On the three ball theorem for solutions of the Helmholtz equation

Let $u_k$ be a solution of the Helmholtz equation with the wave number $k$, $Δu_k+k^2 u_k=0$, 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 $M_{u_k}(2r)\le C(k,r,α)M_{u_k}(r)^αM_{u_k}(4r)^{1-α}$ is well known, it holds for some $α\in (0,1)$ and $C(k,r,α)>0$ independent of $u_k$. We show that the constant $C(k,r,α)$ 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.

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