Three-spheres theorem for harmonic functions (non-concentric case)

TL;DR

This paper extends Hadamard's three-circle theorem to harmonic functions on non-concentric spheres using inversion techniques, with applications to propagation of smallness and uniqueness.

Contribution

It introduces a new three-spheres theorem for harmonic functions on non-concentric spheres, expanding prior concentric cases through inversion methods.

Findings

Established a three-spheres inequality for non-concentric spheres

Extended the concentric case to non-concentric, non-touching spheres

Applied results to propagation of smallness and uniqueness in harmonic functions

Abstract

A direct analog of Hadamard's three-circle theorem is obtained for harmonic functions (in weighted L^2-norm) in case of (n-1)-dimensional non-concentric spheres in R^n. The result extends the concentric case to correlated non-concentric, non-touching spheres via an inversion technique. Applications to propagation of smallness and uniqueness for harmonic functions are given.