Use of the operator \(\Delta^n\) in a Riemannian space (Q1286536)

Let us consider the continued \(n\)-products of the Laplacian operator \(\Delta\), defined by \[ \Delta^n=\Delta \Delta \Delta \ldots \Delta,\qquad n \text{ times } \] where \(n\) is a positive integer. Let us also write for a smooth differentiable function \(f\) of class \(C^\infty\) on a Riemannian manifold of dimension \(m\geq 2:\) \[ \Delta^n f= \Delta(\Delta(\ldots(\Delta(\Delta f)))). \] If \(n\) equals zero then \(\Delta^0 f=f\). We set \(\Delta^n f\) as the ``generalized Laplacian of \(f\)'', it is the Laplacian iterated operator. The author also considers the Riemannian space \(( M, g)\) where \(M\) is a Riemannian manifold of dimension \(m\geq n\) associated with a positive definite metric \(g\). The aim of the author is to obtain integral formulas in the Riemannian space and to use them to establish some of the global results. Moreover, it is shown that the results obtained by the author include the following Green's theorem and Hopf lemma. Let \(M\) be a closed orientable Riemannian manifold. Then \[ \int_{M} g^{ij} {\nabla}_j {\nabla}_i f d V=0, \quad \text{ or } \int_{M} \Delta f d V=0 \] where \(d V\) is the volume element of \(M\). The above theorem is the well-known Green's theorem. The second result, Hopf's lemma, assures that under the same assumption of Green's theorem and if \(f\) is a differentiable function on \(M\) such that \(\Delta f \geq 0\) everywhere on \(M\), or \(\Delta f\leq 0\) everywhere on \(M\), then \(f\) is constant on \(M\), and it follows \(\Delta f=0\).