Advances in Microlocal Analysis by Gunter Bengel (auth.), H. G. Garnir (eds.)

By Gunter Bengel (auth.), H. G. Garnir (eds.)

The 1985 Castel vecchio-Pas coli NATO complicated research Institute is aimed to accomplish the trilogy with the 2 former institutes I equipped : "Boundary price challenge for Evolution Partial Differential Operators", Liege, 1976 and "Singularities in Boundary price Problems", Maratea, 1980. It was once certainly essential to list the enormous development learned within the box of the propagation of singularities of Schwartz Distri­ butions which led lately to the beginning of a brand new department of Mathema­ tical research referred to as Microlocal research. such a lot of this concept was once normally outfitted to be utilized to distribution recommendations of linear partial differential difficulties. a wide a part of this institute nonetheless went during this course. yet, however, it used to be additionally time to discover the hot pattern to take advantage of microlocal research In non linear differential difficulties. i am hoping that the Castelvecchio NATO ASI reached its reasons with assistance from the extra recognized specialists within the box. The assembly used to be held in Tuscany (Italy) at Castelvecchio-Pascoli, little village within the mountains north of Lucca on September 2-12, 1985. It was once hosted via "11 Ciocco" a global holiday middle, In a comfy inn positioned in awesome mountain atmosphere and supplied with all convention and activity facilities.

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Les elements de 36 J-M. BONY Op(E O, I) sont transformes en eux-memes par conjugaison par ce diffeo- morphisme singulier, et il en est de meme des champs de vecteurs singuliers d'apres la formule de changement de variables. Dans l'image de x. t,. sont des symboles d'elements de l'image J 1 J , e t ils engendrent ceux de l'image de ~ Cela etablit a). , de ~ b) II suffit maintenant de prouver que, pour a(T) ::: EA. 4) 0,0) et il suffit de Ie demontrer microloca- lement dans un voisinage conique W de ,T ).

V' y. J u = ( a/at)j u u) (0. I) (x,O) =4>. (x) , J pour une equation semi-lineaire strictement hyperbolique d'ordre m. Nous cherchons a determiner les singularites d'une solution u EH s (s > n/2+m) pour t petit, connaissant les singularites des 4> •• Notons que, pour ces valeurs de s, l'existence de solutions pour J t petit est bien connue (voir [Ill ), et que, si la solution se prolonge pour t grand, l'etude de ses singularites releve des theoremes de propagation et d'interaction ([I], [2], [4], [5], [7], [10]).

On a ~ verifie (C. ). AlaI's. pour tout J-M. BONY 26 =) [P311] avec 4. 9) ICkI A I-1 LAC E 1jJo. 1), un theoreme general de regularite des solutions. Nous considererons ensuite deux applications : le cas d'une equation du second ordre avec donnees singulieres sur une sous~n, le cas d'une equation d'ordre quelconque avec variete lisse de donnees singulieres sur une hypersurface. 1. 2)3 des sousmodules de 1jJ1 (JRn+1) et 1jJ1 (JRn ). On suppose que cI& verifie la condition de corronutation (C. C.

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