Nicola Abatangelo

A Notion of Nonlocal Curvature

We consider a nonlocal (or fractional) curvature and we investigate similarities and differences with respect to the classical local case. In particular, we show that the nonlocal mean curvature can be seen as an average of suitable nonlocal directional curvatures and there is a natural asymptotic convergence to the classical case. Nevertheless, different from the classical cases, minimal and maximal nonlocal directional curvatures are not in general attained at perpendicular directions and, in fact, one can arbitrarily prescribe the set of extremal directions for nonlocal directional curvatures. Also the classical directional curvatures naturally enjoy some linear properties that are lost in the nonlocal framework. In this sense, nonlocal directional curvatures are somewhat intrinsically nonlinear.

Very large solutions for the fractional Laplacian: Towards a fractional Keller-Osserman condition

Abstract We look for solutions of ( - △ ) s ⁢ u + f ⁢ ( u ) = 0 {{\left(-\triangle\right)}^{s}u+f(u)=0} in a bounded smooth domain Ω, s ∈ ( 0 , 1 ) {s\in(0,1)} , with a strong singularity at the boundary. In particular, we are interested in solutions which are L 1 ⁢ ( Ω ) {L^{1}(\Omega)} and higher order with respect to dist ( x , ∂ Ω ) s - 1 {\operatorname{dist}(x,\partial\Omega)^{s-1}} . We provide sufficient conditions for the existence of such a solution. Roughly speaking, these functions are the real fractional counterpart of large solutions in the classical setting.

Integral representation of solutions to higher-order fractional Dirichlet problems on balls

We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power

Large

s>0

s

of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders using explicit Poisson-type kernels and a new notion of higher-order boundary operator, which recovers normal derivatives if

-harmonic functions and boundary blow-up solutions for the fractional Laplacian

s

Getting acquainted with the fractional Laplacian

is a natural number. Our results unify and generalize previous approaches in the study of polyharmonic operators and fractional Laplacians. As applications, we show a novel characterization of

Green function and Martin kernel for higher-order fractional Laplacians in balls

s

Nonhomogeneous boundary conditions for the spectral fractional Laplacian

-harmonic functions in terms of Martin kernels, a higher-order fractional Hopf Lemma, and examples of positive and sign-changing Green functions.