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:

1. Take the Fourier transform.
2. Multiply by |ξ|^2s.
3. 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=W^1,2. 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 (A^s)
• 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)