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 (A(g) + alpha)(k) in a closed Riemannian manifold (M, g), of dimension n > 2k, k is an element of N, k >= 1, with Laplace-Beltrami operator Delta(g) = - div(g)(del<middle dot>), 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. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.