Boundary singularity for
fractional elliptic and parabolic problems

Euro-japanese conference on nonlinear diffusion, ICMAT

David Gómez-Castro
https://gomezcastro.xyz

Universidad Complutense de Madrid

October 17, 2023

Co-authors

Juan Luis Vázquez

U. Autónoma de Madrid, Real Academia de Ciencias

Nicola Abatangelo

U. Bologna

Hardy Chan

U. Basel

Fractional Laplacian in \(\mathbb R^d\)

Fractional Laplacian

\[ (-\Delta)^s u (x) = {C(d,s)} \,\, \underbrace{\lim_{\varepsilon \to 0} \int_{\mathbb R^d \setminus B_\varepsilon (x) }}_{\text{ P.V. }\int _{\mathbb R^d}} \frac{u(x) - u(y)}{|x-y|^{d+2s}} d y. \]

Equivalent definitions (up-to ten [Kwasnicki, 2017])

  • The unique operator such that

    \[ \mathcal F[ (-\Delta)^s u ] = |\xi|^{2s} \mathcal F[u] \]

    i.e. the spectral fractional power of \(-\Delta\).

  • The infinitesimal generator of a Lévy process \(X_h\), characterised by “long jumps”

    \[ (-\Delta)^s u (x) = \lim_{h \to 0} \frac{ \mathbb E[ f(x) - f(x - X_h) ] }{ h } \]

Laplace equation

\[ \begin{cases} (-\Delta)^s u (x) = f(x) & \textrm{for all } x \in \mathbb R^d \\ u(x) \to 0 & \textrm{as } |x| \to \infty \end{cases} \]

Transforming the equation \(|\xi|^{2s} \widehat u(\xi) = \widehat f(\xi)\).

Formally, we can recover \(u = \mathcal F^{-1}[|\xi|^{-2s}] * f\).

For \(d > 2s\), \(\mathcal F^{-1}[|\xi|^{-2s}]\) is the Riesz kernel so we get

\[ u(x) = C(d,-2s) \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-2s}} dy. \]

For any \(d, 2s\) we can also look for this solution by the minisation of the energy functional

\[ J(u) = \frac {C(d,s)} 2 \int_{\mathbb R^d} \int_{\mathbb R^d } \frac{|u(x) - u(y)|^2}{|x-y|^{d+2s}} dx \, dy - \int_\Omega f u \]

in homogeneous fractional Sobolev spaces [Brasco, Gómez-Castro & Vázquez, 2021].

Fractional Elliptic/Heat/Porous Medium in \(\mathbb R^d\)

There is a different Porous-Medium-type fractional equation \[ \partial_t u = \mathrm{div} (u^{m-1} \nabla (-\Delta)^{-s} u) \]

We will not discuss it:

\(m = 2\) [Caffarelli & Vazquez, 2011], \(m \ne 2\) [Stan, Del Teso & Vázquez, 2016]

Numerics

Finite Difference schemes \[ (-\Delta)^s_h u(x) = \sum_{i \in \mathbb Z^d} \omega_{i} (u(x) - u(x+ih)) \]

[Jakobsen, Karlsen & La Chioma, 2008], [Biswas, Jakobsen & Karlsen, 2010],

\(d = 1\): [Huang & Oberman, 2014].

See also: [Del Teso, Endal & Jakobsen, 2018]

Discontinuous Galerkin: [Cifani, Jakobsen & Karlsen, 2011]

Fractional LaplacianS
in bounded domains

Restricted fractional Laplacian

Singular integral operator: \[ (-\Delta)^s_{\mathrm {RFL}} u (x)= C(d,s) \, \mathrm{P.V.}\int_{ \mathbb R^d } \frac{u(x) - u(y)}{|x-y|^{d+2s}} \; dy. \]

If we work only in \(\Omega\), we must prescribe \(u\) in \(\Omega^c = \mathbb R^d \setminus \Omega\).

Spectral fractional Laplacian

Operational power. \[ -\Delta \varphi_m = \lambda_m \varphi_m \textrm{ in } \Omega, \qquad \varphi_m = 0 \textrm{ on } \partial \Omega. \]

one defines \[ u(x) = \sum_{m=1}^{+\infty} {u_m} \varphi_m(x) \qquad \longmapsto \qquad (-\Delta)_{\mathrm{SFL}}^s u (x) = \sum_{m=1}^{+\infty} \lambda_m^s u_m \varphi_m (x). \]

The “boundary condition” is \(u = 0\) on \(\partial \Omega\).

Censored fractional Laplacian (CFL)

For \(s > \frac 1 2\) \[\begin{equation} \tag{CFL} (-\Delta)^s_{\mathrm{CFL}} u (x) = C(d,s) \, \mathrm{P.V.} \int_{ \Omega } \frac{u(x) - u(y)}{|x-y|^{n+2s}} \; dy, \end{equation}\]

We do not integrate over \(\Omega^c\), so it makes sense to pick simply \(u=0\) on \(\partial\Omega.\)

Laplace equation

\[ \begin{dcases} \mathcal L u = f & \Omega \\ u = 0 & \partial \Omega \text{ or } \mathbb R^d \setminus \overline \Omega \\ \end{dcases} \]

We observe

  • \(\mathcal L = (-\Delta)^s_{\mathrm{RFL}}, (-\Delta)^s_{\mathrm{CFL}}\) are sub-differentials of energies \[ J(u) = \int_A \int_A \frac{|u(x) - u(y)|^2 }{|x-y|^{d+2s}} dx \, dy. \]

  • \(\mathcal L = (-\Delta)^s_{\mathrm{SFL}}\) is just a power.

    The inverse is naturally \((-\Delta_\Omega)^{-s}\), and in works between “powers” of \(H_0^1(\Omega)\).

Self-adjoint compact operators. Furthermore \(\lambda_1 > 0\).

Nice theory of energy solutions.

Higher regularity: RFL [Ros-Oton & Serra, 2014], CFL [Chen, 2018], [Fall & Ros-Oton, 2021]

Green kernels

The solution of the Laplace equation \[ \begin{dcases} \mathcal L u = f & \Omega \\ u = 0 & \partial \Omega \text{ or } \mathbb R^d \setminus \overline \Omega \\ \end{dcases} \]

Allows us to define \(\mathrm G : L^2 (\Omega) \to L^2(\Omega)\) we can represent it by a kernel \[ u(x) = \mathrm G[f] (x) = \int_\Omega \mathbb G(x,y) f(y) dy. \]

The probabilistic approach provides in each of our cases a similar shape \[ \mathbb G(x,y) \asymp \frac{1}{|x-y|^{d-2s}} \left(1 \wedge \frac{\delta(x)}{|x-y|} \right)^\gamma\left(1 \wedge \frac{\delta(y)}{|x-y|} \right)^\gamma \]

Since we deal only with self-adjoint \(\mathcal L\), then \(\mathbb G(x,y) = \mathbb G(y,x)\).

The probabilistic approach provides in each of our cases a similar shape \[ \mathbb G(x,y) \asymp \frac{1}{|x-y|^{d-2s}} \left(1 \wedge \frac{\delta(x)}{|x-y|} \right)^\gamma\left(1 \wedge \frac{\delta(y)}{|x-y|} \right)^\gamma \]

And \(\gamma \in (0,1]\) depends on the setting

Numerics

Finite Differences

  • For the RFL and CFL we can re-use the weights of the whole space.

  • For the SFL: Make \(A\) numerical matrix for \(-\Delta\) problem and take \(A^s\): \[ \textrm{e.g. } A = \begin{pmatrix} 2 & -1 & \\ -1 & 2 & - 1 \\ & \ddots & \\ &&-1&2 \end{pmatrix} \]

Finite Elements

Functional set-up

Optimal set of data

Given that \(\mathbb G(x,y) \asymp \frac{1}{|x-y|^{d-2s}} \left(1 \wedge \frac{\delta(x)}{|x-y|} \right)^\gamma\left(1 \wedge \frac{\delta(y)}{|x-y|} \right)^\gamma\)

For \(K \Subset \Omega\) \[ \int_K |G(f)| \le C_K \int_\Omega |f| \delta^\gamma. \]

So \(G: L^1 (\Omega, \delta^\gamma) \to L^1_{loc} (\Omega)\)

If \(f \ge 0\) then, using the kernel estimates \[ \tag{Hopf} G[f] (x) \ge c \delta^\gamma(x) \int_\Omega f(y) \delta(y)^\gamma dy. \]

If \(0\le f \notin L^1 (\Omega, \delta^\gamma)\), then \(G[f] = +\infty\).

Laplace equation
Weak dual formulation

Multiplying by \(\varphi\) and integrating \[ \int_\Omega ( \mathcal L u ) \varphi = \int_\Omega f \varphi \]

If \(\varphi\) is in the suitable class of homogeneous boundary conditions \[ \int_\Omega u ( \mathcal L \varphi )= \int_\Omega f \varphi \]

Choosing the set of test functions \(\varphi\) depends on the problem.

In particular, if we take \(\varphi = \mathrm{G} (\psi)\) we a weak dual solution if \(u \in L^1_{loc} (\Omega)\) and

\[\begin{equation} \tag{WDF} \int_\Omega u \psi = \int_\Omega f \mathrm{G} (\psi) \qquad \forall \psi \in L^\infty_c(\Omega) \end{equation}\]

See [Bonforte, Sire & Vázquez, 2015], [Bonforte, Figalli & Vázquez, 2018]

Large solutions

Definition

By large solution we mean solutions that satisfy \[ \mathcal L u = F(x,u) \text{ in } \Omega \]

By large solution we mean \(\| u \|_{L^\infty} = \infty\).

We are mostly here in boundary blow-up: \[ u(x) \to \infty \text{ as } \mathrm{dist}(x, \partial \Omega) \to 0. \]

From now on

\[ \delta(x) = \mathrm{dist}(x, \partial \Omega) . \]

For the usual Laplacian

For the problems \[ -\Delta u + F(u) = 0 \]

Keller-Osserman condition:

Canonical example: \(-\Delta u + u^p = 0\) with certain \(p\).

This kind of singular behaviour exists also in the fractional case: [Chen & Véron, 2014], [Abatangelo, 2015], [Nguyen & Véron, 2018], [Chen & Véron, 2022]

Large solutions with boundary blow-up for \(\mathcal L\)

[Abatangelo, GC & Vázquez, 2022] denoting \(\delta(x) = \mathrm{dist}(x, \partial \Omega)\)

We can compute \(\mathrm G[\delta^\beta] \asymp \delta^\alpha\) with

\[ \mathrm G[\delta^\beta] \asymp \begin{dcases} \delta^\gamma & \beta +2s > \gamma \\ \delta^{\gamma} \log|\delta| & \beta +2s = \gamma \\ \delta^{\beta + 2s} & \beta +2s < \gamma \\ &\text{ and } \beta > -1-\gamma \\ \infty & \beta \le -1-\gamma \end{dcases} \]

\(\infty \not \equiv G[\delta^\beta] \notin L^\infty \iff -1-\gamma < \beta < -2s\).

For \(-\Delta\) we have \(\gamma = 1 = s\). Either \(G[\delta^\beta] \in L^\infty\) or \(G[\delta^\beta] \equiv + \infty\)

Large solution of the Dirichlet problem for RFL

Large solution of the Dirichlet problem for SFL