The Spatially Variant Fractional Laplacian

The goal of this work is twofold: (i) introduce the spectral fractional Laplacian $(-\Delta)^{s(\cdot)}$ associated with a homogeneous Dirichlet condition on a bounded domain $\Omega\subset\mathbb{R}^{N}$, $N\geq 1$, in the case the fractional order $s(\cdot)$ is spatially variable and possibly attains the values $0$ and $1$; (ii) study the well-posedness of the equation

for some classes of data $h$, and where $v=0$ is understood in an appropriate sense.

Motivated by the extension approach in $\mathbb{R}^{N}$ by Caffarelli and Silvestre [5], or in bounded domains by Stinga and Torrea [17], we define $(-\Delta)^{s(\cdot)}$ to be the Lagrange multiplier associated to a suitable variational problem defined in an extended domain, for measurable functions $s(\cdot)$ with range contained in the interval $[0,1]$. For a general class of functions $s(\cdot)$, the domain of $(-\Delta)^{s(\cdot)}$ can be identified with a quotient space $\mathscr{X}(\Omega,w)$ involving weighted Sobolev spaces,

where $\mathcal{C}=\Omega\times(0,+\infty)$ is the open semi-infinite cylinder (the extended domain) with base $\Omega$, and $w$ is a specific weight function. Roughly speaking, the spaces $\mathscr{L}_{0,L}^{1,2}(\mathcal{C},w)$ and $\mathscr{L}_{0}^{1,2}(\mathcal{C},w)$ are composed of functions that vanish on the lateral boundary of $\mathcal{C}$, and on the whole boundary (including the base $\Omega$), respectively. Equation (1.1) is then solvable for every $h$ in the dual space of $\mathscr{X}(\Omega,w)$. For a smaller class of $s(\cdot)$, the domain can be identified as a subset of a weighted Lebesgue space $L^{2}(\Omega,\tilde{w})$ for some function $\tilde{w}$, and the equation (1.1) is solvable when the right hand side is in $L^{2}(\Omega,\tilde{w})$. For an even smaller class of functions $s(\cdot)$, this result is further improved since the domain of $(-\Delta)^{s(\cdot)}$ is identified with a subset of a weighted Sobolev space of functions with spatially variable smoothness, related to $s(\cdot)$.

The main application that has motivated this work, in addition to the natural theoretical interest, is the recent paper [2]. There, initial results on an extension approach in Hilbert spaces on an open cylinder with base $\Omega$ are given. However, the authors stopped short of defining $(-\Delta)^{s(\cdot)}$ due to the lack of a proper functional framework. The current paper aims to fill this gap. It is worth mentioning that none of the existing results in the literature are applicable to our case and new PDE and variational analysis tools are needed to study the current situation. For example, the extension approaches in [5, 17] assume $s\in(0,1)$ to be a constant and avoid the extreme cases of $0$ and $1$. In this setting, the nonlocal problem $(-\Delta)^{s}v=h$ in $\Omega$, where $(-\Delta)^{s}$ is the $s$-power of the realization of $-\Delta$ in $L^{2}(\Omega)$ with zero Dirichlet boundary conditions, can be equivalently formulated as a local one on a Sobolev space with a Muckenhoupt weight. On the other hand, our $s(\cdot)$ is a function which is allowed to touch the extreme cases 0 and 1 and therefore, the associated weights do not fulfill the Muckenhoupt property [2, Proposition 1]. In particular, fundamental results of type “$H=W$” or Poincaré inequalities are not known in our case, leading to a more complex functional analytic framework.

The literature concerning possible definitions of $(-\Delta)^{s(\cdot)}$ with non-constant $s$ is restricted to the stochastic processes and stochastic calculus approaches and considers always the unbounded case $\Omega=\mathbb{R}^{N}$; see the monograph [3] and the references therein. By means of the Lévy-Khintchine representation formula, and the Fourier transform, the operator is determined to be of Lévy type. However, strong additional assumptions on $s(\cdot)$ are required to show that the operator is associated to a Feller or a Markov process. To name a few, these include assuming that $s(\cdot)$ is Lipschitz continuous and satisfies $\varepsilon\leq s(\cdot)\leq 1-\varepsilon$ for some $\varepsilon\in(0,1)$; see [3, Example 3.5.9]. Neither of these restrictions are present in this work.

The paper is further motivated by several applications. The extension approach with spatially varying $s(\cdot)$ has shown remarkable potential in image denoising: A rough choice of $s(\cdot)$ performs better than an optimal selected regularization parameter in total variation approaches; see [2]. This is indeed a game changer, especially the variable $s(\cdot)$ approach can enable one to replace the nonlinear (and degenerate) Euler-Lagrange equations in case of total variation by a linear one in the case of the variable fractional. The variable $s(\cdot)$ approach can be also applied in geophysics: Models governed by a fractional Helmholtz equation (with constant fractional order) have shown good qualitative agreement with available magnetoteluric data, see [20]. Given the spatially long-range correlated heterogeneity of the medium, nonlocal models with spatially varying fractional order $s(\cdot)$ appear as an atractive tool to further obtain quantitative agreement.

Our main results begin from Section 4, where we introduce a definition of $(-\Delta)^{s(\cdot)}$ on the quotient space $\mathscr{X}(\Omega,w)$. Also in this section we prove the existence and uniqueness of a solution $v\in\mathscr{X}(\Omega,w)$ to the associated Poisson problem (1.1) for every $h$ in the dual space of $\mathscr{X}(\Omega,h)$. It is worth mentioning that the results in Section 4 require minimal conditions on the function $s(\cdot)$, the weight $w$, and the domain $\Omega$. The results given in Section 4, however, do not provide conditions for solvability of the Poisson problem when the right hand side of the elliptic equation is a (regular) real valued function defined only on $\Omega$.

In a second approach, we are able to better identify the domain of $(-\Delta)^{s(\cdot)}$ as a quotient space also, now on a Sobolev space $\mathscr{H}_{0,L}^{1,p}(\mathcal{C},w)$ that consist of functions in $W^{1,p}(\mathcal{C},w)$ that formally vanish on the lateral boundary of $\mathcal{C}$. Differently from the construction given in Section 4, this second approach requires some extra conditions on both, $s(\cdot)$ and $\Omega$. These conditions are intimately related with the existence of $\Omega$-trace results for functions in $\mathscr{H}_{0,L}^{1,2}(\mathcal{C},w)$, as well as with the existence of a Poincaré inequality in $\mathscr{H}_{0,L}^{1,2}(\mathcal{C},w)$; thus, we postpone the second construction until Section 7.

In Section 5, we first study the $\Omega$-traces of functions in $\mathscr{H}_{0,L}^{1,p}(\mathcal{C},w)$, for $2\leq p<\infty$. In particular, we are able to characterize $s(\cdot)$-dependent integrability and differential regularity of restrictions of functions in $\mathscr{H}_{0,L}^{1,p}(\mathcal{C},w)$ to $\Omega$. Subsequently, we are able to prove the existence of a Poincaré inequality for $\mathscr{H}_{0,L}^{1,p}(\mathcal{C},w)$ in Section 6, for a special class of non-constant $s(\cdot)$ functions.

Our results finish in Section 7, where the details on the second definition of $(-\Delta)^{s(\cdot)}$ are given. Here, we identify the domain of $(-\Delta)^{s(\cdot)}$ with a subset of a weighted Lebesgue space $L^{2}(\Omega,\tilde{w})$ for some weight $\tilde{w}$, provided $s(\cdot)$ vanishes only on a set of zero measure and a Poincaré inequality holds for functions in $\mathscr{H}_{0,L}^{1,2}(\mathcal{C},w)$. Further, we improve this result for the case when $\Omega$ is the $N$-dimensional unit square and $s(\cdot)$ satisfies some extra conditions. In this latter case we identify the domain of $(-\Delta)^{s(\cdot)}$ with a subset of a Sobolev space of functions with variable smoothness on $\Omega$. The paper closes with Section 8 that includes, in addition to conclusions, a number of open questions and future research directions.

More general elliptic operators of the form

with spatially variable fractional order $s(\cdot)$, can be defined by extending the ideas in this paper in a natural way.

We assume that $\Omega\subset\mathbb{R}^{N}$, $N\geq 1$, is a non-empty bounded open set with a Lipschitz boundary $\partial\Omega$ (except in Section 4, where no condition is imposed on the $\Omega$ boundary). We denote by $\mathcal{C}$ the open semi-infinite cylinder with base $\Omega$, by $\partial_{L}\mathcal{C}$ the lateral boundary of $\mathcal{C}$, and by $\mathcal{C}_{\Omega}$ the cylinder $\mathcal{C}$ with the base $\Omega$, that is,

A generic point $X$ in $\mathbb{R}^{N+1}$ is denoted by $(x,y)$, where $x\in\mathbb{R}^{N}$ and $y\in\mathbb{R}$.

A function $\rho$ is said to be a weight if $\rho$ is positive and finite almost everywhere. For an open set $U$, and a weight $\rho$, we denote by $L^{p}(U,\rho)$ the space of measurable functions $u:U\to\mathbb{R}$ such

The space $L^{p}(U,\rho)$ endowed with the norm $\|\cdot\|_{L^{p}(U,\rho)}$ is a Banach space. Further, given $p\in[2,+\infty)$ we say that a weight $\rho$ satisfies the $B_{p}$ condition, and write $\rho\in B_{p}$, if $\rho^{-1/{p-1}}$ is locally integrable, that is,

For a weight $\rho\in B_{p}$, we define the weighted Sobolev space $W^{1,p}(U,\rho)$ as the subset of $L^{p}(U,\rho)$ of functions $u$ with weak gradients $\nabla u$ such that $|\nabla u|\in L^{p}(U,\rho)$. Endowed with the norm

$W^{1,p}(U,\rho)$ is a Banach space; see [12]. Notice that $B_{p}$ is a larger class of weights than the Muckenhoupt weights $A_{p}$. The latter is also used to define weighted Sobolev spaces; see [19]. Throughout the paper we assume $p\in[2,\infty)$ and denote the (Hölder) conjugate exponent of $p$ by $p^{\prime}$.

The measurable function $s(\cdot):\Omega\to\mathbb{R}$, which will characterize the spatially variable order of the fractional Laplacian, is assumed to satisfy:

2. (H1)

   $s(x)\in[0,1]$ for almost all $x\in\Omega$.

We use the notation $s(\cdot)$ to emphasize the dependence of the function $s:\Omega\to\mathbb{R}$ on the spatial variable $x\in\Omega$, and use $s$ to denote a constant in the interval $(0,1)$.

Throughout the paper we consider the function $w:\mathcal{C}\to\mathbb{R}$ defined by

and such that for a given $s(\cdot)$, and $p$, the function $\mathrm{G}_{s}:\Omega\to\mathbb{R}$ satisfies that

2. (H2)

   $\mathrm{G}_{s}\in B_{p}$, and if $s(\cdot)=s\in(0,1)$ constant, then

   $$\mathrm{G}_{s}(x)=\frac{2^{2s-1}\Gamma(s)}{\Gamma(1-s)},$$

   for all $x\in\Omega$. Here $\Gamma$ is the standard Euler-Gamma function.

Assumptions (H1) and (H2) imply that $w\in B_{p}$. However, it is known that (in general) $w$ is not expected to be of Muckenhoupt type, see [2, Proposition 1].

Given $\tau>0$, we denote by $\mathcal{C}^{\tau}$ the truncated cylinder $\mathcal{C}$ of height $\tau$, that is,

and define the sets $\partial_{L}\mathcal{C}^{\tau}$ and $\mathcal{C}^{\tau}_{\Omega}$ accordingly. The restriction of the weight $w$ to $\mathcal{C}^{\tau}$ is also denoted by $w$.

This section is devoted to briefly review the well-known extension domain approach to define the spectral fractional Laplacian, see for instance [5, 17, 8]. Throughout this section, we assume that $s\in(0,1)$ is constant.

We denote by $\{\lambda_{n}\}$ the sequence of eigenvalues of the Laplace operator supplemented with a Dirichlet boundary condition, and consider an orthonormal basis $\{\varphi_{n}\}$ of $L^{2}(\Omega)$ of associated eigenfunctions. The spectral fractional Laplacian is defined by

on the space

For extensions of (3.1) to non-homogeneous boundary conditions, we refer to [1]. It is worth mentioning that $H=\mathbb{H}^{s}_{0}(\Omega)$ if $s\in\left(0,\frac{1}{2}\right)\text{ or }s\in\left(\frac{1}{2},1\right)$ and $H=\mathbb{H}^{s}_{00}(\Omega)$ for $s=\frac{1}{2}$. Here, $\mathbb{H}_{0}^{s}(\Omega)$ is the closure in $\mathbb{H}^{s}(\Omega)$ of the space of infinitely continuous differentiable functions with compact support in $\Omega$, and $\mathbb{H}^{s}_{00}(\Omega)$ is the Lions-Magenes space [18]. Moreover, $\mathbb{H}^{s}(\Omega)$ is the fractional Sobolev space of order $s$,

endowed with the norm

The extension approach introduced by Caffarelli and Silvestre [6], see [17, 7] for the case of bounded domains, establishes that if $h\in H^{\prime}$ (dual space of $H$) then the unique solution to the elliptic equation

is given by $v=\mathrm{tr}_{\Omega}\>\,u$, where $u\in H^{1}_{0,L}(\mathcal{C},y^{1-2s})$ satisfies

see [7, Lemma 2.2]. Here, $\langle\cdot,\cdot\rangle_{H^{\prime},H}$ denotes the dual pairing between $H^{\prime}$ and $H$. Moreover, $\mathrm{tr}_{\Omega}\>$ is the $\Omega$-trace operator for functions in the space

More precisely,

is the unique bounded linear operator that satisfies $\mathrm{tr}_{\Omega}\>u=u(\,\cdot\,,0)$ for every $u\in C^{\infty}(\bar{\mathcal{C}})$ that vanishes on $\partial_{L}\mathcal{C}$; which is also onto over $H$, that is

see [7, Proposition 2.1].

Additionally, since the minimization problem

admits a unique solution $u\in H^{1}_{0,L}(\mathcal{C},y^{1-2s})$ for any $v\in\mathrm{tr}_{\Omega}\>H^{1}_{0,L}(\mathcal{C},w)$, the harmonic extension operator

where $u$ is the solution to problem (3.4), is well-defined, linear, and bounded. Then one finds that the spectral fractional Laplacian given by (3.1) satisfies

for all $\psi\in H^{1}_{0,L}(\mathcal{C},y^{1-2s})$ and all $v\in H$, which provides an equivalent definition for $(-\Delta)^{s}$. This second approach is our starting point to study the fractional Laplacian with spatially variable order: We identify a space of traces on which we can define the fractional Laplacian $(-\Delta)^{s(\cdot)}$ by a formula analogous to (3.5).

We consider in this section an abstract derivation of the spatially variable fractional Laplacian $(-\Delta)^{s(\cdot)}$. The advantage of this initial approach is that it requires minimal assumptions, namely (H1) and (H2), which are primarily sufficient conditions to have $w\in B_{p}$; this leads to an appropriate definition of the associated weighted Sobolev spaces. Also, it is worth noticing that the arguments in this section do not require any assumption on the regularity of the $\Omega$ boundary $\partial\Omega$. This path starts with the proper derivation of the trace space for the weighted Sobolev spaces in study. For this matter, we consider the space

and endow it with the semi-norm

Note that $u\mapsto\|u\|_{L^{1,2}(\mathcal{C},w)}$ is a norm on the subset of $C^{1}$ functions in $L^{1,2}(\mathcal{C},w)$ that vanish at $\partial\mathcal{C}$ or $\partial_{L}\mathcal{C}$. Subsequently, we define $\mathscr{L}_{0,L}^{1,2}(\mathcal{C},w)$ and $\mathscr{L}_{0}^{1,2}(\mathcal{C},w)$ as the completion in $L^{1,2}(\mathcal{C},w)$ of the infinitely differentiable functions in $L^{1,2}(\mathcal{C},w)$ with compact support in $\mathcal{C}_{\Omega}$ and $\mathcal{C}$, respectively, that is:

where

The only portion of the boundary where functions in $C^{\infty}_{c}(\mathcal{C}_{\Omega})$ do not necessarily vanish is the $\Omega$ cap. A few words are in order concerning $\mathscr{L}_{0,L}^{1,2}(\mathcal{C},w)$ and $\mathscr{L}_{0}^{1,2}(\mathcal{C},w)$. Note that $C^{\infty}_{c}(\mathcal{C}_{\Omega})\cap L^{1,2}(\mathcal{C},w)$ and $C^{\infty}_{c}(\mathcal{C})\cap L^{1,2}(\mathcal{C},w)$ are both pre-Hilbert spaces when endowed with the inner product

It follows then that their completion, $\mathscr{L}_{0,L}^{1,2}(\mathcal{C},w)$ and $\mathscr{L}_{0}^{1,2}(\mathcal{C},w)$, are Hilbert spaces; in particular for $z_{1},z_{2}\in\mathscr{L}_{0,L}^{1,2}(\mathcal{C},w)$ there exist Cauchy sequences $\{z_{1}^{n}\}$ and $\{z_{2}^{n}\}$ in $C^{\infty}_{c}(\mathcal{C}_{\Omega})\cap L^{1,2}(\mathcal{C},w)$ such that

If there is no risk of confusion, and in order to simplify notation, occasionally we simply write

and analogously we treat $\mathscr{L}_{0}^{1,2}(\mathcal{C},w)$.

Given that $C^{\infty}_{c}(\mathcal{C})\cap L^{1,2}(\mathcal{C},w)\subset C^{\infty}_{c}(\mathcal{C}_{\Omega})\cap L^{1,2}(\mathcal{C},w)$, then we observe that $\mathscr{L}_{0}^{1,2}(\mathcal{C},w)$ is a closed subspace of $\mathscr{L}_{0,L}^{1,2}(\mathcal{C},w)$. Thus, we can define an abstract space of traces on $\Omega$ of functions in $\mathscr{L}_{0,L}^{1,2}(\mathcal{C},w)$ as the quotient space

We then define

i.e., the abstract trace on $\Omega$ of a function $u\in\mathscr{L}_{0,L}^{1,2}(\mathcal{C},w)$ is identified with the equivalence class $[u]$ that contains $u$. The space $\mathscr{X}(\Omega,w)$ is then endowed with the usual norm

Note that

is a linear and bounded operator, and that $\mathscr{X}(\Omega,w)$ is a Hilbert space, given that $\mathscr{L}_{0,L}^{1,2}(\mathcal{C},w)$ and $\mathscr{L}_{0}^{1,2}(\mathcal{C},w)$ are also Hilbert spaces. We denote its inner product as $(\cdot,\cdot)_{\mathscr{X}}$. Further notice that, by definition, $\mathrm{Tr}_{\Omega}\,\mathscr{L}_{0,L}^{1,2}(\mathcal{C},w)=\mathscr{X}(\Omega,w)$. Unless it is not clear from the context, we denote the class $[v]\in\mathscr{X}(\Omega,w)$ simply by $v$. The following result establishes the existence of the harmonic extension operator.

Theorem 4.1 ensures the existence of the abstract weighted harmonic extension operator

where $u$ is the solution to ($\mathbb{P}_{v}$). In addition, the map $S$ is linear and bounded: Linearity follows directly from the examination of the first order conditions. For boundedness, consider (4.3) with $u-z$ instead of $\tilde{u}$, where $u$ solves ($\mathbb{P}_{v}$) and $z\in\mathscr{L}_{0}^{1,2}(\mathcal{C},w)$, to obtain $J_{\mu}(u_{\mu},v)\leq J(u-z)$. Then, by taking the limit as $\mu\to\infty$ we observe

Then, by considering the infimum over all $z\in\mathscr{L}_{0}^{1,2}(\mathcal{C},w)$, we obtain

The well-posedness of the map $S$ allows us to establish a definition for the fractional Laplacian with spatially variable order.

The operator $(-\Delta)^{s(\cdot)}:\mathscr{X}(\Omega,w)\to\mathscr{X}(\Omega,w)^{\prime}$ is well-defined as we see next, and it can be seen as the Lagrange multiplier associated to the harmonic extension problem.