Sets of determination for solutions of the Helmholtz equation

Jarmila Ranošová

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 2, page 309-328
  • ISSN: 0010-2628

Abstract

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Let α > 0 , λ = ( 2 α ) - 1 / 2 , S n - 1 be the ( n - 1 ) -dimensional unit sphere, σ be the surface measure on S n - 1 and h ( x ) = ∫ S n - 1 e λ 〈 x , y 〉 d σ ( y ) . We characterize all subsets M of ℝ n such that inf x ∈ ℝ n u ( x ) h ( x ) = inf x ∈ M u ( x ) h ( x ) for every positive solution u of the Helmholtz equation on ℝ n . A closely related problem of representing functions of L 1 ( S n - 1 ) as sums of blocks of the form e λ 〈 x k , . 〉 / 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.

How to cite

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Ranošová, Jarmila. "Sets of determination for solutions of the Helmholtz equation." Commentationes Mathematicae Universitatis Carolinae 38.2 (1997): 309-328. <http://eudml.org/doc/248073>.

@article{Ranošová1997,
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 $h(x) = \int _\{S^\{n-1\}\} e^\{\lambda \langle x,y\rangle \}\,d\sigma (y)$. We characterize all subsets $M$ of $\mathbb \{R\}^n $ such that \[ \inf \limits \_\{x\in \mathbb \{R\}^n\}\{u(x)\over h(x)\} = \inf \limits \_\{x\in M\}\{u(x)\over h(x)\} \] for every positive solution $u$ of the Helmholtz equation on $\mathbb \{R\}^n$. 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.},
author = {Ranošová, Jarmila},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Helmholtz equation; set of determination; decomposition of $L^1$; decomposition of },
language = {eng},
number = {2},
pages = {309-328},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Sets of determination for solutions of the Helmholtz equation},
url = {http://eudml.org/doc/248073},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Ranošová, Jarmila
TI - Sets of determination for solutions of the Helmholtz equation
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 2
SP - 309
EP - 328
AB - 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 $\mathbb {R}^n $ such that \[ \inf \limits _{x\in \mathbb {R}^n}{u(x)\over h(x)} = \inf \limits _{x\in M}{u(x)\over h(x)} \] for every positive solution $u$ of the Helmholtz equation on $\mathbb {R}^n$. 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.
LA - eng
KW - Helmholtz equation; set of determination; decomposition of $L^1$; decomposition of
UR - http://eudml.org/doc/248073
ER -

References

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