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

We show existence, uniqueness and positivity for the Green’s function of the operator $({\mathrm{\Delta }}_g+\alpha {)}^k$ in a closed Riemannian manifold $(M,g)$, of dimension $n>2k$, $k\in \mathbb{N}$, $k\ge 1$, with Laplace-Beltrami operator ${\mathrm{\Delta }}_g=-{\mathrm{div}}_g(\mathrm{\nabla }\cdot )$, and where $\alpha >0$. We are interested in the case where $\alpha$ is large : We prove pointwise estimates with explicit dependence on $\alpha$ 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 $\alpha$.