In this paper we consider the first eigenvalue λ1(Ω) of the Grushin operator ΔG:=Δx1+|x1|2sΔx2 with Dirichlet boundary conditions on a bounded domain Ω of Rd=Rd1+d2. We prove that λ1(Ω) 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 Ω∗1⊆Rd1 and Ω∗2⊆Rd2. Moreover, we provide a lower bound for |Ω∗1| and for λ1(Ω∗1×Ω∗2). Finally, we consider the limiting problem as s tends to 0 and to +∞.
The first Grushin eigenvalue on cartesian product domains / Luzzini, Paolo; Provenzano, Luigi; Stubbe, Joachim. - In: ADVANCES IN CALCULUS OF VARIATIONS. - ISSN 1864-8258. - 0:0(2023). [10.1515/acv-2022-0015]
The first Grushin eigenvalue on cartesian product domains
Luigi Provenzano;
2023
Abstract
In this paper we consider the first eigenvalue λ1(Ω) of the Grushin operator ΔG:=Δx1+|x1|2sΔx2 with Dirichlet boundary conditions on a bounded domain Ω of Rd=Rd1+d2. We prove that λ1(Ω) 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 Ω∗1⊆Rd1 and Ω∗2⊆Rd2. Moreover, we provide a lower bound for |Ω∗1| and for λ1(Ω∗1×Ω∗2). Finally, we consider the limiting problem as s tends to 0 and to +∞.| File | Dimensione | Formato | |
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