Sets of determination for solutions of the Helmholtz equation

Ranošová, Jarmila

Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) no. 2, pp. 309-328.

Résumé

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.

Détail

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