Variation of Green Functions: Normal Derivatives

TL;DR

This paper derives a variational formula for the normal derivatives of Green functions associated with Schrödinger and Laplace--Beltrami operators, and explores implications for elliptic growth phenomena.

Contribution

It introduces a new variational formula for Green function derivatives and applies it to characterize elliptic growth driven by Green functions.

Findings

Derived a variational formula for Green function normal derivatives

Connected Green function derivatives to elliptic growth modeling

Provided a framework for analyzing domain growth via Green functions

Abstract

We derive a variational formula for the outward normal derivative of the Green function for the Schr\"odinger and Laplace--Beltrami operators, viewed as perturbations of the Laplacian. As an application we begin to characterize elliptic growth---the growth of a domain pushed outward by its own Green function.