A Study on the First Eigenvalue of the P-Laplacian and Critical Sets of Harmonic Functions by Defining Geometric P-Laplacian on Riemannian Manifolds

Exploring the Spectrum and Critical Sets of Harmonic Functions on Riemannian Manifolds

Abstract

In mathematics, the p-Laplacian, or the p-Laplace administrator, is a quasilinear elliptic incomplete differential administrator of second request. It is a nonlinear speculation of the Laplace administrator, where p is permitted to go over . The properties of the spectrum of the weighted p-Laplacian on a complete Riemannian complex with evolving geometry it is a notable component that spectrum as an invariant amount advances as the space does under any geometric flow. The variation recipes, monotonicity, and differentiability for the first eigen value of the p-Laplacian on a n-dimensional shut Riemannian complex whose measurement develops by a summed up geometric flow. the spectrum of the Laplacian on noncompact non-complete manifolds also attracts attention of mathematicians and physicists in the past three decades, since the investigation of the spectral properties of the Dirichlet Laplacian in infinitely extended regions has applications in elasticity and so on. The PDEs involving p-Laplacian are considered in differential geometry in the investigation of critical points for p-harmonic maps between Riemannian manifolds and the eigenvalue problems for p-Laplacian on Riemannian manifolds serve for estimations of the diameter of the manifolds. By using the theory of self-adjoint operators, the spectral properties of the linear Laplacian on a domain in a Euclidean space or a manifold have been concentrated broadly.