On volume constraint problems related to the fractional Laplacian

Abstract

In this work we study volume constraint problems involving the nonlocal operator (−Δ)𝑠 𝛿 depending upon a parameter 𝛿 > 0 called horizon. We analyze the associated linear and spectral problems and the behavior of these volume constraint problems when 𝛿 ! 0+ and 𝛿 ! +1. We prove spectral convergence to the classical Laplacian as 𝛿 ! 0+ under a suitable scaling and spectral convergence to the fractional Laplacian as 𝛿 ! +1.