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 $k$, $\Delta u_k+k^2 u_k=0$, on (a small ball in) either $\mathbb{R}^n$, $\mathbb{S}^n$, or $\mathbb{H}^n$. For a fixed point $p$, we define \[M_{u_k}(r)=\max_{d(x,p)\le r}|u_k(x)|.\] It is known that the following three ball inequality \[M_{u_k}(2r)\le C(k,r,\alpha)M_{u_k}(r)^{\alpha}M_{u_k}(4r)^{1-\alpha}\] 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). Moreover, we compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation.


Introduction
In the present work we study constants in the three ball inequality for solutions of the Helmholtz equation. We begin by recalling Hadamard's celebrated three circle theorem. Let f be a holomorphic function in the disk D R = {z ∈ C : |z| < R}. Then its maximum function M f (r) = max |z|≤r |f (z)| satisfies the convexity condition for any r 0 , r 1 < R and α ∈ (0, 1). The proof of (1.1) is based on the fact that log |f | is a subharmonic function. Note that by the maximum principle (1.1) also holds when the maximum is taken over circles. Surprisingly, Hadamard's theorem generalizes to other classes of functions, such as solutions of second order elliptic equations (for this case the maximum should be taken over balls) and their gradients. We refer the reader to the article [11] of Landis and to the survey [1]. Three spheres theorems for the gradients of harmonic functions and, more generally, harmonic differential forms can be found in [15]. The three ball theorem for solutions of the Helmholtz equation on Riemannian manifolds was studied in [16]. This has various applications, for example it was one of the tools used to estimate the Hausdorff measure of the nodal sets of Laplace eigenfunctions, see [12,13].
We consider the Helmholtz equation We study properties of functions that satisfy the Helmholtz equation on some geodesic ball in the manifold. Fix a point p ∈ M and denote by B(p, r) the geodesic ball of radius r centered at p. Then for a function u we define The following doubling inequality holds for Laplace eigenfunctions on a closed manifold for small r, some fixed α ∈ (0, 1), and constants C 3 and C 4 only depending on the curvature. Further results on the propagation of smallness for eigenfunctions were obtained in [14]. In this article we show that (1.4) is sharp in the following sense: The coefficient C 3 e C 4 kr in (1.4) cannot be replaced by a function growing subexponentially in kr as k grows. This is done by constructing special families of solutions of the Helmholtz equation on Euclidean spaces, hyperbolic spaces, and the standard spheres. We also compare (1.4) with the increased stability for solutions of the Cauchy problem for the Helmholtz equation studied in [8,10,3]. Roughly speaking, the idea is that one can estimate the solution in the interior of some convex domain from an a priori bound and an estimate of the Cauchy data on some part of the boundary. Moreover, the estimate does not depend on k. For solutions of the Helmholtz equation in a geodesic ball B (p, R) we prove for r 1 < R 1 < R that (1.5)  [9,3]. The structure of the paper is as follows. We prove the sharpness of the three ball inequality (1.4) in Section 2. In Section 2.1 we present the argument for the Euclidean space, while the arguments for the hyperbolic space and the sphere are given in Section 2.2. We prove inequality (1.5) in Section 3. Finally, we list some properties of the Bessel functions, and collect some comparison theorems for solutions of the Sturm-Liouville equations in Appendix.
Acknowledgements. The authors are grateful to the anonymous referee for useful comments and suggestions.
solves the Helmholtz equation (1.2). Moreover, any solution of (1.2) in R n (or in the unit ball) can be decomposed into a series of such solutions.
In order to study the constant in the three ball inequality (1.4) that involves the maximum function, we analyze the behavior of the Bessel functions. From now on we assume that n = 2 for simplicity. Our results can be easily extended to all dimensions n ≥ 2.
Lemma 2.1. Let 0 < γ < δ < 1 and set β = √ 1 − δ 2 . Then there exists a constant C, only depending on γ and δ, such that for any positive number m we have Proof. The strategy is to apply the Sturm comparison theorem, see Theorem B.2. We apply the theorem to the Bessel function J m solving the Bessel equation and a solution of the Euler equation Let y be the solution of (2.2) satisfying the initial conditions y(γm) = J m (γm) and y ′ (γm) = J ′ m (γm).
We know that J m is positive and increasing on [0, m]. The latter can be verified by using the second derivative test and inserting the argument of the first maximum of J m into the equation Hence all the conditions in the comparison theorem are satisfied and we conclude that Any solution of the Euler equation (2.2) is on the form Using that We can now prove the main result of this section.
Theorem 2.2. Assume that there is an α ∈ (0, 1) and a constant C(k, r, α) such that for any solution u k of the Helmholtz equation (1.2) the following three ball inequality holds Then C(k, r, α) grows at least exponentially in kr. More precisely, C (k, r, α) ≥ ce dαkr , where c and d are absolute constants.
Proof. Consider solutions of the Helmholtz equation on the form The maximal function then simplifies to We now use the fact that for m > 0 the maximum of J m (x) is attained in the interval (m, m (1 + ε (m))), where ε (m) → 0 as m → ∞. This is a well known result on the asymptotic of the first zero of the Bessel functions, for the convenience of the reader we include a simple proof in Appendix A. We choose m 0 such that ε (m) ≤ 1/3 when m ≥ m 0 . Assume first that Then given r we can find m ≥ m 0 such that 6kr/5 < m < 3kr/2.

2.2.
Solutions of the Helmholtz equation on the sphere and hyperbolic space. In this section we repeat the argument of the sharpness of the three ball inequality on the hyperbolic space and sphere. We show in particular that assumptions on the sign of the curvature do not lead to better behavior of the constant in the three ball inequality. Again, we use the spherical symmetry of the spaces and separation of variables to construct a solution of (1.2) that is the product of a radial and a spherical factor. On the sphere the radial part is given by Legendre polynomials. For the hyperbolic space the radial part is also explicitly known, see [4, p. 4222 eq. (2.26)]. Once again, in our argument we only use the differential equation for the radial part. We define Furthermore, we use the associated functions cos K (r) = (sin K (r)) ′ , cot K (r) = cos K (r) sin K (r) , and tan K (r) = 1 cot K (r) . Then the Laplacian of a simply connected ndimensional Riemannian manifold (M, g) with constant sectional curvature K is given in polar coordinates by In this section we work in dimension two. Assume that u k (r, θ) = R (r) Θ(θ) is a solution of the Helmholtz equation. Then R (r) satisfies the equation Let κ = K/k 2 and let L κ,m (ρ) be the solution of the differential equation We begin by estimating the maximum point of L κ,m from below. Let Note that for ρ ≤ R κ we have that sin κ (ρ) is increasing, or equivalently that cos κ (ρ) ≥ 0.
Applying Lemma 2.5 together with gives (2.10), since We want to estimate the ratio of the values of L κ,m at two points ρ 2 > ρ 1 > sin −1 κ (m). In contrast with the Bessel functions, we do not locate the maximum precisely.
Proof. Consider the family of functions where m is a non-negative integer. By construction, u k,m solves the Helmholtz equation. Thus for any m we have the inequality Note that choosing m = 0 gives C α (k, r, K) ≥ 1 by Remark 2.4. Thus if we assume that kr < C 1 for some constant C 1 , we may choose c 2 and c 1 small enough such that the inequality holds. Assume first that K < 0 so that (M, g) is the hyperbolic plane. If kr > C 1 we choose a positive integer m such that 10m < 18kr < 11m. We apply (2.10) with ρ 1 = kr and ρ 2 = µkr < 2kr, where µ = 19/17. Then ρ 2 < 2/3mδ with δ < 1. We obtain where β = √ 1 − δ 2 and A = sin κ (ρ 2 ) /ρ 2 < sin κ (2kr) /(2kr) < 4 sinh(π/4)/π < 10/9. Therefore q = µ/A > 1 is an absolute constant and we have Thus there are c 1 > 0 and c 2 > 0 such that On the other hand, we have 2kr > ξm for ξ = 10/9. Applying (2.11) we get where C 0 is an absolute constant. Note also that m kr. Then (2.13) follows for negative curvature.

The reverse three ball inequality
The question of stability of the solution to the Cauchy problem for the Helmholtz equation and the dependence of the estimates on the wave number k was studied by many authors, see e.g. [8,19,10,3]. We include a special case of the results to demonstrate the difference between the usual three ball theorem and the reverse one. Let B be a geodesic ball with radius less than the injectivity radius of (M, g).
Additionally, in the case that K > 0 we assume that the diameter of B is strictly less than π 2 √ K . Then there exists a constant C 0 > 0 such that for any function w ∈ C 2 0 (B) and k ≥ 0 we get Proof. The proof relies on the following weighted inequality proved in [3, where φ is a strictly convex weight with D 2 φ > 0 and grad φ = 0 onB and t > t 0 . The powerful property of the last inequality is that C 1 and t 0 do not depend on k. An example of a convex function φ is the distance function φ(x) = dist(p, x) from a point p ∈ B. For the case when K > 0 we choose p such that the diameter of the set B ∪ {p} is still less than π Suppose now that u k is a solution of the Helmholtz equation (1.2) in a ball B that satisfies the conditions in Lemma 3.1. We apply inequality (3.1) to w = u k χ, where χ ∈ C 2 0 (B) is compactly supported on B and identically one on a ball B 1 ⊂⊂ B. This gives the inequality The last inequality implies that for any r < R such that B R = B(x, R) and B r = B(x, r) are geodesic balls satisfying the conditions in Lemma 3.1, there is a constant C 2 (r, R) such that (3.2) Br |u k | 2 + | grad u k | 2 dvol ≤ C 2 (r, R) Inequality (3.2) shows that if u 2 k +| grad u k | 2 is small on the annulus B R \B r , then it is small on the whole ball B R . For the Euclidean space an alternative proof can be obtained by decomposing a solution u k into series of products of Bessel functions and spherical harmonics. From this, one can deduce (3.2) from the Debye asymptotic of the Bessel functions.
To compare with the previous section, we can also use Caccioppoli's inequality to control the Sobolev norm of u k by its L 2 norm. Assume that u k ∈ C 2 (Ω) and ∆ M u k + k 2 u k = 0 in Ω + . Then there exists a constant C = C(M ) such that Proof. There exists a smooth function ϕ + with compact support in Ω + that satisfy ϕ + = 1 on Ω and |grad ϕ + | ≤ C ε and |∆ M ϕ + | ≤ C ε 2 . Then, using the divergence theorem, we have Hence Ω |grad u k | 2 dvol ≤ k 2 + Cε −2 On the other hand, choosing a similar function ϕ − ∈ C ∞ 0 (Ω) such that ϕ − = 1 on Ω − , we conclude that We now go back to the inequality (3.2), and apply the Caccioppoli inequality. Rename R 1 = R + ε and r 1 = r − ε. This gives the following