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

In this work we introduce volume constraint problems involving the nonlocal operator $(-\Delta)_{\delta}^{s}$, closely related to the fractional Laplacian $(-\Delta)^{s}$, and depending upon a parameter $\delta>0$ called horizon. We study the associated linear and spectral problems and the behavior of these volume constraint problems when $\delta\to0^+$ and $\delta\to+\infty$. Through these limit processes on $(-\Delta)_{\delta}^{s}$ we derive spectral convergence to the local Laplacian and to the fractional Laplacian as $\delta\to 0^+$ and $\delta \to +\infty$ respectively, as well as we prove the convergence of solutions of these problems to solutions of a local Dirichlet problem involving $(-\Delta)$ as $\delta\to0^+$ or to solutions of a nonlocal fractional Dirichlet problem involving $(-\Delta)^s$ as $\delta\to+\infty$.