A restricted nonlocal operator bridging together the Laplacian and the fractional Laplacian

In this work we introduce volume constraint problems involving the nonlocal operator ( - Δ ) δ s , closely related to the fractional Laplacian ( - Δ ) s , and depending upon a parameter δ > 0 called horizon. We study the associated linear and spectral problems and the behavior of these volume constraint problems when δ → 0 + and δ → + ∞ . Through these limit processes on ( - Δ ) δ s we derive spectral convergence to the local Laplacian and to the fractional Laplacian as δ → 0 + and δ → + ∞ respectively, as well as we prove the convergence of solutions of these problems to solutions of a local Dirichlet problem involving ( - Δ ) as δ → 0 + or to solutions of a nonlocal fractional Dirichlet problem involving ( - Δ ) s as δ → + ∞ .