# Courant nodal domain theorem for fractional Laplacian

Let $$\lambda_k$$ and $$\varphi_k$$ be the $$k$$-th eigenvalue and a corresponding eigenfunction of the fractional Laplacian in a bounded domain $$\Omega \subset \mathbb{R}^N$$, $$N \geq 2$$. That is, $$\lambda_k$$ and $$\varphi_k$$ satisfy \left\{ \begin{aligned} (-\Delta)^{\alpha/2} \varphi_k &= \lambda_k \varphi_k &&\text{in}~ \Omega,\\ \varphi_k&=0 &&\text{in}~ \mathbb{R}^N \setminus \Omega. \end{aligned} \right. Here the fractional Laplacian $$(-\Delta)^{\alpha/2}$$ is given by $$(-\Delta)^{\alpha/2} u = - C\lim_{\varepsilon \to 0+} \int_{\mathbb{R}^N \setminus B_\varepsilon(0)} \frac{u(y)-u(x)}{|y-x|^{N+\alpha}} \, dy,$$ where $$C>0$$ is a some explicit constant and $$\alpha \in (0,2)$$.

If $$\alpha=2$$, then the fractional Laplacian becomes the classical Laplace operator and it is very well known that the number of nodal domains of $$\varphi_k$$ is bounded from above by $$k$$. This is the content of the Courant nodal domain theorem.

However, in the purely fractional case $$\alpha \in (0,2)$$ much less seems to be known in this respect. Some upper bounds on the number of nodal domains in 1D-case and $$\alpha=1$$ can be found in [Bañuelos, Kulczycki, 2004].

I'm interested in the following very particular case of a weak version of the Courant nodal domain theorem:

Does any second eigenfunction $$\varphi_2$$ have a finite number of nodal domains?

Maybe the answer is well-known for experts or, at least, can be relatively easy to obtain (e.g., by using some variational properties of $$\lambda_2$$)? Perhaps, the answer is known in the radially-symmetric case? I'll appreciate any hint.

As far as I can tell, although it is widely believed that the Courant–Hibert estimate holds for the fractional Laplacian, even finiteness of the number of nodal domains of the second eigenfunctions is open for a general domain. The only known result is that the "harmonic extension to the upper half-space" (in the sense of Caffarelli–Silvestre for $$\alpha \ne 1$$) has at most two nodal domains.
• This is proved by Bañuelos and Kulczycki (perhaps only when $\alpha = 1$, I do not remember well). A closely related result (for operators with potentials rather than in a domain) is used by Frank, Lenzmann and Silvestre in DOI:10.1002/cpa.21591, and there is a short discussion after the statement of Theorem III in my paper with Jacek Mucha, DOI:10.1007/s00028-018-0444-4. – Mateusz Kwaśnicki Jan 8 '20 at 19:18