On the three ball theorem for solutions of the Helmholtz equation

2020/09/19 by Berge, Stine Marie, Malinnikova, Eugenia
#35J05 #35J15 #47A75 #53C20 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2009.09225

Abstract

Let u_k be a solution of the Helmholtz equation with the wave number k, Δu_k+k² u_k=0, on a small ball in either ℝⁿ, \mathbbSⁿ, or ℍⁿ. For a fixed point p, we define M_uk(r)=max_{d(x,p)≤ r}|u_k(x)|. The following three ball inequality M_uk(2r)≤ C(k,r,α)M_uk(r)^αM_uk(4r)^1-α is well known, it holds for some α∈ (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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