Carleman estimate for second order elliptic equations with Lipschitz leading coefficients and jumps at an interface

TL;DR

This paper establishes a local Carleman estimate for second order elliptic equations with Lipschitz coefficients and interface jumps, using elementary methods that require minimal assumptions, advancing the analysis of such PDEs.

Contribution

It introduces a novel elementary approach to derive Carleman estimates for elliptic equations with complex coefficients and interfaces, avoiding microlocal analysis techniques.

Findings

Proves a local Carleman estimate under minimal regularity assumptions.

Handles equations with anisotropic Lipschitz coefficients and interface jumps.

Provides a framework applicable to more general elliptic problems.

Abstract

In this paper we prove a local Carleman estimate for second order elliptic equations with a general anisotropic Lipschitz coefficients having a jump at an interface. Our approach does not rely on the techniques of microlocal analysis. We make use of the elementary method so that we are able to impose almost optimal assumptions on the coefficients and, consequently, the interface.