Approximate Green's Function for the Conductivity Equation With Conormal Coefficient

TL;DR

This paper develops an approximate Green's function for a conductivity operator with conormal coefficients, using Fourier Integral Operators to analyze its composition and canonical relations.

Contribution

It introduces a novel construction of an approximate Green's function for the conductivity equation with conormal coefficients using FIOs and analyzes their composition.

Findings

Constructed an approximate Green's function for the conductivity operator.

Analyzed the composition of Fourier Integral Operators with intersecting canonical relations.

Provided insights into the structure of solutions for the conductivity equation with conormal coefficients.

Abstract

We construct an approximate Green's function for $L_{\gamma}=\nabla \cdot \gamma(x) \nabla u(x)$, which belongs to a class of Fourier Integral Operators (FIOs) associated to two canonical relations. This leads to analysis of the composition of two FIOs, associated to a canonical relation with a zero section problem. The resulting composition is a sum of two FIOs, each associated to two intersecting canonical relations.