Unique continuation property of solutions to general second order elliptic systems

TL;DR

This paper establishes the weak unique continuation property for solutions to general second order strongly elliptic systems using Carleman estimates and the Holmgren transform, under natural and technical assumptions.

Contribution

It introduces a novel approach combining Carleman estimates with operator factorization and freezing coefficients for second order elliptic systems.

Findings

Proves weak unique continuation under broad assumptions.

Develops a new Carleman estimate for the leading part of the system.

Uses operator factorization and parametrix construction for the analysis.

Abstract

This paper concerns about the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumption which called {\sl basic assumptions}, but also some technical assumptions which we called {\sl further assumptions}. It is shown as usual by first applying the Holmgren transform to this inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate given via a partition of unity and Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator.