The Green's function of polyharmonic operators with diverging coefficients: Construction and sharp asymptotics

Abstract

We show existence, uniqueness and positivity for the Green's function of the operator $(Δ_g + α)^k$ in a closed Riemannian manifold $(M,g)$, of dimension $n>2k$, $k\in \mathbb{N}$, $k\geq 1$, with Laplace-Beltrami operator $Δ_g = -\operatorname{div}_g(\nabla \cdot)$, and where $α>0$. We are interested in the case where $α$ is large : We prove pointwise estimates with explicit dependence on $α$ for the Green's function and its derivatives. We highlight a region of exponential decay for the Green's function away from the diagonal, for large $α$.