Fractional-type problems and Sobolev spaces

A description of my research in non-local problems.

On this page I present my lines of research on the non-local problems of fractional Laplacian type, and related topics. Most of the work presented was developed in collaboration with J.L. Vázquez (U. Autónoma de Madrid) and other colleagues.

The fractional Laplacian of order \(s\) in the whole Euclidean space can be seen either as the operator of Fourier simbol \(\mid \xi \mid ^{2s}\) , or as an operator with singular kernel \(\mid y \mid^{-n-2s}\) , i.e.

\[(-\Delta)^s u (x) = \mathcal F^{-1} \left[ |\xi|^{2s} \mathcal F[u] \right] (x) = c_{n,s} \lim_{\varepsilon \to 0} \int_{\mathbb R^d \setminus B_\varepsilon (x) } \frac{u(x) - u(y)}{|x-y|^{n+2s}} {\mathrm d} y,\]

For a detailed explanation on the definition on the whole space and bounded domains, please visit

Fractional Laplacian(s)

Linear equation with singular potential

The first work with Vázquez continues the line of problems with singular potentials. Together with Díaz we published a first article [1] for the restricted operator with value \(0\) outside the domain where we dealt with the problem

\[(-\Delta)^s_{\text{RFL}} u + V u = f \text{ in } \Omega \qquad \textrm{and} \qquad u = 0 \text{ in } \Omega^c\]

through solutions of a very weak type. In this work we were mostly interested in potentials which are singular towards the boundary. Later, we we extend the results to the rest of \((-\Delta)^s\) in [2]. Furthermore, we cover also existence/non-existence caused by interior singularities of \(V\) and \(f\), which is even allowed to be a measure.

For this last article, we used the theory of dual weak solutions. The solutions of the Laplace problem \(L u = f\) for these operators can be expressed by integration with kernels \(\mathbb G\) that belong to a family of functions with two parameters. By replacing \(\varphi = \mathrm G [\psi]\) we can write

\[\int_\Omega u \psi + \int_\Omega V \mathrm G [\psi] = \int_\Omega f \mathrm G[\psi ] , \qquad \forall \psi \in L^\infty_c (\Omega).\]

This extends the notion of very weak solution introduced by Brezis in the 1970s.

Singular boundary data

The work on weak dual formulations led us to realize that there were some subtleties of the theory that had not been studied. In recent years, several articles presented the possibility of specifying singular boundary conditions like

\[\frac {u(x)} {\mathrm{dist}(x,\partial \Omega)^{-\alpha}} \to h (z) \qquad \qquad \textrm{ as } \Omega \ni x \to z \in \partial \Omega.\]

The \(\alpha\) and techniques were specific of each example of \((-\Delta)^s\). Different authors wondered if there was a unified way to deal with these problems for the different operators. Vázquez, Abatangelo (U. Bologna), and myself collaborated in writing [3] answering this question.

The key is to build these solutions as \(\mathrm G [f_n]\) with \(f_n\) concentrating on the boundary properly. We later extended these results with H. Chan (then in ETH Zurich now at ICMAT) to Schrödinger-type problems (missing reference) and to the parabolic case (missing reference) We continue working in this direction.

Fractional Sobolev spaces and numerical analysis

I have also worked on theoretical study of the convergence of numerical methods for non-local problems in domains. The main difficulty of this study is the behaviour of the solutions at the boundary which is, at most, of the Hölder type. We observed that some of the available functional analysis techniques were not sufficient for our work.

In a first step we focused on studying the properties of fractional Sobolev spaces \(H ^ s\), which are the natural energy spaces for fractional Laplacians in domains. This work culminated in [4] We prove an extension of the translation estimate in \(W^{s,p}\), i.e.

\[\|u( \cdot + y ) - u \|_{L^p} \le C |y|^s [u]_{W^{s,p}},\]

and discuss the direct implications the consistency of Finite-Difference schemes.

Characterisation of homogeneous Sobolev spaces

One of the technical problems that arose from the previous work is the characterisation of the completion of \(C_c^\infty (\mathbb R^d)\) endowed with the \([u]_{W^{s,p}}\) norm. We tackled this problem with L. Brasco (U. de Ferrara) in [5] We use the fractional Sobolev and prove a new Morrey inequality to embed this space into subspaces of \(L^p\), \(BMO\), or \(C^\alpha\) spaces depending on \(sp - n\).

Fractional \(p\)-Laplacian

Finally, we observed that some of the techniques we had developed allowed interesting representations of the fractional \(p\)-Laplacian (the subdifferential of the \(W^{s,p}\)), i.e.

\[(-\Delta)^s_p u(x) = C(d,s,p) \textrm{P.V.} \int_{\mathbb R^d} \frac{|u(x) - u(y)|^{p-2}(u(x) - u(y))}{|x-y|^{d+sp}} d y,\]

which we present in [6] We are currently studying how to take advantage of this representation. This line continues to be very active.

References

  1. Díaz, J. I. and Gómez-Castro, D. and Vázquez, J. L.. (2018) The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach. Nonlinear Anal.. Link
  2. Gómez-Castro, D. and Vázquez, J.L.. (2019) The fractional Schrödinger equation with singular potential and measure data. Discret. Contin. Dyn. Syst. - A. Link
  3. Abatangelo, Nicola and Gómez-Castro, D. and Vázquez, J.L.. (2019) Link
  4. del Teso, Félix and Gómez-Castro, D. and Vázquez, J.L.. (2020) Estimates on translations and Taylor expansions in fractional Sobolev spaces. Nonlinear Anal.. Link
  5. Brasco, Lorenzo and Gómez-Castro, D. and Vázquez, J.L.. (2021) Characterisation of homogeneous fractional Sobolev spaces. Calculus of Variations and Partial Differential Equations. Link
  6. del Teso, Félix and Gómez-Castro, D. and Vázquez, J.L.. (2021) Three representations of the fractional p-Laplacian: Semigroup, extension and Balakrishnan formulas. Fract. Calc. Appl. Anal.. Link