Jarmila Rano\v{s}ov\'a
Sets of determination for solutions of the Helmholtz equation

Comment.Math.Univ.Carolinae 38,2 (1997) 309-328.

Abstract:Let $\alpha > 0$, $\lambda = (2\alpha )^{-1/2}$, $S^{n-1}$ be the $(n-1)$-dimensional unit sphere, $\sigma $ be the surface measure on $S^{n-1}$ and $h(x) = \int _{S^{n-1}} e^{\lambda \delimiter "426830A x,y\delimiter "526930B } d\sigma (y)$. \par We characterize all subsets $M$ of $\Bbb R^n $ such that $$ \inf \limits _{x\in \Bbb R^n}{u(x)\over h(x)} = \inf \limits _{x\in M}{u(x)\over h(x)} $$ for every positive solution $u$ of the Helmholtz equation on $\Bbb R^n$. A closely related problem of representing functions of $L_1(S^{n-1})$ as sums of blocks of the form $ e^{\lambda \delimiter "426830A x_k,.\delimiter "526930B }/h(x_k)$ corresponding to points of $M$ is also considered. The results provide a counterpart to results for classical harmonic functions in a ball, and for parabolic functions on a slab, see References.

Keywords: Helmholtz equation, set of determination, decomposition of $L^1$
AMS Subject Classification: 35J05, 31B10