The fundamental difference between nonlocal models and classical local models is that the latter only involve differential operators, whereas the former rely on integral operators. Nonlocal models can describe phenomena not well represented by classical PDEs, including problems characterized by long-range interactions and discontinuities. In addition, the use of fractional-order or nonlocal operators yields a relaxation of the formulations that give rise to additional stability of solutions and modeling capabilities.
The research lines of the project include the following.
Coupling local and nonlocal scalar/vectorial models: peridynamics, integrable kernels, and applications to elasticity.
Electromagnetism: interface problems between dielectrics and metamaterials.
Nonlocal formulations for models in nanoplasmonics.
A game-theoretic approach to non-local/local coupled problems.
Two-species competition with non-local/local diffusions.
Mixing local and nonlocal evolutions.
Dirichlet-to-Neumann maps on trees.
The Dirichlet problem on a tree in a random environment.