Fractional Laplacian

Gudrun talks with Debajyoti Choudhuri. He is staying at KIT as a
short term guest. He is Associate Professor in the School of Basic
Sciences at IIT Bhubaneswar, India. He did his M.Sc. and Ph.D. in
Mathematics at the University of Hyderabad. His research interest
lies in the analysis of elliptic PDEs using Functional Analytic and
topological methods. In this he touches and has a slight overlap
with the research of Gudrun. The conversation starts with the
discussion about a small paper which Debajyoti put on the archiv.
It is about understanding how to work with the Fractional
Laplacian. This means extending the classical Laplace operator Δ to
non-integer powers. This operator is the main part in PDEs which
model, e.g, anomalous diffusion, probability theory, image
processing, finance, and nonlocal mechanics.

(-Δ)s, where 0 < s < 1.
What makes It different to the ordinary Laplacian? While the
traditional Laplace operator is local, i.e. it depends only on
values of u and its derivatives near x, the fractional Laplacian is
nonlocal, it depends on values of u everywhere in space. Thus, for
the analytical and numerical treatment one needs very different
methods. There are several possible definitions. Some of them can
be found in the Wikipedia article which is cited below. On ℝn, the
cleanest definition is the Fourier definition which follows the
idea: Take the Fourier transform. Multiply by |ξ|2s. Transform
back. In the short paper which is discussed the singular integral
definition is used:

For 0 < s < 1:
(-Δ)^s u(x) = C(n,s) PV ∫ [u(x) - u(y)] / |x - y|^(n + 2s) dy

This makes the nonlocality explicit: every point y contributes to
the value at x.
The method central in studying Laplace problems is variational.
It considers an (infinite) family of generalised problems and works
on the existence of so-called weak solutions. These problems are
formulated with the help of Sobolev spaces. The weak solution for
the Laplace problem is an element of the space H1=W1,2. This means
the solution and its (generalised) gradient are bounded in L2 in
the domain in which the problem is solved. This has physical
meaning and due to known properties (embedding) of Sobolev spaces
the pointwise (strong) solutions often can be constructed when
enough regularitiy of the weak solutions is proved. Fractional
Laplacians naturally live in fractional Sobolev spaces. These are
not that easy to connect to physical properties and a few of the
equivalent definitions in the context of classical Sobolev spaces
are not equivalent any more everywhere.

Common approaches for numerics for PDEs including the fractional
Laplacian are:


Fourier spectral methods (periodic domains)

Finite element methods for fractional PDEs

Matrix-function methods (As)

Caffarelli–Silvestre extension methods

Quadrature approximations of singular integrals

The Extension trick introduced by Caffarelli and Silvestre in
2007 (their original paper is cited below) is also discussed as
part of the short note. p-laplacian augurs well in the sense
because the unicity of the definitions of the s-laplacian is still
lacking. The conversation then turns to how Debajyoti found his way
into mathematics and the topic of PDEs and how life and work feel
like in his university. More information:

Webpage of Debajyoti Choudhuri

Debajyoti Choudhuri: A quick sneak-peek at the s-fractional
Laplacian operator (2022)

Wikipedia on the Fractional Laplace operator

Mateusz Kwaśnicki: Ten equivalent definitions of the
fractional Laplace operator (2015)

E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide
to the fractional Sobolev spaces, Bull. Sci. Math., 136(5),
521–573 (2012)

L. Caffarelli, L. Silvestre, An extension problem related to
the fractional Laplacian, Communications in Partial Differential
Equations, 32, 1245–1260 (2007)