| Abstract: |
The p-Laplacian is a nonlinear generalization of the classical Laplace operator and has become a central object of study in differential geometry, nonlinear analysis, and elliptic partial differential equations. Its applications extend to geometric analysis, nonlinear elasticity, fluid mechanics, image processing, and mathematical physics. On Riemannian manifolds, the p-Laplacian provides a natural framework for investigating nonlinear eigenvalue problems, variational principles, and the qualitative behavior of harmonic and p-harmonic functions. Particular attention has been devoted to estimating the first eigenvalue of the p-Laplacian, as it encodes important geometric and analytical information about the underlying manifold. Moreover, the study of critical sets of harmonic functions has deepened the understanding of regularity theory, singular structures, and solutions to nonlinear elliptic partial differential equations. This review presents an overview of the mathematical foundations of the p-Laplacian on Riemannian manifolds, including its variational formulation, eigenvalue problems, and connections with harmonic functions. The paper also highlights important analytical techniques and identifies emerging research directions in nonlinear geometric analysis. |
