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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 $δ\to0^+$ and $δ\to+\infty$. Through these limit processes on $(-Δ)_δ^{s}$ we derive spectral convergence to the local Laplacian and to the fractional Laplacian as $δ\to 0^+$ and $δ\to +\infty$ respectively, as well as we prove the convergence of solutions of these problems to solutions of a local Dirichlet problem involving $(-Δ)$ as $δ\to0^+$ or to solutions of a nonlocal fractional Dirichlet problem involving $(-Δ)^s$ as $δ\to+\infty$.