On the characteristics of partial differential equations

Transcription

1 Sur les caractérstques des équatos au dérvées artelles Bull Soc Math Frace 5 (897) 8- O the characterstcs of artal dfferetal equatos By JULES BEUDON Traslated by D H Delhech I a ote that was reseted to l Académe des Sceces ( ) I radly dcated a eteso of the oto of characterstc to artal dfferetal equatos of order hgher tha oe ad to more tha two deedet varables Here I roose to gve the roof of the facts that were aouced; however before I do that I wll ot out a termology that wll be of great utlty ( ) If z s a aalytc fucto of varables the oe wll have relatos of the form: (a) d α α d (α α α α α ) Oe ca cosder the quattes z ( vares from to ) to be α α deedet varables Oe calls ay system of values that are attrbuted to those symbols a elemet of order -dmesoal sace Two ftely-close elemets that verfy those relatos are sad to be uted Ay system of equatos the coordates of a elemet that verfes equatos (a) defes a multlcty M I each of those systems there are some relatos betwee ust z that defe a ot-le multlcty that oe calls the suort of M There are multlctes M such that each ot of the suort corresods to ust oe elemet of order If q s the umber of dmesos of the suort the we shall call them multlctes M q Frst cosder a artal dfferetal equato that s lear wth resect to the secod-order dervatves; let: () A ϕ Φ z ( ) J BEUDON Sur les sgulartés des équatos au dérvées artelles Comtes redus des séaces 4 (9 March 897) ( ) For more detals see J BEUDON Sur les systèmes d équatos au dérvées artelles dot les caracterérstques déedet d u ombre f de aramètres A de l E N S Su 896

2 Beudo O the characterstcs of artal dfferetal equatos whch the A ad ϕ are gve fuctos of the z for such a equato From the geeral theorems of Cauchy o the estece of tegrals of dfferetal systems oe ows that ay aalytc tegral of that equato ca be defed by gvg a multlcty M that t must cota The followg equatos: () z ( ) whch z ad are arbtrary fuctos of rereset a multlcty M I order to calculate the secod dervatves as fuctos of oe mst mae use of the formulas: () π from whch oe wll deduce that: (4) (5) Uo substtutg ths to the roosed equato () oe wll fally get: A A A A A ϕ I order for determacy to est oe must have: (6) (7) A A A A A ϕ

3 Beudo O the characterstcs of artal dfferetal equatos I shall call the multlctes M that are defed by equatos () (6) ad (7) sgular multlctes It s easy to see ther degree of geeralty whle remag qute geeral Ideed f oe chooses arbtrarly as a fucto of the equato (6) wll ermt oe to obta ad as a result Uo relacg those symbols wth ther values equato (7) oe wll have a lear secod-order artal dfferetal equato that defes z If oe ows a tegral surface of equato () the the suorts of the sgular multlctes that are laced o that tegral surface wll be defed by the frst-order artal dfferetal equato (6) ad the oretatos of the frst-order elemets of those sgular multlctes wll be defed by equato (7) As oe ca easly see those two equatos have the same characterstcs We shall et see that those characterstcs have great mortace I shall ow erform a chage of varables that leaves ualtered but s such that wll become a fucto of ad the ew varable y I wll get formulas () () ad: z (9) y y y I deduce the relatos: () y y y y y from that Uder those codtos z wll be fuctos of ad y I shall determe the chage of varables such a way that those fuctos wll rereset a sgular multlcty M for ay value that s attrbuted to y Suose that oe has a sgular multlcty for y y ; equato () wll the be verfed I would le to wrte dow that the dervatve of ts left-had sde wth resect to y s zero To smlfy otatos I wll set: d d z Uo tag equatos (9) ad () to accout I wll have: A A A y da dϕ A A y d d

4 Beudo O the characterstcs of artal dfferetal equatos 4 The coeffcet of / y s zero by hyothess ad sce / y s ot detcally zero oe wll have: () A ϕ A A I say that equato () reresets the codto that must be verfed by the oretato of the secod-order elemets of ay tegral that cotas the multlcty M o that sgular multlcty I order to rove that fact I shall see to calculate the values of the thrd-order dervatves but frst I shall troduce the otato: d d ( ) z Whe the roosed equato s dfferetated wth resect to that wll gve: () A ϕ A We remar that oe wll have: () o the multlcty M so those relatos wll ermt oe to calculate all of the as fuctos of ; uo substtutg that to equato () oe wll get: A A dϕ A A d The coeffcet of ths equato s recsely the left-had sde of equato (6) It s zero sce we are o a sgular multlcty M The coeffcet of / s the left-had sde of equato () whch we assume to be verfed What wll fally rema s: A dϕ A A d

5 Beudo O the characterstcs of artal dfferetal equatos 5 That equato s verfed sce the secod-order elemets that oe fers from formulas () ad () wll satsfy the roosed equato detcally whe oe laces oeself uo a sgular multlcty We have the establshed that f the oretatos of the chose secod-order elemets o the sgular multlcty verfy equato () the the calculato of the thrd-order dervatves wll be determate 4 Equatos () () (6) (7) ad () defe a famly of multlctes M that I shall call sgular multlctes If oe cludes formulas () the oe wll get a ftude of multlctes M that verfy the roosed equatos ad ts derved oes However they are ot all located o a -dmesoal tegral multlcty I order to fd the codtos uder whch that stuato wll reset tself I shall mae a chage of varables that s aalogous to the oe that was emloyed o ; e such that oe wll be dealg wth sgular multlctes M for ay y sce z are fuctos of ad the ew varable y I must aed the formulas: ad y y Uder those codtos oe wll have: Φ Φ because the multlctes M verfy equatos () detcally I order for that to be true for ay y oe must have: Φ Φ Φ y y y Now Φ / s verfed detcally so t wll the suffce that oe should have: Φ y Ths s what oe obtas whe oe taes the tegrablty codtos to accout: (4) da d A d ϕ A A d d d whch oe sets:

6 Beudo O the characterstcs of artal dfferetal equatos 6 df d z Whe equato (4) s combed wth equatos () that wll eress the codto for the multlctes M to be cotaed -dmesoal tegral multlctes Oe ca verfy that by otg that the fourth-order elemets are determate We thus obta the sgular multlctes M Uo roceedg that way ste-by-ste oe wll rove the estece of sgular multlctes M for all values of the order Oe wll also see that oce a sgular multlcty M has bee chose the set of multlctes M that corresod to t wll deed uo a arbtrary fucto of argumets We shall rove a lttle later that those sgular multlctes deed cota tegral multlctes; e that the recedg codtos that we have recogzed to be ecessary wll be fact suffcet as well 5 I shall ow evso a olear secod-order artal dfferetal equato For the sae of smlcty I shall suose that t s ratoal the symbols that eter to t; let that equato be: (5) f ( z ) The degree of geeralty s defed the same maer as t s for lear equatos e by formulas () I order to calculate the secod-order dervatves we mae use of formulas () I order for there to be determacy t s ecessary that: (6) However that codto whch s ecessary s o loger suffcet If equato (6) o loger cludes the the dscusso wll roceed as t dd for lear equatos; we the suose that ths s ot true Equatos () () (5) ad (6) defe a famly of multlctes M However those multlctes are ot all laced o the -dmesoal tegral multlctes of the roosed equato I order to recogze the cases whch that wll be true we aga mae use of the chage of varables that has roved useful to us ad after some calculatos that I shall omt we wll fd that oe must have: (7) z Equatos () () (5) (6) ad (7) defe a famly of multlctes that I shall call the sgular multlctes M Oe verfes determacy etrely the same maer

7 Beudo O the characterstcs of artal dfferetal equatos 7 as above the calculato of the thrd-order dervatves ad oe wll rove the estece of sgular multlctes M for all values of If a sgular multlcty M s gve the the sgular multlctes that corresod to t wll deed uo a arbtrary fucto of argumets 6 It ow remas for me to establsh that f oe s gve a sgular multlcty M the there wll deed be a ftude of -dmesoal tegral multlctes that cota t ( ) I ca always erform a chage of varables such that the suort of that sgular multlcty has the equatos: Oe wll the have tur: z f ( ) ( ) ϕ ( ) ad f oe sets: the oe wll come dow to: z z f ϕ z z ( ) Sce oe s dealg wth a sgular multlcty equatos (6) ad (7) must be verfed as well as the equatos that are obtaed by dfferetatg the roosed equatos wth resect to Oe wll the have: for the recedg values of the argumets Oe ca the ut the roosed equato to the form: (8) α α α β γ ; the uwrtte terms have hgher degree Uo relacg wth λ oe ca mae the term dsaear Before gog further I would le to rove the followg theorem: ( ) For that roof I was sred by the method that was dcated by Goursat Leços sur les équatos au dérvées artelles du secod ordre 88

8 Beudo O the characterstcs of artal dfferetal equatos 8 If oe s gve the equato: F ( ) such that F / ad F / are zero for the followg values: z ψ ( ) ψ ( ) ψ ϕ ψ ( ) ψ ( ) ϕ ψ ϕ the t wll admt a tegral that s holomorhc a eghborhood of that reduces to: Ψ ψ ( ) ( ) ψ ( ) for Φ ψ ( ) ( ) ϕ ( ) for Oe ca wthout coveece suose that all of the tal gves are zero I order to that t wll suffce to set: z z ψ ( ) ( ) ψ ( ) ϕ It results from ths that f oe calculates the values of the successve dervatves for ste-by-ste the oe wll have by hyothess: ( )

9 Beudo O the characterstcs of artal dfferetal equatos 9 The roosed equato the gves all of the by dfferetatg wth resect to The absece of terms of degree oe ad ermts oe to calculate the terms uambguously ad as a result the terms as well ad so o I order to rove the covergece of the develomet thus-obtaed I shall emloy the method of maorzg fuctos ad cosder the followg aulary equato: (9) M z R R R R M ad R are the rad of the crcles of covergece for F for the corresodg symbols ad M s the mamum modulus of F The rght-had sde s obvously a maorzg fucto for F It wll the suffce to establsh the covergece for that equato; for that reaso I shall set: u v Equatos (8) wll become: v M z z ( ) ( ) ( ) u v z ( ) z z u v u R R u v R v or rather: v M R v ( ) ( ) 4M M R u R u v R R v

10 Beudo O the characterstcs of artal dfferetal equatos whch the uwrtte terms are ether of order less tha or of degree greater tha f they are of order equal to or greater tha res That equato admts a holomorhc tegral such that z ad z / v reduce to for v Oe wll have: u u v for u v for ay moreover so oe ca uambguously calculate the values of the the value of u v ad so o whch s easy to recogze ad all of the coeffcets that oe obtas v wll be ostve It results mmedately that the develomet that s gve by equato (9) s coverget alog wth the oe that s rovded by the roosed equato I shall ow retur to equato (8) whch s the rcal obect of that aragrah; the alcato of the recedg theorem s mmedate There s a holomorhc tegral that reduces to zero for ad to: 4 ϕ ϕ 4 for whch ϕ ϕ 4 are arbtrary fuctos of Oe sees that: for ad sce oe has o the other had: for z oe wll deduce that: It wll the result that whe that holomorhc tegral s develoed owers of t wll cota oly terms of degree three There s the a ftude of tegrals that cota the sgular multlcty M The arbtrary fuctos ϕ ϕ 4 corresod recsely to the arbtrary fuctos that 4 eter to the defto of the sgular multlctes M M

11 Beudo O the characterstcs of artal dfferetal equatos CONCLUSIONS If oe s gve the secod-order artal dfferetal equato: f ( z ) the a tegral of that equato s geerally defed by a multlcty a eceto whe the multlcty s sgular; e f the equatos: M ; there wll be z z f are verfed Oe the obtas a sgular multlcty M that s cotaed a ftude of tegrals Ay order wll corresod to sgular multlctes M that eoy the same roerty Ay sgular multlcty M corresods to a ftude of sgular multlctes M that deed uo a frst-order artal dfferetal equato As a result f two sgular multlctes M corresod to the same sgular multlcty M ad f they have a elemet of order commo at a ot the the same thg wll be true all alog a curve that asses through that ot