Research Projects

In recent years, partial and integro-differential equations have attracted considerable attention due to their wide range of applications in fields such as image processing, quantum mechanics, phase transition problems, biology. When developing mathematical models, establishing well-posedness is essential. This involves proving the existence of a unique solution and ensuring that the solution depends continuously on the model parameters. Additionally, investigating the regularity properties of the solution, assuming it exists, is a critical aspect. While the theory of regularity and existence for partial and integro-differential equations is well established in Euclidean spaces, nonlocal problems in the Heisenberg group (a non-Euclidean setting) remain relatively unexplored. A key feature of the Heisenberg group is its non-commutativity. By "singularity," we refer to cases where the nonlinearity becomes unbounded near the origin. Such problems have been extensively studied in Euclidean settings for both local and nonlocal cases. In non-Euclidean contexts, local singular equations modeled on the subLaplacian have been examined in Garain-Ukhlov. In the nonlocal case, Manfredini-Palatucci-Piccinini-Polidoro have demonstrated some regularity results for the non-singular case, and recently, Ghosh, Kumar, and Ruzhansky have addressed the singular case. To our knowledge, both singular and non-singular/regular nonlocal problems are not well understood in the Heisenberg group, even in linear cases. The primary goal of this proposed project is to investigate the regularity and existence properties of nonlocal problems in the Heisenberg group with singular nonlinearities. In pursuit of this, we will also examine non-singular nonlocal problems, as we believe that understanding their regularity will be beneficial for tackling singular problems.