The first non-zero Neumann pp-fractional eigenvalue

In this work we study the asymptotic behavior of the first non-zero Neumann pp-fractional eigenvalue λ1(s,p)λ1(s,p) as s→1−s→1− and as p→∞p→∞. We show that there exists a constant KK such that K(1−s)λ1(s,p)K(1−s)λ1(s,p) goes to the first non-zero Neumann eigenvalue of the pp-Laplacian. While in the limit case p→∞p→∞, we prove that λ1(1,s)1/pλ1(1,s)1/p goes to an eigenvalue of the Hölder ∞∞-Laplacian.