  We study the semigroup of the symmetric $\alpha$-stable process in bounded
domains in $\R^2$. We obtain a variational formula for the spectral gap, i.e.
the difference between two first eigenvalues of the generator of this
semigroup. This variational formula allows us to obtain lower bound estimates
of the spectral gap for convex planar domains which are symmetric with respect
to both coordinate axes. For rectangles, using "midconcavity" of the first
eigenfunction, we obtain sharp upper and lower bound estimates of the spectral
gap.
