Imad El Bouchairi

Université de Caen Normandie | UNICAEN · Département de Mathématiques

In this Element, the authors consider fully discretized p-Laplacian problems (evolution, boundary value and variational problems) on graphs. The motivation of nonlocal continuum limits comes from the quest of understanding collective dynamics in large ensembles of interacting particles, which is a fundamental problem in nonlinear science, with appl...

In this paper we study continuum limits of the discretized [[EQUATION]] -Laplacian evolution problem on sparse graphs with homogeneous Neumann boundary conditions. This goes far beyond known results by handling much more general class of kernels, possibly singular, and graph sequences whose limit are the so-called [[EQUATION]] -graphons. More preci...

The nonlocal p-Laplacian operator, the associated evolution equation and boundary value problem, governed by a given kernel, have applications in various areas of science and engineering. In particular, they have become modern tools for massive data processing (including signals, images, geometry), and machine learning tasks such as semi-supervised...

In this paper we study continuum limits of the discretized $p$-Laplacian evolution problem on sparse graphs with homogeneous Neumann boundary conditions. This extends the results of [24] to a far more general class of kernels, possibly singular, and graph sequences whose limit are the so-called $L^q$-graphons. More precisely, we derive a bound on t...

In this paper, we first introduce a new family of operators on weighted graphs called p-bilaplacian operators, which are the analogue on graphs of the continuous p-bilaplacian operators. We then turn to study regularized variational and boundary value problems associated to these operators. For instance, we study their well-posedness (existence and...