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Enrico Valdinoci

Publications and source records attributed to Enrico Valdinoci.

At least 91 records · Page 5Linked to original sources
arXiv2023

Boundary continuity of nonlocal minimal surfaces in domains with singularities and a problem posed by Borthagaray, Li, and Nochetto

Differently from their classical counterpart, nonlocal minimal surfaces are known to present boundary discontinuities, by sticking at the boundary of smooth domains. It has been observed numerically by J. P. Borthagaray, W. Li, and R. H. Nochetto ``that stickiness is larger near the concave portions of the boundary than near the convex ones, and that it is absent in the corners of the square'', leading to the conjecture ``that there is a relation between the amount of stickiness on $\partialΩ$ and the nonlocal mean curvature of $\partialΩ$''. In this paper, we give a positive answer to this conjecture, by showing that the nonlocal minimal surfaces are continuous at convex corners of the domain boundary and discontinuous at concave corners. More generally, we show that boundary continuity for nonlocal minimal surfaces holds true at all points in which the domain is not better than $C^{1,s}$, with the singularity pointing outward, while, as pointed out by a concrete example, discontinuities may occur at all point in which the domain possesses an interior touching set of class $C^{1,α}$ with $α>s$.

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arXiv2023

Some perspectives on (non)local phase transitions and minimal surfaces

We present here some classical and modern results about phase transitions and minimal surfaces, which are quite intertwined topics. We start from scratch, revisiting the theory of phase transitions as put forth by Lev Landau. Then, we relate the short-range phase transitions to the classical minimal surfaces, whose basic regularity theory is presented, also in connection with a celebrated conjecture by Ennio De Giorgi. With this, we explore the recently developed subject of long-range phase transitions and relate its genuinely nonlocal regime to the analysis of fractional minimal surfaces.

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arXiv2023

On the Harnack inequality for antisymmetric $s$-harmonic functions

We prove the Harnack inequality for antisymmetric $s$-harmonic functions, and more generally for solutions of fractional equations with zero-th order terms, in a general domain. This may be used in conjunction with the method of moving planes to obtain quantitative stability results for symmetry and overdetermined problems for semilinear equations driven by the fractional Laplacian. The proof is split into two parts: an interior Harnack inequality away from the plane of symmetry, and a boundary Harnack inequality close to the plane of symmetry. We prove these results by first establishing the weak Harnack inequality for super-solutions and local boundedness for sub-solutions in both the interior and boundary case. En passant, we also obtain a new mean value formula for antisymmetric $s$-harmonic functions.

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arXiv2022

A Faber-Krahn inequality for mixed local and nonlocal operators

We consider the first Dirichlet eigenvalue problem for a mixed local/nonlocal elliptic operator and we establish a quantitative Faber-Krahn inequality. More precisely, we show that balls minimize the first eigenvalue among sets of given volume and we provide a stability result for sets that almost attain the minimum.

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arXiv2022

Symmetry and quantitative stability for the parallel surface fractional torsion problem

We study symmetry and quantitative approximate symmetry for an overdetermined problem involving the fractional torsion problem in a bounded domain $Ω\subset \mathbb R^n$. More precisely, we prove that if the fractional torsion function has a $C^1$ level surface which is parallel to the boundary $\partial Ω$ then the domain is a ball. If instead we assume that the solution is close to a constant on a parallel surface to the boundary, then we quantitatively prove that $Ω$ is close to a ball. Our results use techniques which are peculiar to the nonlocal case as, for instance, quantitative versions of fractional Hopf boundary point lemma and boundary Harnack estimates for antisymmetric functions. We also provide an application to the study of rural-urban fringes in population settlements.

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arXiv2022

Dispersive effects in a scalar nonlocal wave equation inspired by peridynamics

We study the dispersive properties of a linear equation in one spatial dimension which is inspired by models in peridynamics. The interplay between nonlocality and dispersion is analyzed in detail through the study of the asymptotics at low and high frequencies, revealing new features ruling the wave propagation in continua where nonlocal characteristics must be taken into account. Global dispersive estimates and existence of conserved functionals are proved. A comparison between these new effects and the classical local {\it scenario} is deepened also through a numerical analysis.

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arXiv2022

Classification of global solutions of a free boundary problem in the plane

We classify nontrivial, nonnegative, positively homogeneous solutions of the equation \begin{equation*} Δu=γu^{γ-1} \end{equation*} in the plane. The problem is motivated by the analysis of the classical Alt-Phillips free boundary problem, but considered here with negative exponents $γ$. The proof relies on several bespoke results for ordinary differential equations.

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arXiv2022

Regularity results for solutions of mixed local and nonlocal elliptic equations

We consider the mixed local-nonlocal semi-linear elliptic equations driven by the superposition of Brownian and Lévy processes \begin{equation*} \left\{ \begin{array}{ll} - Δu + (-Δ)^s u = g(x,u) & \hbox{in $Ω$,} u=0 & \hbox{in $\mathbb{R}^n\backslashΩ$.} \\ \end{array} \right. \end{equation*} Under mild assumptions on the nonlinear term $g$, we show the $L^\infty$ boundedness of any weak solution (either not changing sign or sign-changing) by the Moser iteration method. Moreover, when $s\in (0, \frac{1}{2}]$, we obtain that the solution is unique and actually belongs to $C^{1,α}(\overlineΩ)$ for any $α\in (0,1)$.

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arXiv2022

Efficiency functionals for the Lévy flight foraging hypothesis

We consider a forager diffusing via a fractional heat equation and we introduce several efficiency functionals whose optimality is discussed in relation to the Lévy exponent of the evolution equation. Several biological scenarios, such as a target close to the forager, a sparse environment, a target located away from the forager and two targets are specifically taken into account. The optimal strategies of each of these configurations are here analyzed explicitly also with the aid of some special functions of classical flavor and the results are confronted with the existing paradigms of the Lévy foraging hypothesis. Interestingly, one discovers bifurcation phenomena in which a sudden switch occurs between an optimal (but somehow unreliable) Lévy foraging pattern of inverse square law type and a less ideal (but somehow more secure) classical Brownian motion strategy. Additionally, optimal foraging strategies can be detected in the vicinity of the Brownian one even in cases in which the Brownian one is pessimizing an efficiency functional.

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arXiv2022

The stickiness property for antisymmetric nonlocal minimal graphs

We show that arbitrarily small antisymmetric perturbations of the zero function are sufficient to produce the stickiness phenomenon for planar nonlocal minimal graphs (with the same quantitative bounds obtained for the case of even symmetric perturbations, up to multiplicative constants). In proving this result, one also establishes an odd symmetric version of the maximum principle for nonlocal minimal graphs, according to which the odd symmetric minimizer is positive in the direction of the positive bump and negative in the direction of the negative bump.

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arXiv2022

The Fractional Malmheden Theorem

We provide a fractional counterpart of the classical results by Schwarz and Malmheden on harmonic functions. From that we obtain a representation formula for $s$-harmonic functions as a linear superposition of weighted classical harmonic functions which also entails a new proof of the fractional Harnack inequality. This proof also leads to optimal constants for the fractional Harnack inequality in the ball.

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arXiv2022

Radial symmetry of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities

We prove radial symmetry for bounded nonnegative solutions of a weighted anisotropic problem. Given the anisotropic setting that we deal with, the term "radial" is understood in the Finsler framework. In the whole space, J. Serra obtained the symmetry result in the isotropic unweighted setting. In this case we provide the extension of his result to the anisotropic setting. This provides a generalization to the anisotropic setting of a celebrated result due to Gidas-Ni-Nirenberg and such a generalization is new even for in the case of linear operators whenever the dimension is greater than 2. In proper cones, the results presented are new even in the isotropic and unweighted setting for suitable nonlinear cases. Even for the previously known case of unweighted isotropic setting, the present paper provides an approach to the problem by exploiting integral (in)equalities which is new for $N>2$: this complements the corresponding symmetry result obtained via the moving planes method by Berestycki-Pacella.

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arXiv2022

Nonlocal capillarity for anisotropic kernels

We study a nonlocal capillarity problem with interaction kernels that are possibly anisotropic and not necessarily invariant under scaling. In particular, the lack of scale invariance will be modeled via two different fractional exponents $s_1, s_2\in (0,1)$ which take into account the possibility that the container and the environment present different features with respect to particle interactions. We determine a nonlocal Young's law for the contact angle and discuss the unique solvability of the corresponding equation in terms of the interaction kernels and of the relative adhesion coefficient.

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arXiv2022

Integral operators defined "up to a polynomial''

We introduce a suitable notion of integral operators (comprising the fractional Laplacian as a particular case) acting on functions with minimal requirements at infinity. For these functions, the classical definition would lead to divergent expressions, thus we replace it with an appropriate framework obtained by a cut-off procedure. The notion obtained in this way quotients out the polynomials which produce the divergent pattern once the cut-off is removed. We also present results of stability under the appropriate notion of convergence and compatibility results between polynomials of different orders. Additionally, we address the solvability of the Dirichlet problem. The theory is developed in general in the pointwise sense. A viscosity counterpart is also presented under the additional assumption that the interaction kernel has a sign, in conformity with the maximum principle structure.

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arXiv2021

The Bernstein technique for integro-differential equations

We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully nonlinear equations of order smaller than two -- for which we prove uniform estimates as their order approaches two. Our method is robust enough to be applied to some Pucci-type extremal equations and to obstacle problems for fractional operators, although several of the results are new even in the linear case. We also raise some intriguing open questions, one of them concerning the "pure" linear fractional Laplacian, another one being the validity of one-sided second derivative estimates for Pucci-type convex equations associated to linear operators with general kernels.

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arXiv2021

Mixed local and nonlocal elliptic operators: regularity and maximum principles

We start in this paper a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.

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