Research Projects

The broad objective of this project is to explore and establish quantitative regularity results for a family of nonlocal operators. Specifically, we will focus on: --The fractional p-Laplacian (with and without gradient perturbations), --Nonlinear nonlocal operators with rough kernels, --Fractional porous medium equations. Our goal is twofold: extending existing results and developing new tools to advance the theory. Key Research Directions: 1. Fractional p-harmonic Functions:
 We aim to improve upon recent regularity results, which we believe are not yet optimal. Additionally, we will investigate the variant of the fractional p-Laplacian introduced by Chasseigne and Jakobsen—a problem that remains unexplored in the literature. 2. Gradient-Perturbed Fractional p-Laplacian:
The regularity theory for this class of operators presents significant challenges, as standard variational methods are not directly applicable. This project will seek new techniques to address this gap. 3. General Nonlocal Operators:
Beyond the fractional p-Laplacian, we will study broader classes of nonlocal operators, with a focus on deriving quantitative regularity estimates that deepen the understanding of their analytical properties. This research aligns with active developments in nonlocal PDEs, where sharp regularity estimates are crucial for both theory and applications.