Sharp three sphere inequality for perturbations of a product of two second order elliptic operators and stability for the Cauchy problem for the anisotropic plate equation

TL;DR

This paper establishes a sharp three sphere inequality for solutions to perturbed elliptic operators and derives stability estimates for the Cauchy problem in anisotropic plate equations, with applications to detecting inclusions.

Contribution

It introduces a new sharp three sphere inequality for third order perturbations of elliptic operators and applies it to stability and size estimation problems in anisotropic plates.

Findings

Proves a sharp three sphere inequality for perturbed elliptic operators.

Derives quantitative unique continuation estimates for anisotropic plate equations.

Provides stability estimates for the Cauchy problem and applications to inclusion detection.

Abstract

We prove a sharp three sphere inequality for solutions to third order perturbations of a product of two second order elliptic operators with real coefficients. Then we derive various kinds of quantitative estimates of unique continuation for the anisotropic plate equation. Among these, we prove a stability estimate for the Cauchy problem for such an equation and we illustrate some applications to the size estimates of an unknown inclusion made of different material that might be present in the plate. The paper is self-contained and the Carleman estimate, from which the sharp three sphere inequality is derived, is proved in an elementary and direct way based on standard integration by parts.