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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  1. Aikawa H., Sets of determination for harmonic functions in an NTA domains, J. Math. Soc. Japan, to appear. MR1376083
  2. Bauer H., Harmonische Räume und ihre Potentialtheorie, Springer-Verlag Berlin-Heidelberg-New York (1966). (1966) Zbl0142.38402MR0210916
  3. Bonsall F.F., Decomposition of functions as sums of elementary functions, Quart J. Math. Oxford (2) 37 (1986), 129-136. (1986) MR0841422
  4. Bonsall F.F., Domination of the supremum of a bounded harmonic function by its supremum over a countable subset, Proc. Edinburgh Math. Soc. 30 (1987), 441-477. (1987) Zbl0658.31001MR0908454
  5. Bonsall F.F., Some dual aspects of the Poisson kernel, Proc. Edinburgh Math. Soc. 33 (1990), 207-232. (1990) Zbl0704.31001MR1057750
  6. Caffarelli L.A., Littman W., Representation formulas for solutions to Δ u - u = 0 in ℝ n , Studies in Partial Differential Equations, Ed. W. Littman, MAA Studies in Mathematics 23, MAA (1982). (1982) MR0716508
  7. Gardiner S.J., Sets of determination for harmonic function, Trans. Amer. Math. Soc. 338 (1993), 233-243. (1993) MR1100694
  8. Korányi A., A survey of harmonic functions on symmetric spaces, Proc. Symposia Pure Math. XXV, part 1 (1979), 323 -344. (1979) MR0545272
  9. Korányi A., Taylor J.C., Fine convergence and parabolic convergence for the Helmholtz equation and the heat equation, Illinois J. Math. 27.1 (1983), 77-93. (1983) MR0684542
  10. Ranošová J., Sets of determination for parabolic functions on a half-space, Comment. Math. Univ. Carolinae 35 (1994), 497-513. (1994) MR1307276
  11. Ranošová J., Characterization of sets of determination for parabolic functions on a slab by coparabolic (minimal) thinness, Comment. Math. Univ. Carolinae 37 (1996), 707-723. (1996) Zbl0887.35064MR1440703
  12. Rudin W., Functional analysis, McGraw-Hill Book Company (1973). (1973) Zbl0253.46001MR0365062
  13. Taylor J.C., An elementary proof of the theorem of Fatou-Naïm-Doob, 1980 Seminar on Harmonic Analysis (Montreal, Que., 1980), CMS Conf. Proc., vol.1, Amer. Math. Soc., Providence, R.I (1981), 153-163. (1981) Zbl0551.31004MR0670103
  14. Watson G.N., Theory of Bessel functions, 2nd ed., Cambridge Univ. Press Cambridge (1944). (1944) Zbl0063.08184MR0010746

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