On the three ball theorem for solutions of the Helmholtz equation

TL;DR

This paper investigates the three ball inequality for Helmholtz equation solutions, revealing that the associated constant grows exponentially with the wave number, and compares this with stability results on Riemannian manifolds.

Contribution

It demonstrates the exponential growth of the three ball inequality constant with respect to the wave number for Helmholtz solutions.

Findings

Constant grows exponentially with wave number k

Provides comparison with stability in Riemannian settings

Extends understanding of solution behavior in different geometries

Abstract

Let $u_k$ be a solution of the Helmholtz equation with the wave number $k$, $\Delta u_k+k^2 u_k=0$, on a small ball in either $\mathbb{R}^n$, $\mathbb{S}^n$, or $\mathbb{H}^n$. For a fixed point $p$, we define $M_{u_k}(r)=\max_{d(x,p)\le r}|u_k(x)|.$ The following three ball inequality $M_{u_k}(2r)\le C(k,r,\alpha)M_{u_k}(r)^{\alpha}M_{u_k}(4r)^{1-\alpha}$ is well known, it holds for some $\alpha\in (0,1)$ and $C(k,r,\alpha)>0$ independent of $u_k$. We show that the constant $C(k,r,\alpha)$ grows exponentially in $k$ (when $r$ is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds.