Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9521105 | Indagationes Mathematicae | 2005 | 41 Pages |
Abstract
Let k = (kα)αεâ, be a positive-real valued multiplicity function related to a root system â, and Îk be the Dunkl-Laplacian operator. For (x, t) ε âN, à â, denote by uk(x, t) the solution to the deformed wave equation Îkuk,(x, t) = δttuk(x, t), where the initial data belong to the Schwartz space on âN. We prove that for k ⩾ 0 and N ⩾ l, the wave equation satisfies a weak Huygens' principle, while a strict Huygens' principle holds if and only if (N â 3)/2 + Σαεâ+kα ε â. Here â+ â â is a subsystem of positive roots. As a particular case, if the initial data are supported in a closed ball of radius R > 0 about the origin, the strict Huygens principle implies that the support of uk(x, t) is contained in the conical shell {(x, t), ε âN à â | |t| â R ⩽ âxâ ⩽ |t| + R}. Our approach uses the representation theory of the group SL(2, â), and Paley-Wiener theory for the Dunkl transform. Also, we show that the (t-independent) energy functional of uk is, for large |t|, partitioned into equal potential and kinetic parts.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Salem Ben Saïd, Bent Ãrsted,