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

Norair U. Arakelian, Norayr Matevosyan

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.

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

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.

Paper Structure

This paper contains 6 sections, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminary constructions
  3. Correlated non-concentric spheres
  4. Log-Convexity
  5. A THREE-SPHERES THEOREM.
  6. Three balls theorem (non-concentric case)