On the three ball theorem for solutions of the Helmholtz equation

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.

17 pages