arXiv subjects

Enrico Valdinoci

Publications and source records attributed to Enrico Valdinoci.

At least 109 records · Page 6Linked to original sources

arXiv2021

A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators

Given a bounded open set $Ω\subseteq{\mathbb{R}}^n$, we consider the eigenvalue problem of a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of $Ω$. We prove that the second eigenvalue $λ_2(Ω)$ is always strictly larger than the first eigenvalue $λ_1(B)$ of a ball $B$ with volume half of that of $Ω$. This bound is proven to be sharp, by comparing to the limit case in which $Ω$ consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing sequence is given by the union of two disjoint balls of half volume whose mutual distance tends to infinity.

math.AP↗

arXiv2021

A fractional glance to the theory of edge dislocations

We revisit some recents results inspired by the Peierls-Nabarro model on edge dislocations for crystals which rely on the fractional Laplace representation of the corresponding equation. In particular, we discuss results related to heteroclinic, homoclinic and multibump patterns for the atom dislocation function, the large space and time scale of the solutions of the parabolic problem, the dynamics of the dislocation points and the large time asymptotics after possible dislocation collisions.

math.AP↗

arXiv2021

$(s,p)$-harmonic approximation of functions of least $W^{s,1}$-seminorm

We investigate the convergence as $p\searrow1$ of the minimizers of the $W^{s,p}$-energy for $s\in(0,1)$ and $p\in(1,\infty)$ to those of the $W^{s,1}$-energy, both in the pointwise sense and by means of $Γ$-convergence. We also address the convergence of the corresponding Euler-Lagrange equations, and the equivalence between minimizers and weak solutions. As ancillary results, we study some regularity issues regarding minimizers of the $W^{s,1}$-energy.

math.AP↗

arXiv2021

Global gradient estimates for a general type of nonlinear parabolic equations

We provide global gradient estimates for solutions to a general type of nonlinear parabolic equations, possibly in a Riemannian geometry setting. Our result is new in comparison with the existing ones in the literature, in light of the validity of the estimates in the global domain, and it detects several additional regularity effects due to special parabolic data. Moreover, our result comprises a large number of nonlinear sources treated by a unified approach, and it recovers many classical results as special cases.

math.AP↗

arXiv2021

Qualitative aspects in nonlocal dynamics

In this paper we investigate, through numerical studies, the dynamical evolutions encoded in a linear one-dimensional nonlocal equation arising in peridynamcs. The different propagation regimes ranging from the hyperbolic to the dispersive, induced by the nonlocal feature of the equation, are carefully analyzed. The study of an initial value Riemann-like problem suggests the formation of a singularity.

nlin.PS↗

arXiv2021

Description of an ecological niche for a mixed local/nonlocal dispersal: an evolution equation and a new Neumann condition arising from the superposition of Brownian and Lévy processes

We propose here a motivation for a mixed local/nonlocal problem with a new type of Neumann condition. Our description is based on formal expansions and approximations. In a nutshell, a biological species is supposed to diffuse either by a random walk or by a jump process, according to prescribed probabilities. If the process makes an individual exit the niche, it must come to the niche right away, by selecting the return point according to the underlying stochastic process. More precisely, if the random particle exits the domain, it is forced to immediately reenter the domain, and the new point in the domain is chosen randomly by following a bouncing process with the same distribution as the original one. By a suitable definition outside the niche, the density of the population ends up solving a mixed local/nonlocal equation, in which the dispersion is given by the superposition of the classical and the fractional Laplacian. This density function satisfies two types of Neumann conditions, namely the classical Neumann condition on the boundary of the niche, and a nonlocal Neumann condition in the exterior of the niche.

math.AP↗

arXiv2021

A quantitative rigidity result for a two-dimensional Frenkel-Kontorova model

We consider a Frenkel-Kontorova system of harmonic oscillators in a two-dimensional Euclidean lattice and we obtain a quantitative estimate on the angular function of the equilibria. The proof relies on a PDE method related to a classical conjecture by E. De Giorgi, also in view of an elegant technique based on complex variables that was introduced by A. Farina. In the discrete setting, a careful analysis of the reminders is needed to exploit this type of methodologies inspired by continuum models.

math.AP↗

arXiv2021

Global gradient estimates for nonlinear parabolic operators

We consider a parabolic equation driven by a nonlinear diffusive operator and we obtain a gradient estimate in the domain where the equation takes place. This estimate depends on the structural constants of the equation, on the geometry of the ambient space and on the initial and boundary data. As a byproduct, one easily obtains a universal interior estimate, not depending on the parabolic data. The setting taken into account includes sourcing terms and general diffusion coefficients. The results are new, to the best of our knowledge, even in the Euclidean setting, though we treat here also the case of a complete Riemannian manifold.

math.AP↗

arXiv2021

On divergent fractional Laplace equations

We consider the divergent fractional Laplace operator presented in [Dipierro-Savin-Valdinoci, Rev. Mat. Iberoam.] and we prove three types of results. Firstly, we show that any given function can be locally shadowed by a solution of a divergent fractional Laplace equation which is also prescribed in a neighborhood of infinity. Secondly, we take into account the Dirichlet problem for the divergent fractional Laplace equation, proving the existence of a solution and characterizing its multiplicity. Finally, we take into account the case of nonlinear equations, obtaining a new approximation results. These results maintain their interest also in the case of functions for which the fractional Laplacian can be defined in the usual sense.

math.AP↗

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.

math.AP↗

arXiv2020

Non-symmetric stable operators: regularity theory and integration by parts

We study solutions to $Lu=f$ in $Ω\subset\mathbb R^n$, being $L$ the generator of any, possibly non-symmetric, stable Lévy process. On the one hand, we study the regularity of solutions to $Lu=f$ in $Ω$, $u=0$ in $Ω^c$, in $C^{1,α}$ domains~$Ω$. We show that solutions $u$ satisfy $u/d^γ\in C^{\varepsilon_\circ}\big(\overlineΩ\big)$, where $d$ is the distance to $\partialΩ$, and $γ=γ(L,ν)$ is an explicit exponent that depends on the Fourier symbol of operator $L$ and on the unit normal $ν$ to the boundary $\partialΩ$. On the other hand, we establish new integration by parts identities in half spaces for such operators. These new identities extend previous ones for the fractional Laplacian, but the non-symmetric setting presents some new interesting features. Finally, we generalize the integration by parts identities in half spaces to the case of bounded $C^{1,α}$ domains. We do it via a new efficient approximation argument, which exploits the Hölder regularity of $u/d^γ$. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian.

math.AP↗

arXiv2020

(Dis)connectedness of nonlocal minimal surfaces in a cylinder and a stickiness property

We consider nonlocal minimal surfaces in a cylinder with prescribed datum given by the complement of a slab. We show that when the width of the slab is large the minimizers are disconnected and when the width of the slab is small the minimizers are connected. This feature is in agreement with the classical case of the minimal surfaces. Nevertheless, we show that when the width of the slab is large the minimizers are not flat discs, as it happens in the classical setting, and, in particular, in dimension $2$ we provide a quantitative bound on the stickiness property exhibited by the minimizers. Moreover, differently from the classical case, we show that when the width of the slab is small then the minimizers completely adhere to the side of the cylinder, thus providing a further example of stickiness phenomenon.

math.AP↗

arXiv2020

Minimisers of a fractional seminorm and nonlocal minimal surfaces

The recent literature has intensively studied two classes of nonlocal variational problems, namely the ones related to the minimisation of energy functionals that act on functions in suitable Sobolev-Gagliardo spaces, and the ones related to the minimisation of fractional perimeters that act on measurable sets of the Euclidean space. In this article, we relate these two types of variational problems. In particular, we investigate the connection between the nonlocal minimal surfaces and the minimisers of the $W^{s,1}$-seminorm. In particular, we show that a function is a minimiser for the fractional seminorm if and only if its level sets are minimisers for the fractional perimeter, and that the characteristic function of a nonlocal minimal surface is a minimiser for the fractional seminorm; we also provide an existence result for minimisers of the fractional seminorm, an explicit non-uniqueness example for nonlocal minimal surfaces, and a Yin-Yang result describing the full and void patterns of nonlocal minimal surfaces.

math.AP↗

arXiv2020

Minimizing cones for fractional capillarity problems

We consider a fractional version of Gauss capillarity energy. A suitable extension problem is introduced to derive a boundary monotonicity formula for local minimizers of this fractional capillarity energy. As a consequence, blow-up limits of local minimizers are shown to subsequentially converge to minimizing cones. Finally, we show that in the planar case there is only one possible fractional minimizing cone, the one determined by the fractional version of Young's law.

math.AP↗

arXiv2020

On the mean value property of fractional harmonic functions

As well known, harmonic functions satisfy the mean value property, namely the average of the function over a ball is equal to its value at the center. This fact naturally raises the question on whether this is a characterizing feature of balls, namely whether a set for which all harmonic functions satisfy the mean value property is necessarily a ball. This question was investigated by several authors, and was finally elegantly, completely and positively settled by Ülkü Kuran, with an artful use of elementary techniques. This classical problem has been recently fleshed out by Giovanni Cupini, Nicola Fusco, Ermanno Lanconelli and Xiao Zhong who proved a quantitative stability result for the mean value formula, showing that a suitable "mean value gap" (measuring the normalized difference between the average of harmonic functions on a given set and their pointwise value) is bounded from below by the Lebesgue measure of the "gap" between the set and the ball (and, consequently, by the Fraenkel asymmetry of the set). That is, if a domain "almost" satisfies the mean value property, then it must be necessarily close to a ball. Here we investigate the nonlocal counterparts of these results. In particular we will prove a classification result and a stability result, establishing that: if fractional harmonic functions enjoy a suitable exterior average property for a given domain, then the domain is necessarily a ball, a suitable "nonlocal mean value gap" is bounded from below by an appropriate measure of the difference between the set and the ball. Differently from the classical case, some of our arguments rely on purely nonlocal properties, with no classical counterpart, such as the fact that "all functions are locally fractional harmonic up to a small error".

math.AP↗

arXiv2020

Semilinear elliptic equations involving mixed local and nonlocal operators

In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for concreteness we focus on the case in which the operator takes the form $-Δ+(-Δ)^s$, with $s\in(0,1)$. We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons.

math.AP↗

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)].

math.AP↗

arXiv2020

A Serrin-type problem with partial knowledge of the domain

We present a quantitative estimate for the radially symmetric configuration concerning a Serrin-type overdetermined problem for the torsional rigidity in a bounded domain $Ω$, when the equation is known on $Ω\setminus \overlineω$ only, for some open subset $ω\Subset Ω$. The problem has concrete motivations in optimal heating with malfunctioning, laminar flows and beams with small inhomogeneities.

math.AP↗