The first Grushin eigenvalue on cartesian product domains

In this paper, we consider the first eigenvalue.1(O) of the Grushin operator.G :=.x1 + |x1|2s.x2 with Dirichlet boundary conditions on a bounded domain O of Rd = R d1+ d2. We prove that.1(O) admits a unique minimizer in the class of domains with prescribed finite volume, which are the cartesian product of a set in Rd1 and a set in Rd2, and that the minimizer is the product of two balls Omega()(1).subset of R-d1 and O- (2)subset of R-d2. Moreover, we provide a lower bound for | Omega() (1) | and for lambda(1)( O- (1) x O-* (2)). Finally, we consider the limiting problem as s tends to 0 and to +8.