Introduction

The theor y o f partia l differentia l equation s has see n a remarkable developmen t i n th e

last twent y years . Ne w questions have bee n aske d an d ne w an d powerfu l technique s de-

veloped, leading often t o deepe r understandin g an d resolutio n o f ol d problems.

In thes e lecture s w e take u p several topic s i n th e theor y o f linea r partia l differentia l

equations, beginning with rather elementary, expository , materia l an d goin g on t o som e of

the curren t development s an d techniques . Th e lecture s ar e mean t fo r th e nonexpert-a s a n

introduction t o som e o f th e curren t question s an d ideas . I t contain s ol d a s well a s new re-

sults. Sinc e I wish t o includ e som e dee p results I have had t o b e technica l o n occasion , bu t

I have endeavoured t o describ e th e necessar y background .

Chapter 1 deals with th e rathe r classica l question o f reducin g (firs t order ) operator s t o

simple, canonical forms , suc h a s th e Cauchy-Riemann equations , b y suitabl e chang e o f inde-

pendent variables . Excep t fo r a few technica l requirements , which ar e described , i t i s more

or les s self-contained. I n § 1 we present a variant o f th e Lew y exampl e o f a n inhomogeneou s

partial differentia l equatio n withou t solutions . Th e questio n o f transformatio n o f equation s

to canonical form s i s related t o th e existenc e o f nontrivia l solution s o f th e homogeneou s

equations. I n §§ 2 an d 3 examples o f homogeneou s equation s having only trivial , o r re-

stricted, solution s ar e described; thes e result s ar e new . Th e question o f transformin g a sys-

tem o f n equation s i n R

2n

t o th e Cauchy-Rieman n operator s i n C

n

i s treated i n § 4 b y

the metho d o f Malgrang e [25] . Thoug h w e ad d nothin g ne w t o hi s argument, i t i s so strik-

ing, to m y mind , tha t I feel i t wort h presentin g agai n t o a (possibly) wider audienc e i n thes e

lectures.

In Chapter 2 we tak e u p som e o f th e technica l tool s tha t ar e no w provin g s o useful ,

namely pseudo-differentia l operators , an d th e notio n o f wav e fron t se t (or singula r spectrum )

of a distribution , an d presen t severa l applications . § 5 give s th e definitio n an d a brie f

survey o f a (somewhat restricted ) clas s of pseudo-differentia l operators . I n §§ 6 an d 7 , as a

rather remarkabl e illustratio n o f thei r use , we prove a theorem o f Caldero n [2 ] o n loca l

uniqueness fo r th e initia l valu e problem , an d als o a slight extensio n o f hi s result. (Thi s is

perhaps th e mos t technicall y involve d materia l o f th e lectures. ) Th e wav e front se t i s then

introduced i n § 8 an d Hormander' s elegan t theore m [15] o n propagatio n o f singularitie s is

proved. Finally , i n § 9 w e extend hi s result t o includ e th e behaviou r a t a boundary, du e t o

* The autho r wa s partl y supporte d b y a National Scienc e Foundatio n Gran t GP-3462 0 an d b y a

contract wit h th e Offic e o f Nava J Researc h No . N00014-67A-0467-0024.

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http://dx.doi.org/10.1090/cbms/017/01