Prof. El Haj Laamri

During his residency at the UM6P Institute for Advanced Studies, Prof. Laamri completed three research works in the fields of partial differential equations, nonlinear analysis, and fractional operators.
The first work, currently submitted to Fractional Calculus and Applied Analysis, is devoted to the regularity theory for fractional parabolic equations and to applications to nonlinear models of Kardar–Parisi–Zhang type. In the first part of the paper, the global regularity of the unique solution to the fractional heat equation posed in a bounded domain is established, first in an appropriate parabolic Bessel potential space and subsequently in the corresponding parabolic fractional Sobolev space. The proof relies on a new pointwise estimate for the fractional gradient of the associated kernel. The compactness of the corresponding solution operator is also established. As a major application, the second part of the work addresses a class of Kardar–Parisi–Zhang equations involving fractional diffusion and a nonlocal gradient term. The analysis establishes both the existence and regularity of solutions, while also yielding several auxiliary results of independent interest. Altogether, the work contributes new analytical tools to the study of nonlocal evolution equations and nonlinear phenomena governed by fractional diffusion.
The second work concerns a class of singular nonlinear elliptic problems with gradient dependence, including both scalar equations and coupled elliptic systems. Prior to this work, only partial results were available even in the case of a single equation, and only under a subquadratic growth assumption on the gradient term. Under suitable assumptions on the data, the authors prove existence and regularity results for weak solutions while allowing arbitrary growth of the gradient terms, including, in particular, the superquadratic regime. Their approach combines new regularity estimates for the associated Poisson problem with the Schauder fixed-point theorem, applied within a suitable functional framework. These results significantly extend the range of nonlinear singular elliptic problems for which existence and regularity can be established. This work has been accepted for publication in Communications on Pure and Applied Analysis.
The third work investigates the existence and qualitative properties of weak solutions to a class of nonlinear fractional equations driven by the spectral fractional Laplacian under mixed Dirichlet–Neumann boundary conditions. The existence of nontrivial weak solutions is established by means of appropriate topological and variational methods. The analysis combines a topological and variational principle with a localization argument involving carefully chosen compactly supported cutoff functions. More specifically, the proof relies on a delicate local comparison argument based on localized plateau-type test functions and explicit asymptotic estimates near the origin. This method provides a flexible analytical framework that may potentially be adapted to a variety of related nonlinear and nonlocal problems. As further applications, simplified existence criteria in the autonomous case are derived through the use of the Chebyshev radius of the domain, and concrete examples illustrating the applicability of the abstract theory are presented. This work is already available on HAL (hal-05685120) and arXiv (http://arxiv.org/abs/2607.07166) and has been submitted to the Electronic Journal of Differential Equations.
The three works were carried out in collaboration with other researchers, highlighting the collaborative nature of Prof. Laamri’s research and the importance of scientific exchange in these active areas of mathematical analysis.
Taken together, these three works reflect Prof. Laamri’s broader research programme at the intersection of nonlinear partial differential equations, fractional analysis, regularity theory, and the study of singular and nonlocal phenomena. They illustrate the depth and diversity of contemporary research on fractional and nonlinear equations, while contributing new methods and results to several active areas of mathematical analysis.