| Abstract: |
The nonlinear p-Laplacian operator plays a fundamental role in geometric analysis, nonlinear spectral theory, and elliptic partial differential equations. Estimating its first eigenvalue is of particular importance because it reflects both the analytical and geometric properties of the underlying Riemannian manifold. This paper proposes a variational framework for studying the first eigenvalue of the p-Laplacian through the geometry of critical sets of harmonic functions. The manuscript presents the mathematical formulation, theoretical background, and analytical framework that can be used to derive new eigenvalue estimates under suitable curvature assumptions. The relationships among variational methods, Sobolev inequalities, comparison geometry, and nonlinear elliptic equations are discussed, together with possible directions for original theorem development. |
