It is not difficult to show that the Poisson problem for the Neumann spectral fraction Laplacian

\[(-\Delta_N)_{SFL}^s u = f, \qquad \text{in } \Omega.\]

preserves $L^2$ and $H^1$ regularity in bounded domains $\Omega$ such that $\partial \Omega$ is Lipschitz. This is useful, in particular, for numerical analysis where the domain is a polygon.

It is known (see [1] and there references therein) that Lipschitz domains have the $H^1$ extension property, meaning that there is an operator $E: H^1 (\Omega) \to H^1 (\mathbb R^n)$ such that $E(u) = u$ in $\Omega$. Hence, it is not difficult to show that the resolvent of the classical Neumann problem

\[\begin{cases} -\Delta u = f & \text{in } \Omega, \\ \frac{\partial u}{\partial n} = 0 & \text{on } \partial \Omega \end{cases}\]

is an operator $(-\Delta_N)^{-1}: L^2 (\Omega) \to L^2 (\Omega)$ which is compact. Hence it admits an eigen-decomposition, which we take $L^2$ orthonormal. We have $\lambda_1 = 0$ and $\varphi_1$ constant.

With this $(-\Delta_N)^{\pm s}$ is defined as follows. If $\int_\Omega f = 0$, using the spectral decomposition the solution of with $\int_\Omega u = 0$ you get

\[u =(-\Delta_N)^{- s}f =\sum_{i>1}\lambda_i^{- s} \langle f,\varphi_i \rangle \varphi_i.\]

In other words, $\langle u, \varphi_1\rangle = 0$, and for $i > 1$ we have $\langle u, \varphi_i \rangle = \lambda_i^{-s} \langle f,\varphi_i\rangle$. In this setting, you can observe

\[\begin{aligned} ||v||_{L^2}^2 &= \sum _{i \ge 1} \langle v, \varphi_i \rangle^2 \\ ||D v||_{L^2}^2 &= \langle v, -\Delta v \rangle = \sum _{i > 1} \lambda_i \langle v, \varphi_i \rangle^2 \end{aligned}\]

It is therefore easy to compute

\[||D^k v||^2_{L^2} \le \lambda_2^{-s} ||D^k f||^2 _{L^2}, \qquad \text{for } k = 0,1 .\]

To recover improvements of regularity, for example into “$H^{1+s}$ spaces”, one has to be careful. There are several different definitions (Sobolev-Slobodeckij, Hardy, … ). These can cause difficulties with the regularity of $\Omega$.

We point to [2].


  1. Bucur, Dorin and Mazzoleni, Dario and Pratelli, Aldo and Velichkov, Bozhidar. Lipschitz Regularity of the Eigenfunctions on Optimal Domains. Link
  2. Grubb, Gerd. Regularity of Spectral Fractional Dirichlet and Neumann Problems. Link