# zbMATH — the first resource for mathematics

Propagation de la polarisation pour des problèmes aux limites convexes pour les bicaractéristiques. (Propagation of the polarization for convex boundary value problems for the bicharacteristics). (French) Zbl 0593.58037
The author considers the boundary value problem $D_ xu=G(x,y,D_ y)u,\quad x>0,\quad y\in R^ n,\quad \beta (y,D_ y)u=0,\quad x=0\quad y\in R^ n$ where u is vector-valued. $$D_ x-G$$ is of real principal type, the principal symbol of G can be reduced to a suitable block form, and $$\beta$$ satisfies the Lopatinskij-Shapiro condition. The following results are established. The polarized wavefront set propagates along Hamiltonian orbits above null bicharacteristic strips transversal to $$x=0$$ (when $$x>0$$, this is an earlier result of Denker). In the gliding case, the polarized wavefront set propagates along boundary Hamiltonian orbits (which lie above gliding rays). Interaction of polarization between transversal and gliding rays is also studied.
Reviewer: P.Godin

##### MSC:
 58J32 Boundary value problems on manifolds 58J47 Propagation of singularities; initial value problems on manifolds
Full Text:
##### References:
 [1] Dencker N., On the propagation of polarization sets for systems ofreal principal type 46 pp 351– (1982) · Zbl 0487.58028 [2] Anderson K.G., The propagation of singularities along gliding rays pp 197– (1977) [3] Gerard C., Thèse de 3ème cycle (1984) [4] Taylor M.E., Reflection of singularities of solutions to systems of differential equations 28 pp 457– (1975) · Zbl 0332.35058 [5] Eskin, G. 1977. ”Seminaire Goulaouic-Schwartz”. Vol. 12, exposè [6] Melrose R.B., Transformations of boundary value problems 147 pp 149– (1981) · Zbl 0477.35087 [7] Melrose R.B., Equivalence of glancing hypersurfaces> 37 pp 165– (1976) · Zbl 0354.53033 [8] Melrose R.B., Singularities of boundary value problems II 5 pp 129– (1982) · Zbl 0546.35083 [9] Melrose R.B., In Singularities of Boundary Value Problems · Zbl 0452.35120 · doi:10.1002/cpa.3160310504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.