Variational Formulas for the Green Function
Charles Z. Martin
TL;DR
This paper reviews Hadamard's formula for Green function variation and explores approximations of Green function changes under perturbations of the Laplacian to other operators like Helmholtz and Schrödinger.
Contribution
It introduces new approximation methods for Green function variations when the Laplacian is perturbed into different differential operators.
Findings
Derived approximation formulas for Green function changes
Extended Hadamard's formula to various operator perturbations
Provided insights into Green function behavior under operator perturbations
Abstract
The Green function has a complex dependence upon its underlying domain and differential operator. We briefly review Hadamard's formula for the first variation of the Green function due to a perturbation of the domain. We then take a different avenue and approximate the change in the Green function when the Laplacian is perturbed into a number of different operators: Helmholtz, Schr\"odinger and Laplace--Beltrami.
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Taxonomy
TopicsAdvanced Mathematical Modeling in Engineering · Numerical methods in inverse problems · Electromagnetic Scattering and Analysis