and the nonlocal condition introduced in [S. Dipierro, X. Ros-Oton, E. Valdinoci, Rev. Mat. Iberoam. (2017)]. We will establish the existence of a minimal solution for this problem and provide a throughout discussion on whether it is possible to obtain non-trivial solutions (corresponding to the survival of the population). The investigation will rely on a quantitative analysis of the first eigenvalue of the associated problem and on precise asymptotics for large lower and upper bounds of the resource. In this, we also analyze the role played by the optimization strategy in the distribution of the resources, showing concrete examples that are unfavorable for survival, in spite of the large resources that are available in the environment.

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arXiv2020

Linear theory for a mixed operator with Neumann conditions

We consider here a new type of mixed local and nonlocal equation under suitable Neumann conditions. We discuss the spectral properties associated to a weighted eigenvalue problem and present a global bound for subsolutions. The Neumann condition that we take into account comprises, as a particular case, the one that has been recently introduced in [S. Dipierro, X. Ros-Oton, E. Valdinoci, Rev. Mat. Iberoam. (2017)]. Also, the results that we present here find a natural application to a logistic equation motivated by biological problems that has been recently considered in [S. Dipierro, E. Proietti Lippi, E. Valdinoci, preprint (2020)].

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arXiv2020

Linking over cones for the Neumann Fractional $p-$Laplacian

We consider nonlinear problems governed by the fractional $p-$Laplacian in presence of nonlocal Neumann boundary conditions. We face two problems. First: the $p-$superlinear term may not satisfy the Ambrosetti-Rabinowitz condition. Second, and more important: although the topological structure of the underlying functional reminds the one of the linking theorem, the nonlocal nature of the associated eigenfunctions prevents the use of such a classical theorem. For these reasons, we are led to adopt another approach, relying on the notion of linking over cones.

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arXiv2019

Neumann fractional $p-$Laplacian: eigenvalues and existence results

We develop some properties of the $p-$Neumann derivative for the fractional $p-$Laplacian in bounded domains with general $p>1$. In particular, we prove the existence of a diverging sequence of eigenvalues and we introduce the evolution problem associated to such operators, studying the basic properties of solutions. Finally, we study a nonlinear problem with source in absence of the Ambrosetti-Rabinowitz condition.

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