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.

The first Grushin eigenvalue on cartesian product domains

Luzzini, Paolo;
2023

Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/3508119
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