We prove that the first eigenvalue of the fractional Dirichlet–Laplacian of order s on a simply connected set of the plane can be bounded from below in terms of its inradius only. This is valid for 1 / 2 < s< 1 and we show that this condition is sharp, i.e., for 0 < s≤ 1 / 2 such a lower bound is not possible. The constant appearing in the estimate has the correct asymptotic behavior with respect to s, as it permits to recover a classical result by Makai and Hayman in the limit s↗ 1. The paper is as self-contained as possible.
The fractional Makai–Hayman inequality
Brasco L.
Co-primo
2022
Abstract
We prove that the first eigenvalue of the fractional Dirichlet–Laplacian of order s on a simply connected set of the plane can be bounded from below in terms of its inradius only. This is valid for 1 / 2 < s< 1 and we show that this condition is sharp, i.e., for 0 < s≤ 1 / 2 such a lower bound is not possible. The constant appearing in the estimate has the correct asymptotic behavior with respect to s, as it permits to recover a classical result by Makai and Hayman in the limit s↗ 1. The paper is as self-contained as possible.File in questo prodotto:
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