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

在这项工作中，我们介绍了涉及非局部算子\（（-\ Delta）_ {\\ delta} ^ {s} \）的体积约束问题，与分数拉普拉斯算子\（（-\ Delta）^ {s} \）密切相关，并取决于称为“地平线”的参数\（\ delta> 0 \）。当\（\ delta \ rightarrow 0 ^ + \）和\（\ delta \ rightarrow + \ infty \）时，我们研究了相关的线性和频谱问题以及这些体积约束问题的行为。通过\（（-\ Delta）_ {\\ delta} ^ {s} \）上的这些极限过程，我们得出了局部拉普拉斯算子和分数阶拉普拉斯算子的频谱收敛为\（\ delta \ rightarrow 0 ^ + \）和\ （\ delta \ rightarrow + \ infty \）分别，以及我们证明了这些问题的解对包含\（（-\ Delta）\）为\（\ delta \ rightarrow 0 ^ + \）的局部Dirichlet问题的解或对非局部分数阶解的收敛性Dirichlet问题涉及\（（-\ Delta）^ s \）作为\（\ delta \ rightarrow + \ infty \）。