CHAPTER 1 Reduction o f Operators of Firs t Order to Canonical Form s 1. Firs t order equations. Le t u s begin wit h th e simples t example- a linea r firs t orde r partial differentia l operato r wit h rea l coefficient s actin g o n rea l valued function s w(x) o f X = (x 1 , •• , *") i n a n ope n se t O i n R n a s a directional derivative : (1.1) Pw=fV(x)(aw/dxO . i At every poin t x i n H a smoothly varyin g (C°° ) vecto r a{x) = (A 1 , *• , an) i s given, and th e actio n o f P o n th e functio n w i s to differentiat e w i n the directio n o f a, th e operator P i s also called a vector field . On e solve s the equatio n Pw - f b y finding "in - tegral curves " of th e vecto r field, i.e . curve s wit h th e propert y tha t a t ever y poin t o n each curve th e vecto r a a t tha t poin t i s tangent t o th e curve . I f suc h a curve i s given by x = x(f) with t a s a real paramete r the n x{t) satisfie s th e syste m o f ordinar y differentia l equations , the "characteristi c equations'* , (1.2) dx!/dt=a1(x(t)), j= 1 , , « . From th e theor y o f ordinar y differentia l equation s i t follow s tha t throug h ever y poin t x 0 in n ther e passe s exactly on e integra l curve . I f w e consider th e restrictio n o f a functio n vv(x) t o suc h a curve, on whic h i t become s a function o f f , w e find (using summatio n convention) dw _ dw dx f __ dt ~ " dx ^ It Thus on th e curv e w satisfie s a simple ordinar y differentia l equatio n an d th e valu e of w is then uniquel y determine d (a t leas t locally ) b y give n initia l value s on a hypersurface ( a sur - face of dimensio n n ~- 1) which i s transversal t o th e vecto r field . Locally on e ca n introduc e ne w independen t variable s y = (yx, •• ,'" ) whic h "straighten out " th e integra l curve s an d reduc e th e differentia l operato r t o a particularl y simple form , (1.3) P= ^VT^ **0- The leadin g symbo l a J \. i s thus transforme d t o \r) n . An y solutio n o f Pw = 0 i s then simply a n arbitrar y functio n o f (y l •• , vn _ 1 ) . W e see therefor e tha t th e loca l stud y o f 3 http://dx.doi.org/10.1090/cbms/017/02
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