A local and nonlocal coupling model involving the $p$-Laplacian
Uriel Kaufmann, Raúl Vidal
TL;DR
The paper studies a coupled local and nonlocal $p$-Laplacian problem on a domain $\,\Omega$ split into $\,\Omega_ ext{ell}$ and $\,\Omega_{n\ell}$, connected through source terms and a nonlocal kernel $J$. It builds a variational framework with an energy functional $E$ on the space $W$, and establishes the existence of a minimizer via the direct method, which corresponds to a weak solution of the coupled system; under the additional assumption that $\xi\mapsto f(x,\xi)$ is strictly concave for a.e. $x$, the minimizer is unique. The results extend prior work on the Laplacian to the $p$-Laplacian setting and rely on growth conditions on $f$, compactness properties of the operator $T_{J,q}$, and the partitioned local/nonlocal coupling. This provides a rigorous foundation for local–nonlocal diffusion models and heterogeneous systems with $p$-growth in applications.
Abstract
In this paper we extend some results presented in \cite{julio} to the case of the $p$-Laplacian operator. More precisely, we consider a model that couples a local $p$-Laplacian operator with a nonlocal $p$-Laplacian operator through source terms in the equation. The resulting problem is associated with an energy functional. We establish the existence and uniqueness of a solution, which is obtained via the direct minimization of the corresponding energy functional.