Optimal shapes for the first Dirichlet eigenvalue of the p-Laplacian and dihedral symmetry

Chorwadwala, Anisa M. H.; Ghosh, Mrityunjoy

doi:10.1016/j.jmaa.2021.125901

Cited by 1 publication

(2 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…They have established a family of Faber-Krahn type inequalities (for 1 ≤ q ≤ 2n n−2 ) for the first eigenvalue of the Robin Laplacian. In [3], Bobkov and Kolonitskii studied a monotonicity result (with respect to domain perturbation) for the first Dirichlet eigenvalue of the p-Laplacian with suitable nonlinearity; see [2,7] also for a similar result when q = p. The symmetries of the minimizers of R q subject to different boundary conditions can be found in [19,27,28]. We also refer to the monographs [16,17] for further readings in this direction.…”

Section: Introduction and Statements Of The Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Reverse Faber-Krahn inequalities for Zaremba problems

Anoop¹,

Ghosh²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Let Ω be a multiply-connected domain in R n (n ≥ 2) of the form Ω = Ωout \Ωin. Set ΩD to be either Ωout or Ωin. For p ∈ (1, ∞), and q ∈ [1, p], let τ1,q(Ω) be the first eigenvalue ofUnder the assumption that ΩD is convex, we establish the following reverse Faber-Krahn inequality τ1,q(Ω) ≤ τ1,q(Ω ⋆ ), where Ω ⋆ = BR \Br is a concentric annular region in R n having the same Lebesgue measure as Ω and such that (i) (when ΩD = Ωout) W1(ΩD) = ωnR n−1 , and (Ω ⋆ )D = BR, (ii) (when ΩD = Ωin) Wn−1(ΩD) = ωnr, and (Ω ⋆ )D = Br. Here Wi(ΩD) is the i th quermassintegral of ΩD. We also establish Sz. Nagy's type inequalities for parallel sets of a convex domain in R n (n ≥ 3) for our proof.

show abstract