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 (Δg + α)ᵏ in a closed Riemannian manifold $(M,g)$, of dimension $n>2k$, k∈ N, k≥ 1, with Laplace-Beltrami operator Δg = -divg(∇ ·), 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.