# Carleman estimates for the parabolic transmission problem and H\"older propagation of smallness across an interface

## TL;DR

This paper establishes a H"older propagation of smallness for solutions to second order parabolic equations with jump discontinuities at an interface, using a novel local Carleman estimate for anisotropic operators.

## Contribution

It introduces a new Carleman estimate for parabolic operators with anisotropic coefficients having jumps at interfaces, enabling propagation of smallness results.

## Key findings

- Proved a local Carleman estimate for parabolic operators with interface jumps.
- Established H"older propagation of smallness across interfaces.
- Extended techniques to anisotropic, Lipschitz continuous coefficients.

## Abstract

In this paper we prove a H\"older propagation of smallness for solutions to second order parabolic equations whose general anisotropic leading coefficient has a jump at an interface. We assume that the leading coefficient is Lipschitz continuous with respect to the parabolic distance on both sides of the interface. The main effort consists in proving a local Carleman estimate for this parabolic operator.