Spectral gap for stable process on convex planar double symmetric domains

摘要

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 (Banuelos et al., Potential Anal. 24(3): 205-221, 2006), we obtain sharp upper and lower bound estimates of the spectral gap.