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 geared up : "Boundary worth challenge for Evolution Partial Differential Operators", Liege, 1976 and "Singularities in Boundary price Problems", Maratea, 1980. It was once certainly essential to checklist the huge development discovered within the box of the propagation of singularities of Schwartz Distri­ butions which led lately to the start of a brand new department of Mathema­ tical research referred to as Microlocal research. so much of this concept was once often equipped to be utilized to distribution recommendations of linear partial differential difficulties. a wide a part of this institute nonetheless went during this path. yet, nevertheless, it was once additionally time to discover the recent pattern to take advantage of microlocal research In non linear differential difficulties. i'm hoping that the Castelvecchio NATO ASI reached its reasons with the aid of the extra recognized professionals 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 by way of "11 Ciocco" a world holiday heart, In a comfy resort positioned in fabulous mountain atmosphere and supplied with all convention and activity facilities.

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Alors, pour ME~, 3. 3. TMoreme : On suppose s >m-1/2, tel que on a : Yj Mu pour =L Ej, k, S NS Yk u j,k=O, ... ,m-1, et II est clair que, pour EjkSE1jJ j-k (mod If (JRn )) n (JR). E E1jJo, E u verifie les memes hypotheses que u. 2) It; I = 0, et egal a l'operateur P etant inversible en ces points, on a 1 J-M. BONY 24 s+2 , e t donc Eu EH u par (I-E)u, s+\/2-JS-J' Y. ME u E H CH . On peut donc, en rempla<;ant J supposerque WF(u) ne rencontre pas {(X,t,I;,T)III;I=o}. ]) (modCoo Mu::tM'u+L:ASNSu Y.

Laurent [9] dans Ie cadre analytique. Les resultats en sont rappeles aux paragraphes 5 et 6. 1. relative au cas ou, pour l'equation des ondes en dimension 2 d'espace, les donnees de Cauchy sont singulieres sur plusieurs courbes concourantes. II s'agit de l'analogue, pour Ie probleme de Cauchy, du theoreme demontre dans [7] (voir aussi [10]) sur l'interaction de trois ondes. Les solutions sont alors regulieres hors des surfaces caracteristiques issues des courbes et du cone d'onde issu de leur point de concours.

Sont des algebY'es pOUX' Hs,v(A) =Hs,o(~ ,v) =Hs,o(,£ ,v) s>(n+1)/2. 4. 5. Remarque. A l'exception du cas ou ~A 'V P est un produit d'operateurs du second ordre, n'est pas engendre - meme microlocalement - par des champs de vecteurs. 6. ne peuvent pas s'etendre au cas d'equations d'ordre superieur. En utilisant la seconde microlocalisation relative a T~(lRn+I), on pourrait demontrer l'analogue du theoreme 7. I. pour des donnees de H~. --v j, V , pour V d e d lmenSlon que 1 conque d ans lRn - tou- Cauc h y d ans jours en supposant que la surface cylindro-conique projection de lisse hors de V.

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