Geodesic
Sets of determination for solutions of the Helmholtz equation
Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) no. 2, pp. 309-328 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 31B10, 35J05
Keywords: Helmholtz equation; set of determination; decomposition of $L^1$ 
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     author = {Rano\v{s}ov\'a, Jarmila},
     title = {Sets of determination for solutions of the {Helmholtz} equation},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {309--328},
     year = {1997},
     volume = {38},
     number = {2},
     mrnumber = {1455498},
     zbl = {0887.35035},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1997_38_2_a11/}
}
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Ranošová, Jarmila. Sets of determination for solutions of the Helmholtz equation. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) no. 2, pp. 309-328. http://geodesic.mathdoc.fr/item/CMUC_1997_38_2_a11/