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

Abstract

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 $\delta \to 0^{+}$ 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 $\delta \to 0^{+}$ or to solutions of a nonlocal fractional Dirichlet problem involving $(-\Delta {)}^s$ as $\delta \to +\infty$.

24 pages