The first non-zero Neumann $p-$fractional eigenvalue

In this work we study the asymptotic behavior of the first non-zero Neumann $p-$fractional eigenvalue $λ_1(s,p)$ as $s\to 1^-$ and as $p\to\infty.$ We show that there exists a constant $\mathcal{K}$ such that $\mathcal{K}(1-s)λ_1(s,p)$ goes to the first non-zero Neumann eigenvalue of the $p-$Laplacian. While in the limit case $p\to \infty,$ we prove that $λ_1(1,s)^{1/p}$ goes to an eigenvalue of the Hölder $\infty-$Laplacian.

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