Carleman Estimates for Parabolic Operators with Discontinuous and Anisotropic Diffusion Coefficients, an Elementary Approach

TL;DR

This paper develops Carleman estimates for parabolic operators with anisotropic diffusion coefficients, extending previous work on isotropic cases by employing a pointwise estimate and a specific weight function.

Contribution

It introduces an elementary approach to derive Carleman estimates for anisotropic diffusion coefficients, broadening the applicability of previous isotropic results.

Findings

Established Carleman estimates for anisotropic diffusion coefficients.

Provided a new elementary method using pointwise estimates.

Extended previous isotropic results to more general anisotropic cases.

Abstract

By using some deep tools from microlocal analysis, J. Le Rousseau and L. Robbiano (Invent. Math., 183 (2011), 245--336) established several Carleman estimates for parabolic operators with isotropic diffusion coefficients which have jumps at interfaces. In this paper, we revisit the same problem but for the general case of anisotropic diffusion coefficients. Our main tools are a pointwise estimate for parabolic operators and a suitable chosen weight function.