A Nonlocal $p$-Laplacian Interface Model with Sharp Interface
For researchers in nonlocal PDEs and interface problems, this provides a rigorous framework for nonlocal-to-local convergence with sharp interfaces.
The paper proposes a nonlocal p-Laplacian interface model that retains a sharp interface, enabling extension to other interface problems. It proves Γ-convergence of minimizers to local counterparts and validates with numerical experiments.
We propose an energy-based nonlocal $p$-Laplacian interface problem. Neumann interface conditions are naturally formulated via the energy, while Dirichlet conditions are enforced through a penalty term. A key feature is that the model retains a sharp interface, which facilitates extension to other interface problems; we illustrate this by developing a nonlocal approximation for the $p$-Laplacian interface problem with membrane conditions. By establishing $Γ$-convergence and compactness, we prove that as the nonlocal horizon vanishes, minimizers of the nonlocal functionals converge to those of the local counterparts. Numerical experiments using an efficient finite element method confirm the convergence.