Estimate of the Fundamental Solution for Parabolic Operators with Discontinuous Coefficients

TL;DR

This paper extends fundamental solution estimates for scalar parabolic equations to those with discontinuous coefficients, crucial for inverse problems in heat conduction.

Contribution

It demonstrates that estimates for derivatives of fundamental solutions hold even with discontinuous coefficients, broadening applicability.

Findings

Estimates for derivatives of fundamental solutions are valid with discontinuous coefficients.

The results are applicable to inverse problems involving heat conduction.

Provides theoretical foundation for boundary measurement-based inclusion detection.

Abstract

We will show that the same type of estimates known for the fundamental solutions for scalar parabolic equations with smooth enough coefficients hold for the first order derivatives of fundamental solution with respect to space variables of scalar parabolic equations of divergence form with discontinuous coefficients. The estimate is very important for many applications. For example, it is important for the inverse problem identifying inclusions inside a heat conductive medium from boundary measurements.