LINEAR PARTIA L DIFFERENTIA L EQUATION S 5 where P { an d P 2 ar e linearl y independen t (real) vector field s i n H . Thi s operator i s elliptic. I t i s a remarkable fac t that , locally, thi s operator i s no mor e genera l tha n th e Cauchy-Riemann operator . I n fac t ther e exis t ne w local coordinate s (x , y), z = x + iy t s o that P take s th e for m '-£ with X a nonvanishing factor . Th e equation Pw - f i s then locall y equivalen t t o w- = //X. The operato r P thu s induce s a "complex structure " in O th e solution s o f Pw = 0 be - come holomorphi c function s relativ e t o thi s structure i n particula r the y ar e al l o f clas s C°° . The local coordinate s z ar e obtaine d a s local solution s o f Pz = 0 wit h R e grad z an d Im grad z linearl y independent . I f z i s such a local solutio n the n i n term s o f th e coordin - ates, z an d z th e operato r P mus t the n have th e for m P= X d/9z~ 4 - ji 3/dz , with , how- ever, fjt = Pz = 0 . Th e proo f o f th e existenc e o f suc h loca l solution s z i s not trivial , see for instanc e Couran t an d Hilber t [5 , Chapter 4 , §8] . Turn nex t t o n 2 an d P = P Y + i"P 2 wit h P , an d P 2 linearl y independen t i n O. A t ever y poin t i n O th e tw o vector field s spa n a two-dimensional plane . I t i s reason- able t o tr y t o exten d th e procedur e use d i n handlin g a single vecto r fiel d (1.1) , namely, t o seek a family o f two-dimensiona l integra l surfaces . Thes e hav e th e propert y tha t a t an y poin t on on e o f the m bot h vecto r field s ar e tangen t t o it . Thi s is impossible i n general . Th e class- ical Frobeniu s theore m give s a necessary an d sufficien t conditio n fo r thi s t o b e possible . Th e condition i s that th e commutato r [P v P 2 ] = PXP2 "^i^x ° ^ t ne operator s P x an d P 2 is a linear combinatio n o f th e operator s P x an d P 2 . I f thi s condition hold s then , by Frobenius' theorem , th e se t 1 2 i s foliated (simpl y covered ) by integra l surfaces—on e throug h each point . Th e operator s P { *P2 an d henc e P, the n ac t withi n eac h surface . Tha t mean s that i n eac h surfac e S w e have th e situatio n o f th e precedin g case : P define s a complex structure o n S. Suppose th e Frobeniu s integrabilit y conditio n i s not satisfied , i.e . P p P 2 an d [Pp P 2 ] ar e linearly independent . Strang e phenomen a ca n occur . I n 195 7 Han s Lew y [22] presente d his , now famous , exampl e o f suc h an operato r i n R 3 : having th e propert y tha t fo r mos t (i n som e sense ) C°° function s / th e equatio n Pw = f admits n o solutio n i n an y ope n set . Her e 1 " 2 dx X dx* ' Fl ~ 2 dx* X dx* ' [ v P2] " 3x 3 * Shortly thereafte r Hormande r derive d a general necessar y conditio n fo r loca l solvabilit y o f any linea r partia l differentia l operato r o f an y orde r Pw = f. I n th e cas e tha t P = Px + iP2 + c, wit h P v P 2 vecto r field s (no t necessaril y linearl y independent) , th e necessar y
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