Analysis of nonlocal p-Lévy operators — New trends on Integro-Differential Equations (IDEs)

• Applicant: Dr. Guy Fabrice Foghem Gounoue, Ph.D.
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 556370664

The proposal under submission is centered around fundamental questions regarding the analysis of complement value problems for p-Lévy operators with particular keen interest on on the analysis of Integer-Differential Equations (IDEs). In the linear setting p=2, such operators are well-know as nonlocal Lévy type operators, naturally raising in probability as generator of Lévy stochastic processes. A prototype of such operators includes the fractional Laplace operator when p=2 or the fractional p-Laplace operator in general. The study of the nonlocal p-Lévy operators is interesting in its own right in the general setting where the associated p-Lévy kernel fractional-free, that is, it is not necessarily compared to a fractional one. However, studies related to such nonlocal operators are often both highly challenging and interesting when the associated kernel is fractional-free. Various topics of interest in the nonlocal setting include elliptic and evolution Integro-Differential Equations (IDEs), function spaces, special inequalities (Sobolev, Poincaré, Hardy, etc.), spectral theory, maximum principles, etc., and their connections with their local correspondences. The current proposal is divided into five selected projects and is based on very recent works of mine and/or with my collaborators: - Project A: Optimal stability of complement value problems for p-Lévy operators. - Project B: Spectral stability for p-Lévy operators. - Project C: Complement value problem for 1-Lévy operators. - Project D: Essential self-adjointness of Lévy operators. - Project E: Gradient Flow structure for singular L{\'e}vy operators.

• DFG Programme: WBP Position