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Helmholtz equation; set of determination; decomposition of $L^1$
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 $h(x) = \int_{S^{n-1}} e^{\lambda\langle x,y\rangle }\,d\sigma(y)$. 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\langle x_k,.\rangle }/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.
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