Inequalities for the gradient of eigenfunctions of the Laplace-Beltrami operator

Abstract

In this paper we shall consider propelties of the eigenfunctions of the Laplace-Beltrami operator $\Delta \rho$ and properties of its gradient for a proper domain $D$ with a conformal metric, which density is equal to the reciprocal value of a defining function $\rho(x)$ for this domain i.e. $ds = \rho^{-1}(x)\vert dx \vert$.