27 citations · 54 across the 13 of their papers we have counts for
21 papers · 1 filter
Rigidity of balls in the solid mean value property for polyharmonic functions
Nicola Abatangelo
We show that balls are the only open bounded domains for which the mean value formula for polyharmonic functions holds. We do so by adapting an argument of Ü. Kuran for harmonic fu…
Optimal boundary regularity for mixed local and nonlocal equations
Nicola Abatangelo, Elisa Affili, Matteo Cozzi
We provide sharp boundary regularity estimates for solutions to elliptic equations driven by an integro-differential operator obtained as the sum of a Laplacian with a nonlocal ope…
A gentle invitation to the fractional world
Nicola Abatangelo, Serena Dipierro, Enrico Valdinoci
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 t…
An asymptotic relationship between Lane-Emden systems and the 1-bilaplacian equation
Nicola Abatangelo, Alberto Saldaña, Hugo Tavares
Consider the following Lane-Emden system with Dirichlet boundary conditions: \[ -ΔU = |V|^{β-1}V,\ -ΔV = |U|^{α-1}U \text{ in }Ω,\qquad U=V= 0 \text{ on }\partial Ω, \] in a bounde…
Oscillatory Phenomena for Higher-Order Fractional Laplacians
Nicola Abatangelo, Sven Jarohs
We collect some peculiarities of higher-order fractional Laplacians , , with special attention to the range , which show their oscillatory nature. These inc…
On the shape of the first fractional eigenfunction
Nicola Abatangelo, Sven Jarohs
We show that the first eigenfunction of the fractional Laplacian , , is superharmonic in the unitary ball up to dimension . To this aim, we also rely on…
A Hopf lemma for the regional fractional Laplacian
Nicola Abatangelo, Mouhamed Moustapha Fall, Remi Yvant Temgoua
We provide a Hopf boundary lemma for the regional fractional Laplacian , with a bounded open set. More precisely, given a pointwise or weak sup…
An elliptic boundary value problem with fractional nonlinearity
Nicola Abatangelo, Matteo Cozzi
We investigate existence and uniqueness of solutions to second-order elliptic boundary value problems containing a power nonlinearity applied to a fractional Laplacian. We detect t…
Fractional Laplacians on ellipsoids
Nicola Abatangelo, Sven Jarohs, Alberto Saldaña
We show explicit formulas for the evaluation of (possibly higher-order) fractional Laplacians of some functions supported on ellipsoids. In particular, we derive the explicit expre…
Singular boundary behaviour and large solutions for fractional elliptic equations
Nicola Abatangelo, David Gómez-Castro, Juan Luis Vázquez
We show that the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains strongly differs from the one of solutions to elliptic problems modelled…
Obstacle problems for integro-differential operators: Higher regularity of free boundaries
Nicola Abatangelo, Xavier Ros-Oton
We study the higher regularity of free boundaries in obstacle problems for integro-differential operators. Our main result establishes that, once free boundaries are , the…
Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions
Nicola Abatangelo, Serena Dipierro, Mouhamed Moustapha Fall +2
We present explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the ope…
A remark on nonlocal Neumann conditions for the fractional Laplacian
Nicola Abatangelo
We show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfy…
Getting acquainted with the fractional Laplacian
Nicola Abatangelo, Enrico Valdinoci
These are the handouts of an undergraduate minicourse at the Università di Bari, in the context of the 2017 INdAM Intensive Period "Contemporary Research in elliptic PDEs and relat…
Positive powers of the Laplacian: from hypersingular integrals to boundary value problems
Nicola Abatangelo, Sven Jarohs, Alberto Saldaña
Any positive power of the Laplacian is related via its Fourier symbol to a hypersingular integral with finite differences. We show how this yields a pointwise evaluation which is m…
Integral representation of solutions to higher-order fractional Dirichlet problems on balls
Nicola Abatangelo, Sven Jarohs, Alberto Saldaña
We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power of the Laplacian. We are able to prescribe values…
On the maximum principle for higher-order fractional Laplacians
Nicola Abatangelo, Sven Jarohs, Alberto Saldaña
We study existence, regularity, and qualitative properties of solutions to linear problems involving higher-order fractional Laplacians for any . Using the nonlocal p…
Large solutions for fractional Laplacian operators
Nicola Abatangelo
The thesis studies linear and semilinear Dirichlet problems driven by different fractional Laplacians. The boundary data can be smooth functions or also Radon measures. The goal is…
Nonhomogeneous boundary conditions for the spectral fractional Laplacian
Nicola Abatangelo, Louis Dupaigne
We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the bo…
Very large solutions for the fractional Laplacian: towards a fractional Keller-Osserman condition
Nicola Abatangelo
We look for solutions of in a bounded smooth domain , , with a strong singularity at the boundary. In particular, we are interested in solutions w…
Large -harmonic functions and boundary blow-up solutions for the fractional Laplacian
Nicola Abatangelo
We present a notion of weak solution for the Dirichlet problem driven by the fractional Laplacian, following the Stampacchia theory. Then, we study semilinear problems on bounded d…