  We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian
operators. In particular, we show that there exists a compactly supported
nonnegative Sobolev function $u_0$ that attains the infimum (which will be a
positive real number) of the set \[ \left\{ \int\int_{\{u > 0\}\times\{u>0\}}
\frac{|u(x) - u(y)|^2}{|x - y|^{n + 2 \sigma}}d x d y : u \in \mathring
H^\sigma(\mathbb{R}^n), \int_{\mathbb{R}^n} u^2 = 1, |\{u > 0 \}| \leq
1\right\}. \] Unlike the corresponding problem for the usual fractional
Laplacian, where the domain of the integration is $\mathbb{R}^n \times
\mathbb{R}^n$, symmetrization techniques may not apply here. Our approach is
instead based on the direct method and new a priori diameter estimates. We also
present several remaining open questions concerning the regularity and shape of
the minimizers, and the form of the Euler-Lagrange equations.
