arXiv subjects
Publications and source records attributed to Enrico Valdinoci.
Inspired by a classical identity proved by James Simons, we establish a new geometric formula in a nonlocal, possibly fractional, setting. Our formula also recovers the classical case in the limit, thus providing an approach to Simons' work that does not heavily rely on differential geometry.
In this paper we provide density estimates for a class of functions which includes all the minimizers of the energy $\mathcal{E}_s^p(u,Ω):=(1-s)\left(\frac{1}{2}\int_Ω\int_Ω\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\,dx\,dy +\int_Ω\int_{\mathbb{R}^n \setminus Ω}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\,dx\,dy\right)+\int_ΩW(u(x))\,dx,$ where $p\in (1,+\infty)$, $s \in \left(0,1\right)$ and $W$ is a double-well potential with polynomial growth $m\in \left[p,+\infty\right)$ from the minima. The nonlocal estimates obtained are uniform as $s\to1$. Moreover, making use of a $Γ$-convergence result for $\mathcal{E}_s^p$ as $s\to 1$, we obtain density estimates for the minimizers of the limit energy functional, which takes the form $\mathcal{E}_1^p(u,Ω):=\frac{K_{n,p}}{2p}\int_Ω \left|\nabla u(x)\right|^p+\int_Ω W(u(x))\,dx,$ for a suitable $K_{n,p}\in (0,+\infty)$.
We revisit a Harnack inequality for antisymmetric functions that has been recently established for the fractional Laplacian and we extend it to more general nonlocal elliptic operators. The new approach to deal with these problems that we propose in this paper leverages Bochner's relation, allowing one to relate a one-dimensional Fourier transform of an odd function with a three-dimensional Fourier transform of a radial function. In this way, Harnack inequalities for odd functions, which are essentially Harnack inequalities of boundary type, are reduced to interior Harnack inequalities.
We consider a weighted nonlocal area functional in which the coefficients do not satisfy the triangle inequality. In the context of three phase transitions, this means that one of the weights is larger than the sum of the other two, say $$σ_{-1,1} > σ_{-1,0} + σ_{0,1}.$$ We show that the energy can be reduced by covering interfaces between phases $-1$ and $1$ with a thin strip of phase $0$. Moreover, as the fractional parameter $s\nearrow1$, we prove that the nonlocal energies $Γ$-converge to a local area functional with different weights. The functional structure of this long-range interaction model is conceptually different from its classical counterpart, since the functional remains lower semicontinuous, even in the absence of the triangle inequality.
We investigate existence, uniqueness and asymptotic behavior of minimizers of a family of non-local energy functionals of the type $$ \frac{1}{4}\iint_{\mathbb{R}^{2n}\setminus (\mathbb{R}^n \setminus Ω)^2}|u(x)-u(y)|^2 K(x-y) \,dx dy + \int_ΩW(u(x)) \,dx. $$ Here, $W$ is a possibly degenerate double well potential with a polynomial control on its second derivative near the wells. Also, ${K}$ belongs to a wide class of measurable kernels and is modeled on that of the fractional Laplacian.
We study a critical problem for an operator of mixed order obtained by the superposition of a Laplacian with a fractional Laplacian. In particular, we investigate the corresponding Sobolev inequality, detecting the optimal constant, which we show that is never achieved. Moreover, we present an existence (and nonexistence) theory for the corresponding subcritical perturbation problem.
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.
We provide a new proof of the fractional version of the De Giorgi conjecture for the Allen-Cahn equation in $\mathbb{R}^2$ for the full range of exponents. Our proof combines a method introduced by A. Farina in 2003 with the $s$-harmonic extension of the fractional Laplacian in the half-space $\mathbb{R}^{3}_+$ introduced by L. Caffarelli and L. Silvestre in 2007. We also provide a representation formula for finite-energy weak solutions of a class of weighted elliptic partial differential equations in the half-space $\mathbb{R}^{n+1}_+$ under Neumann boundary conditions. This generalizes the $s$-harmonic extension of the fractional Laplacian and allows us to relate a general problem in the extended space with a nonlocal problem on the trace.
In the Euclidean setting, Napoleon's Theorem states that if one constructs an equilateral triangle on either the outside or the inside of each side of a given triangle and then connects the barycenters of those three new triangles, the resulting triangle happens to be equilateral. The case of spherical triangles has been recently shown to be different: on the sphere, besides equilateral triangles, a necessary and sufficient condition for a given triangle to enjoy the above Napoleonic property is that its congruence class should lie on a suitable surface (namely, an ellipsoid in suitable coordinates). In this article we show that the hyperbolic case is significantly different from both the Euclidean and the spherical setting. Specifically, we establish here that the hyperbolic plane does not admit any Napoleonic triangle, except the equilateral ones. Furthermore, we prove that iterated Napoleonization of any triangle causes it to become smaller and smaller, more and more equilateral and converge to a single point in the limit.
We consider the mixed local/nonlocal semilinear equation \begin{equation*} -ε^{2}Δu +ε^{2s}(-Δ)^s u +u=u^p\qquad \text{in } Ω \end{equation*} with zero Dirichlet datum, where $ε>0$ is a small parameter, $s\in(0,1)$, $p\in(1,\frac{n+2}{n-2})$ and $Ω$ is a smooth, bounded domain. We construct a family of solutions that concentrate, as $ε\rightarrow 0$, at an interior point of $Ω$ having uniform distance to $\partialΩ$ (this point can also be characterized as a local minimum of a nonlocal functional). In spite of the presence of the Laplace operator, the leading order of the relevant reduced energy functional in the Lyapunov-Schmidt procedure is polynomial rather than exponential in the distance to the boundary, in light of the nonlocal effect at infinity. A delicate analysis is required to establish some uniform estimates with respect to $ε$, due to the difficulty caused by the different scales coming from the mixed operator.
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.
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.
This book is intended as a self-contained introduction to selected topics in the fractional world, focusing particularly on aspects that arise in the study of equations driven by the fractional Laplacian. The scope of this work is not intended to be exhaustive or all-encompassing. We have chosen topics that we believe will appeal to readers embarking on their journey into fractional analysis. It requires only fundamental calculus and a basic understanding of measure theory. In Chapter 1, we introduce the primary object of study, the fractional Laplacian. This operator appears in diverse contexts, prompting multiple definitions and viewpoints, many of which we explore, along with some key identities. A notable distinction between local and nonlocal analysis is that in the latter, explicit calculations are often impractical or impossible. There are anyway some fortunate exceptions which are gathered in Chapter 2, providing useful and instructive examples. Chapter 3 presents an introduction to the important aspect of Liouville-type results. A large portion of this book is devoted to the regularity theory of solutions in Lebesgue spaces. Chapter 4 examines global solutions using Riesz and Bessel potential analysis, capturing the impact of both low and high frequencies on smoothness, decay, and oscillations. These spaces are also flexible enough to provide, as a byproduct, a solid regularity theory in the more commonly used fractional Sobolev spaces. In Chapter 5 we derive the corresponding interior regularity theory for solutions within a bounded domain using appropriate cutoffs and localization techniques. Additionally, technical appendices include auxiliary results used in key proofs.
In the setting of fractional minimal surfaces, we prove that if two nonlocal minimal sets are one included in the other and share a common boundary point, then they must necessarily coincide. This strict maximum principle is not obvious, since the surfaces may touch at an irregular point, therefore a suitable blow-up analysis must be combined with a bespoke regularity theory to obtain this result. For the classical case, an analogous result was proved by Leon Simon. Our proof also relies on a Harnack Inequality for nonlocal minimal surfaces that has been recently introduced by Xavier Cabré and Matteo Cozzi and which can be seen as a fractional counterpart of a classical result by Enrico Bombieri and Enrico Giusti. In our setting, an additional difficulty comes from the analysis of the corresponding nonlocal integral equation on a hypersurface, which presents a remainder whose sign and fine properties need to be carefully addressed.
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.
This article is concerned with ``up to $C^{2, α}$-regularity results'' about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators. First of all, an estimate on the $L^\infty$ norm of weak solutions is established for more general cases than the ones present in the literature, including here critical nonlinearities. We then prove the interior $C^{1,α}$-regularity and the $C^{1,α}$-regularity up to the boundary of weak solutions, which extends previous results by the authors [X. Su, E. Valdinoci, Y. Wei and J. Zhang, Math. Z. (2022)], where the nonlinearities considered were of subcritical type. In addition, we establish the interior $C^{2,α}$-regularity of solutions for all $s\in(0,1)$ and the $C^{2,α}$-regularity up to the boundary for all $s\in(0,\frac{1}{2})$, with sharp regularity exponents. For further perusal, we also include a strong maximum principle and some properties about the principal eigenvalue.
In this paper, we deal with the following mixed local/nonlocal Schrödinger equation \begin{equation*} \left\{ \begin{array}{ll} - Δu + (-Δ)^s u+u = u^p \quad \hbox{in $\mathbb{R}^n$,} u>0 \quad \hbox{in $\mathbb{R}^n$,} \lim\limits_{|x|\to+\infty}u(x)=0, \end{array} \right. \end{equation*} where $n\geqslant2$, $s\in (0,1)$ and $p\in\left(1,\frac{n+2}{n-2}\right)$. The existence of positive solutions for the above problem is proved, relying on some new regularity results. In addition, we study the power-type decay and the radial symmetry properties of such solutions. The methods make use also of some basic properties of the heat kernel and the Bessel kernel associated with the operator $- Δ+ (-Δ)^s$: in this context, we provide self-contained proofs of these results based on Fourier analysis techniques.
We study the higher-order asymptotic development of a nonlocal phase transition energy in bounded domains and with prescribed external boundary conditions. The energy under consideration has fractional order $2s \in (0,1)$ and a first-order asymptotic development in the $Γ$-sense as described by the fractional perimeter functional. We prove that there is no meaningful second-order asymptotic expansion and, in fact, no asymptotic expansion of fractional order $μ> 2-2s$. In view of this range value for $μ$, it would be interesting to develop a new asymptotic development for the $Γ$-convergence of our energy functional which takes into account fractional orders. The results obtained here are also valid in every space dimension and with mild assumptions on the exterior data.