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Hadamard’s problem and Coxeter groups: New examples of Huygens’ equations. (English. Russian original) Zbl 0845.35062
Funct. Anal. Appl. 28, No. 1, 3-12 (1994); translation from Funkts. Anal. Prilozh. 28, No. 1, 3-15 (1994).
The main result of the work is the following. By a root system we mean a finite set $${\mathfrak R}= \{\alpha\}\subseteq \mathbb{R}^n$$ of vectors that have pairwise different directions, span $$\mathbb{R}^n$$, and satisfy the condition that the reflections $$s_\alpha(\beta)= \beta- 2\alpha(\alpha, \beta)/(\alpha, \alpha)$$ transform $$\mathfrak R$$ into itself. The reflections $$s_\alpha$$ generate a finite group $$\mathcal W$$ referred to as the Coxeter group. Let us define the potential associated with $$\mathcal R$$ by setting $u(x)= \sum_{\alpha\in {\mathfrak R}_+} {m_\alpha(m_\alpha+ 1)(\alpha, \alpha)\over (\alpha, x)^2},$ where $$m_\alpha$$ are integers such that the function $$m(\alpha)= m_\alpha$$ is $${\mathcal W}$$-invariant, and $${\mathfrak R}_+\subset {\mathfrak R}$$ denotes the set of roots positive with respect to a suitable linear form on $$\mathbb{R}^n$$. Consider the hyperbolic operator ${\mathcal L}= {\partial^2\over \partial t^2}- {\partial^2\over \partial x^2_1}-\cdots- {\partial^2\over \partial x^2_n}- {\partial^2\over \partial y^2_1}-\cdots- {\partial^2\over \partial y^2_m}+ u(x_1,\dots, x_n),$ where $$N= m+ n$$ is an arbitrary odd number satisfying the inequality $$N\geq 3+ 2 \sum_{\alpha\in {\mathfrak R}_+} m_\alpha$$. Theorem. For any Coxeter group $$\mathcal W$$ and integer-valued $${\mathcal W}$$-invariant function $$m$$ on the corresponding root system $$\mathfrak R$$ the equation $${\mathcal L}\varphi= 0$$ satisfies Huygens’ principle.

##### MSC:
 35L10 Second-order hyperbolic equations
##### Keywords:
Hadamard’s problem; Coxeter group; Huygens’ principle
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