Asymptotics of variational eigenvalues for a general nonlocal $p$-Laplacian with varying horizon

TL;DR

This paper introduces a new nonlocal p-Laplacian based on finite horizon kernels, analyzing eigenvalue asymptotics and stability as the horizon varies, connecting nonlocal and local eigenproblems.

Contribution

It defines a novel nonlocal p-Laplacian with variable horizon and proves stability results linking it to local and fractional p-Laplacians.

Findings

Eigenvalues converge to local p-Laplacian eigenvalues as horizon shrinks to zero.

Eigenvalues approach fractional p-Laplacian eigenvalues as horizon grows large.

Stability of solutions established for varying horizon limits.

Abstract

From the recent developing of nonlocal gradients with finite horizon $\delta>0$ based on general kernels, we introduce a new nonlocal $p$-Laplacian and study the eigenvalue problem associated with it. Furthermore, by virtue of $\Gamma$-convergence arguments, we establish stability results of the solutions for varying horizon in the extreme cases $\delta\to 0^+$ and $\delta\to\infty$, recovering the solutions for the local eigenvalue problem associated with the $p$-Laplacian, and the ones associated with the $H^{s,p}$-Laplacian, respectively.