Reverse Faber-Krahn inequalities for Zaremba problems

Let Ω be a domain in Rⁿ (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 of {alignat*}2 -Δ_p u &=τ (∫Ω|u|^q dx )^{({p-q})/{q}} |u|q-2u &&in Ω,\\ u &=0&&on ∂Ω_D, \\ ∂ u/∂ η&=0&& on ∂ Ω ∂ Ω_D. {alignat*} Under the assumption that ΩD is convex, we establish the following reverse Faber-Krahn inequality τ1,q(Ω)≤ τ1,q(Ω⁾,% where Ω⁼BR Bᵣ̄ is a concentric annular region in Rⁿ having the same Lebesgue measure as Ω and such that {enumerate}[(i)] (when ΩD=Ωₒᵤₜ) W₁(ΩD)= ωₙ Rⁿ⁻¹, and (Ω⁾D=BR, (when ΩD=Ωᵢₙ) Wₙ₋₁(ΩD)=ωₙr, and (Ω⁾D=Bᵣ. {enumerate} Here Wᵢ(ΩD) is the iᵗʰ { quermassintegral} of ΩD. We also establish Sz.-Nagy's type inequalities for parallel sets of a convex domain in Rⁿ (n≥ 3) for our proof.