Reverse Faber-Krahn inequalities for Zaremba problems

Let be a domain in Rⁿ (n 2) of the form =₎ₔₓ ₈₍. Set D to be either ₎ₔₓ or ₈₍. For p (1, ), and q 1, p, let ₁, ₐ () be the first eigenvalue of alignat*2 -ₚ u &= (_|u|q dx) ^ ({p-q) /q} |u|^q-2u &&in, \\ u &=0&&on D, \\ u &=0&& on D. alignat* Under the assumption that D is convex, we establish the following reverse Faber-Krahn inequality ₁, ₐ () ₁, ₐ (^), % where ^=BR Bᵣ is a concentric annular region in Rⁿ having the same Lebesgue measure as and such that enumerate (i) (when D=₎ₔₓ) W₁ (D) = ₙ R^n-1, and (^) D=BR, (when D=₈₍) W₍-₁ (D) =ₙr, and (^) D=Bᵣ. enumerate Here W₈ (D) is the i^th quermassintegral of D. We also establish Sz. -Nagy's type inequalities for parallel sets of a convex domain in Rⁿ (n 3) for our proof.