A generazaton of Pcard-Lndeof theorem/ the method of characterstcs to systems of PDE Erfan Shachan b,a,1 a Department of Physcs, Unversty of Toronto, 60 St. George St., Toronto ON M5S 1A7, Canada b Department of Mathematcs and Department of Physcs, Sharf Insttute of Technoogy, Azad St., Tehran, Iran arxv:1812.08925v1 [math.ap] 21 Dec 2018 Abstract We generaze Pcard-Lndeof theorem/ the method of characterstcs to the foowng system of PDE: x,y) y / x + y / x m = x,y). WthaLpschtzorC r, : [ a,a] m [ b,b] n R and nta condton I : [ ā,ā] m 1 b,b), ā a, we obtan a oca unque Lpschtz or C r souton f, respectvey that satsfes the nta condton, f v,0) = I v), v [ ā,ā] m 1. To construct the souton we set bounds on the vaue of the souton by dscretzng the doman of the souton aong the drecton perpendcuar to the nta condton hyperpane. As the number of dscretzaton hyperpanes s taken to nfnty the upper and ower bounds of the souton approach each other, hence ths gves a unque functon for the souton Ufs). A ocaty condton s derved based on the constants of the probem. The dependence of, and I on parameters, the generazaton to nonnear systems of PDE and the appcaton to hyperboc quasnear systems of frst order PDE n two ndependent varabes s dscussed. Keywords: hyperboc quasnear systems of PDE, method of characterstcs, Lpschtz contnuty, upper and ower bounds Contents 1 Introducton and outne 2 2 An aternatve proof of the Pcard-Lndeof theorem of ODE 7 3 A generazaton of Pcard-Lndeof theorem/ the method of characterstcs to systems of PDE 9 3.1 A recurson reaton for the Lpschtz constants L N,k.................. 15 3.2 Loca boundedness of L N,k................................ 17 3.3 Unque functon for souton Ufs) soves Theorem 3.1................. 19 4 Generazatons and appcaton of Theorem 3.1 24 4.1 Dependence of nta condton and coeffcents on parameters............. 24 4.2 Generazaton to nonnear systems of PDE....................... 26 4.3 Appcaton to hyperboc quasnear systems of frst order PDE n two ndependent varabes........................................... 30 Appendx A 31 Appendx B 32 Ema address: erfanshachan@gma.com,eshachan@physcs.utoronto.ca Erfan Shachan) 1 Fu name: M. Erfan Shachan T. Preprnt submtted to Journa of LATEX Tempates December 24, 2018
1. Introducton and outne The method of characterstcs for sovng a frst order parta dfferenta equaton n an unknown functon has been known to mathematcans n the past centures, however, the generazaton of ths method to systems of frst order PDE has remaned unknown e.g.[1]: Chapter VI, Secton 7 t s stated that there s no anaog of the method of characterstcs for systems of frst order PDE). In ths work we w prove theorems, n partcuar Theorem 1.1 beow, that w generaze the resut obtaned usng the method of characterstcs, typcay appcabe to one equaton wth one unknown functon, to systems of frst order PDE whch the parta dervatves of each functon appear n separate equatons. Theorem 1.1 can aso be consdered as the generazaton of the Pcard-Lndeof theorem of ODE to PDE. The man resut of ths work proven s Secton 3 s the foowng Theorem: Theorem 1.1 A generazaton of Pcard-Lndeof theorem/ the method of characterstcs to systems of PDE) Let, : P R, = 1,...,n, = 1,...,m 1, m 2, be Lpschtz contnuous or C r r 1) functons defned on the paraepped P P 1 P 2 wth P 1 x R m x x 0 a,x 0 R m } and P 2 y R n y y 0 b,y 0 R n }. And et the Lpschtz contnuous or C r nta condton functon I : V P 2 for V x P 1 x m = x 0m, x x 0 ā}, 0 < ā a and M I y0 < b wth M I y0 max Iu) y 0 u V} be gven. The foowng system of parta dfferenta equatons C 1 x,y) y y +...+C m 1 x,y) + y = x,y) 1.1) x 1 x m 1 x m has a unque Lpschtz contnuous 1 or C r souton respectvey, f : B P 2 for V B P 1, B contanng a neghbourhood of V nt, wth V nt x P 1 x m = x 0m, x x 0 < ā} and f reducng to the nta condton functon I on V, fu) = Iu) for u V. The proof of Theorem 1.1 s far from trva. The man dffcuty n generazng the method of characterstcs to the system of PDE of the type 1.1 s that the characterstc curves for each equaton are dstnct therefore t cannot be reduced to systems of ODE. One way to gan contro over these characterstcs s to set bounds on the vaue of the souton satsfyng an nta condton and the characterstc curves whch are dstnct for each equaton by dscretzng the hyperpanes aong the drecton perpendcuar to the nta condton hyperpane. If the bounds are set n an approprate and optma way t can be shown that n the mt that the number of dscretzaton hyperpanes s taken to nfnty the bounds for the vaue of the souton and the characterstc curves approach each other, hence ths gves a unque functon for the souton Ufs). It shoud be noted that there s a more genera and abstract theorem n hyperboc systems of parta dfferenta equatons that s reated to the system of PDE of reaton 1.1, however the condtons of that theorem, beng a more genera resut are not as mnma as the condtons of Theorem 1.1. For exampe the dfferentabty assumptons of that theorem have to ncrease proportona to 1 By Lpschtz contnuous souton we mean a Lpschtz contnuous functon that soves the system of PDE 1.1 at ts dfferentabe ponts. By Rademacher theorem for a proof refer to [4]) a Lpschtz contnuous functon s dfferentabe amost everywhere. 2
the number of ndependent varabes used n the hyperboc system of PDE n order for the souton to be a bounded ordnary functon possessng fnte dervatves to a certan order for more detas refer to [3], Chapter VI, Secton 10). On the other hand the condtons of Theorem 1.1 are as mnma as they can be. Another nterestng feature of Theorem 1.1 s the method whch t s proven wth, whch s an eegant generazaton of the method of characterstcs, appcabe to one equaton wth one unknown functon, to the system of PDE of 1.1. The dfference now s that there are many characterstcs comng out of each pont of the doman whch the souton s beng constructed on, therefore t s not possbe to reduce t to systems of ODE. As descrbed n the prevous paragraph one way to gan contro over these characterstcs and the vaue of the souton, s to set bounds on them by dscretzng the hyperpanes parae to the nta condton hyperpane and ater show that these bounds approach each other as the number of dscretzaton hyperpanes goes to nfnty. Aso we derve expct expressons for the ocaty condton and the Lpschtz constant of the souton of the PDE of Theorem 1.1 based on the constants of the probem as foows: 1 α <, c 1 = nl D m 1)L C expθc 1 )c 1 α)nm 1)L C L I +1/n) 1.2) L I +1/n)expc 1 α) L f = 1/n 1 nm 1)L C αl I +1/n)expθc 1 )c 1 α) 1.3) L Ufs = max } L f,m D +L f m 1)M C 1.4) L C and L D refer to the Lpschtz constants of the and functons on P, respectvey. θc 1 ) s the step functon. L I s the Lpschtz constant of the nta condton functons I on V. M D and M C refertoabound for and on P, respectvey. Theextent whch, ngenera, the souton canbe constructedn the x m drectonaboveorbeowthe nta condtonhyperpanes gvenby the ocaty condton of 1.2: α x m x 0m +α. Aso α ᾱ wth ᾱ = mna,b M I y0 )/M D } to make sure the doman and range of the souton e wthn P 1 and P 2, respectvey. Wth L f n reaton 1.3 beng the Lpschtz constant of the souton aong the hyperpanes parae to the nta condton hyperpane, L Ufs n reaton 1.4 gves the tota Lpschtz constant of the souton on ts doman of constructon. One of the appcatons of Theorem 1.1 s n regard to hyperboc quasnear systems of frst order PDE n two ndependent varabes whch, as an exampe, are used to descrbe the one dmensona space fow of fuds. These systems of PDE can be reduced to the PDE of Theorem 1.1 by dfferentatng the system, dagonazng ts coeffcent matrx and performng a change of functon varabes, therefore Theorem 1.1 and the method whch ts souton s constructed ths s dscussed n Secton 3) offer an aternatve way, whch s more drect and convenent especay for fndng a numerca souton, as compared to other methods, e.g. teraton methods [3], for constructng the souton of hyperboc quasnear systems of frst order PDE n two ndependent varabes. In order to ustrate the man dea of provng Theorem 1.1 n a smper context, n Secton 2 we present an aternatve proof of the Pcard-Lndeof theorem of ODE by settng upper and ower bounds on the vaue of the souton of the system of ODE: y = ft,y), yt 0 ) = y 0, by dscretzng 3
the tme nterva [t 0,t 0 +α] nto 2 N parttons at the N th step y N,k,m y t 0 +kα/2 N ) y N,k, k = 1,...,2N, y N,0 = yn,0,m = y 0 1.5) and fnd a recurson reaton for y N,k y N,k yn,k,m y N,k = y N,k 1 1+nL f δt)+cδt 2 +ǫδt, y N,0 = 0 1.6) δt = α/2 N, C a bounded constant, L f the Lpschtz constant of ft,y) and ǫ 0 as δt 0. After sovng reaton 1.6 we fnd y N,k 1/2 N + ǫ therefore as N, the upper and ower bounds for the souton n 1.5 approach each other, hence ths gves a unque functon for the souton to the system of ODE. We w see that ths aternatve way of provng the Pcard-Lndeof theorem s more easy generazabe to the quasnear system of PDE of 1.1. Settng upper and ower bounds on the vaue of the souton enabes us to have more contro over the possbe range of vaues of the souton and the bounds at the N +1 th step of parttonng naturay fa wthn the bounds at the N th step of parttonng, therefore wth denotng the set of possbe ranges of vaues for the souton on the tme nterva at the N th step of parttonng by R N, these sets form a nested sequence R N R N+1 R N+2..., hence n order to show that ths nested sequence converges to the graph of a unque functon for souton we ony need to show that at the N th step of the parttonng the dfference between the upper and ower bounds of the souton s of order 1/2 N. In the current methods whch we make successve approxmatons to the souton wthout fndng bounds for the souton, e.g. by makng successve approxmatons to the souton from the ntegra equaton of the system of ODE as n [1] or consderng the dscretzaton of the system of ODE as when sovng t numercay, n order to show convergence to a souton the dfference between the approxmatons to the souton at the N th step and the N +1 th step have to be found and fnay show that the sequence of approxmatons to the souton at the N th step converges unformy to a souton. In these methods when the exstence of the souton s proven one s not sure about ts unqueness and therefore a unqueness proof has to be presented separatey. In the method descrbed above whch we set bounds on the vaue of the souton the proof of the exstence of the souton s not separate from provng the unqueness of the souton, snce n order to demonstrate exstence t has to be shown that the bounds set on the souton at the N th step form a nested sequence and approach each other as N whch automatcay shows unqueness as we. Ths mpes that ths method s ony appcabe to when the condtons of the theorem are such that we obtan a unque souton e.g. when ft,y) n the system of ODE above s Lpschtz), and t cannot be apped to showthe exstence ofasouton onye.g. t cannotbe appedto when ft,y) s contnuous). In Secton 3 we prove Theorem 1.1. We mpement the same dea used n Secton 2 and descrbed n the paragraph after Theorem 1.1 to prove ths resut. A standard doman S + s defned as S + x P 1 0 x m x 0m α, ā+m C x m x 0m ) x x 0 ā+m C x m x 0m )} 1.7) 4
and the souton s constructed on ths doman. m C and M C refer to a ower and upper bound for for = 1,...,n on P, respectvey. α > 0 s chosen sma enough. Smary an S doman can be defned for beow the nta condton hyperpane 3. The doman between the nta condton hyperpane at x m = x 0m n S + and the hyperpane x m = x 0m +α n S + s dvded nto 2 N equa parttons for N = 0, 1,.... The hyperpanes at x m = x 0m +kα/2 N n S + are denoted by V N,k for k = 1,...,2 N and V N,0 V. Upper and ower bound functons ndependent of the assumed souton are defned on V N,k : f N,k : V N,k R and f N,k,m : V N,k R such that f fx) s a souton to 1.1 satsfyng the nta condton then f N,k,m x) f x) f N,k x), x V N,k 1.8) and f N,0,m = fn,0 I. Next n order to fnd a smar recurson reaton as 1.6 for f N,k f N,k x) fn,k,m x), x V N,k we need to ntroduce the Lpschtz constants L N,k of f N,k and fn,k,m and toshowthat f N,k 0asN weneed to showthat theselpschtz constantsarebounded. Ths s done by fndng a recurson reaton for the Lpschtz constants n Secton 3.1 and showng that they are ocay.e. cose enough to the nta condton hyperpane) bounded n Secton 3.2. The recurson reaton for f N,k, L N,k and a bound for the Lpschtz constants L N,k L N,2N are gven by f N,k = α ) α ) 2 1+C 1 2 N +C 2 2 N 1.9) L N,k =L N,k 1 α 1+m 1)L C 2 N +nl α ) α D 2 N +nm 1)L C L N,k 1 ) 2 α +LD 2 N 2 N 1.10) L L N,2N I +1/n)expc 1 α) 1 nm 1)L C αl I +1/n)expθc 1 )c 1 α) 1/n L f 1.11) wth f N,0 = 0 and L N,0 = L I. C 1 and C 2 are bounded constants. If the ocaty condton of 1.2 s satsfed, t can be shown that L N,k are bounded for a N and k, wth ther bound gven by L f n reaton 1.11. In Appendx B t s shown n deta that the bounds for the souton at the N + 1 step of parttonng of S + e wthn the bounds of the N step of parttonng. Therefore wth denotng the set of possbe ranges of vaues of the souton on S + at the N step of parttonng by P N + we have P N + PN+1 +... and x,fx)) P N + for x S +. Sovng the recurson reaton of 1.9 for f N,k we fnd f N,k 1/2 N hence P N + converges to the graph of a unque functon for the souton Ufs) as N. Fnay n Secton 3.3 t s shown that the Ufs obtaned n the prevous Subsectons soves the system of PDE of Theorem 1.1 at ts dfferentabe ponts subject to the nta condton. When the coeffcents, and the nta condton I are C 1 n order to prove that Ufs s C 1 on the 3 A st of equvaent defntons for when constructng the souton on the S doman s gven n Appendx A. 5
hyperpanes V N,k the foowng functons are defned recursvey f N,k f N,0 x) = x C x ν, x ν ) ) α/2 N) + x ν, x ν ) ) α/2 N 1.12) I, x V N,k, x ν = x ν α 2 N, ν = m C +M C )ê /2+ê m,c = C 1,...,C m 1,1) the functons f N,k x) are defned such that f N,k,m x) fn,k x) f N,k x) for x V N,k. A fxed V N,kN s consdered for k N 1,...,2 N } and q = k N /2 N hed fxed as N. Based on the dscussonabovet s cearthat the sequenceoffunctons f N,kN x) convergesunformy to Ufsx) on V N,kN, furthermore t s shown that the sequence of ther parta dervatves f N,kN / x s bounded and equcontnuous, therefore there s a subsequence of ther parta dervatves that converges unformy. From ths t s concuded that Ufsx) s C 1 on V N,kN, ths s then easy generazed to a hyperpanes parae to the nta condton hyperpane n S +. Based on ths fact t s then shown that Ufs soves the system of PDE of 1.1 subject to the nta condton and s C 1 on S +. Note that reaton 1.12 can be used to sove the system of PDE of 1.1 numercay on S +. One mght attempt to show that the dscretzed functons n 1.12 converge to the souton of the PDE of Theorem 1.1. In ths case one has to evauate the dfference between f N,k x) and f N+1,2k x) and show that ths dfference s of order 1/2 N unformy on V N,k for k = 1,...,2 N, ths s aso a possbty, however as mentoned earer n the approach whch we set bounds on the vaues of the souton thngs are more under contro, therefore t s a more convenent and reabe method hence ths w be the approach we consder n ths work. Secton 4 dscusses the generazatons and appcaton of Theorem 1.1. In Subsecton 4.1 t s shown that the Lpschtz or C r dependence of the nta condton and coeffcents and on parameters s nherted to the souton, Subsecton 4.2 dscusses the generazaton of Theorem 1.1 to non-near systems of PDE and n Subsecton 4.3 the appcaton of Theorem 1.1 n regard to quasnear hyperboc frst order systems of PDE n two ndependent varabes s brefy dscussed. The generazaton of Pcard-Lndeof theorem/ the method of characterstcs to systems of PDE s a resut concernng the cassca theory of parta dfferenta equatons whch has remaned unknown n the past centures. As far as the author s concerned ths resut, n the form stated n Theorem 1.1 wth mnma dfferentabty assumptons and expct expressons for the ocaty condton and the Lpschtz constant of the souton, s not approachabe usng known methods or theorems and the ony way s by drect constructon of the souton. Here our man focus w be on provng ths resut and brefy dscuss some of ts generazatons and appcaton but eave further nvestgatons for future works. 6
2. An aternatve proof of the Pcard-Lndeof theorem of ODE In ths Secton we demonstrate the man dea n provng Theorem 1.1 n the smper context of ordnary dfferenta equatons. Consder Pcard-Lndeof theorem 4 : Theorem 2.1 Pcard-Lndeof theorem) Let y,f R n ; ft,y) contnuous on a paraeepped R : a t t 0 a, y y 0 b and Lpschtz contnuous wth respect to y. Let M f be a bound for ft,y) on R; α = mna,b/m f }. Then y = ft,y), yt 0 ) = y 0 2.1) has a unque souton y = yt) on [t 0 α,t 0 +α]. The standard proofs of ths theorem are textbook matera [1]. Here we present an aternatve way to prove ths theorem. Proof Aternatve proof of Pcard-Lndeof theorem). Lets assume the system of ODE 2.1 has a souton. We can ntegrate 2.1 for ths souton to obtan t y t) = y 0 + f t,y t))d t, t 0 yt 0 ) = y 0 2.2) to frst approxmaton the maxmum and mnmum vaues of ths souton at t = t 0 +α are gven by y 0,1,m y 0 +αm 0,1 f y t 0 +α) y 0 +αm 0,1 f y 0,1, = 1,...,n 2.3) where M 0,1 f and m 0,1 f denote the maxmum and mnmum vaues of f t,y) n the regon R 0,1 t,y) 0 t t0 α, y y 0 M f α }. Next we dvde the nterva [t 0,t 0 + α] n haf. The maxmum and mnmum vaues of the souton at t = t 0 +α/2 are gven by y 1,1,m y 0 +m 1,1 f α/2 y t 0 +α/2) y 0 +M 1,1 f α/2 y 1,1 2.4) where M 1,1 f and m 1,1 f are the maxmum and mnmum vaues of f t,y) n R 1,1 t,y) 0 t t 0 α/2, y y 0 M f α/2}, respectvey. Now we use the bounds n 2.4) for the possbe range of the souton at t = t 0 +α/2 as a range of possbe nta condtons at t = t 0 +α/2 to fnd a better range of vaues for the souton at t = t 0 +α. Ths s gven by y 1,2,m y1,1,m +m1,2 f α/2 y t 0 +α) y 1,1 +M1,2 f α/2 y 1,2 2.5) where M 1,2 f and m 1,2 f are the maxmum and mnmum vaues of f t,y) n R 1,2 t,y) α/2 t t 0 α, y 1,1,m M f α/2 y y 1,1 + M f α/2}, respectvey. We contnue ths process by dvdng the nterva [t 0,t 0 + α] nto 2 N equa ntervas for N = 0,1,2,... and set bounds on the 4 We make use of the maxmum or nfnty norm: x = max t the paper. x t and the 1-norm: x 1 = t xt throughout 7
souton at t = t 0 +kα/2 N for k = 1,...,2 N y N,k,m yn,k 1,m +m N,k f α/2 N y t 0 +kα/2 N ) y N,k 1 R N,k t,y) k 1) α 2 N t t 0 k α 2 N,yN,k 1,m +MN,k f α/2 N y N,k α M f 2 N y y N,k 1 +M f α } 2.6) 2 N wth y N,0,m = yn,0 = y 0 and M N,k f and m N,k f denotng the maxmum and mnmum vaues of f t,y) n R N,k, respectvey. From 2.6 t can be verfed that the bounds for the souton at the N +1 step of the parttonng e wthn the bounds at the N step of the parttonng 5, therefore wth defnng R N k=1 2N RN,k we have, R N R N+1 R N+2... and ceary based on how R N s defned we have t,yt)) R N for t [t 0,t 0 + α], hence f we show that as N, y N,k yn,k,m 0 for k = 1,...,2 N t can be concuded that the regons R N w shrnk to a graph of a unque functon for the souton to 2.1. To show ths consder the foowng recurson reaton y N,k yn,k,m = yn,k 1 yn,k 1,m + M N,k f m N,k f )α/2 N 2.8) by assumpton the functon f satsfes the Lpschtz condton n ts y coordnates and beng a contnuous functon defned on the compact regon R N,k t assumes ts maxmum and mnmum vaues M N,k f and m N,k f at certan ponts n R N,k therefore we have M N,k f m N,k f y N,k 1 +M f α/2 N y N,k 1,m M f α/2 N)} L f +ǫ 2.9) wth L f beng the Lpschtz constant of the functon ft,y) wth respect to y. Snce the functon ft,y) s contnuous and t s defned on a compact set t s unformy contnuous therefore for any ǫ > 0 there s a δ > 0 ndependent of y) such that f t t < δ, f t,y) f t,y) < ǫ. Now we can choose N arge enough such that α/2 N < δ. Ths defnes the ǫ used n reaton 2.9. Usng 2.9 we can derve an upperbound for 2.8 y N,k yn,k,m yn,k 1 + y N,k 1 L f α/2 N +L f M f α/2 N 1 α/2 N} +ǫα/2 N y N,k 2.10) 5 Ths can be seen as foows, wth assumng y N+1,2k 2,m k 1 = 0) we have to show y N+1,2k,m M N+1,2k 1 f y N+1,2k 2 y N+1,2k = y N+1,2k 1 M N,k f, M N+1,2k f y N,k 1 y N,k,m, yn+1,2k +M N+1,2k f M N,k f, therefore ths proves yn+1,2k y N,k, y N,k 1,m, yn+1,2k 2 y N,k 1 note that ths s true for α = yn+1,2k 2 2N+1 + 1 2 It s cear that R N+1,2k 1 R N,k snce by assumpton y N+1,2k 2,m R N+1,2k R N,k snce y N+1,2k 1,m = y N+1,2k 2,m M N+1,2k 1 f ) +M N+1,2k α 2.7) f 2 N snce R N+1,2k 1 R N,k and R N+1,2k R N,k and by assumpton y N,k, the proof of yn+1,2k,m y N,k,m s smar. + m N+1,2k 1 f y N,k 1,m y N+1,2k 1 = y N+1,2k 2 +M N+1,2k 1 f α/2 N+1 y N,k 1 range of R N,k and ther t range s aso ceary a subset of the t range of R N,k. and y N+1,2k 2 and M f α/2 N+1 and smary y N,k 1 α/2 N+1 y N,k 1,m + M f α/2 N+1 hence ther y range s a subset of the y 8
wth y N,k 1 yn,k 1,m y N,k 1. From 2.10 we have y N,k = y N,k 1 1+nL f δt)+cδt 2 +ǫδt 2.11) wth C 2nL f M f and δt α/2 N. Sovng 2.11 wth notng that y N,0 = 0 we fnd y N,k =Cδt 2 +ǫδt)1+1+nl f δt)+...+1+nl f δt) k 1 }=Cδt+ǫ)1+nL f δt) k 1}/nL f ) Cδt+ǫ) exp nl f αk/2 N) 1 } /nl f ) 2.12) From 2.12 t can be easy seen that as N, y N,k 0 for any k = 1,...,2 N hence R N converges to a graph of a unque functon for the souton Ufs) on [t 0,t 0 + α]. It can be shown that Ufs ndeed soves 2.1: Ufs t+ t) Ufs t) = Ufs t+ t) Ufs t)+ tf t,ufst)))+ tf t,ufst)) = O t 2 )+ǫo t)+ tf t,ufst)) = Ufs t) = ft,ufst))) 2.13) wth ǫ 0 as t 0. The second equaty foows from 2.11, for N = 0, k = 1, δt = t 6, y 0,0 = 0, wth consderng y 0 = Ufst) as the nta condton at t [0,α] and notng that y 0,1,m Ufs t + t) y 0,1 and y0,1,m Ufs t) + tf t,ufst)) y 0,1. It s cear that wth a smar procedure we can construct a unque souton on [ α+t 0,t 0 ]. 3. A generazaton of Pcard-Lndeof theorem/ the method of characterstcs to systems of PDE In ths Secton we w appy the dea used n the prevous Secton for provng the Pcard-Lndeof theorem to prove the theorem beow. Theorem 3.1 A generazaton of Pcard-Lndeof theorem/ the method of characterstcs to systems of PDE) Let, : P R, = 1,...,n, = 1,...,m 1, m 2, be Lpschtz contnuous or C r r 1) functons defned on the paraepped P P 1 P 2 wth P 1 x R m x x 0 a,x 0 R m } and P 2 y R n y y 0 b,y 0 R n }. And et the Lpschtz contnuous or C r nta condton functon I : V P 2 for V x P 1 x m = x 0m, x x 0 ā}, 0 < ā a and M I y0 < b wth M I y0 max Iu) y 0 u V} be gven. The foowng system of parta dfferenta equatons C 1 x,y) y y +...+C m 1 x,y) + y = x,y) 3.1) x 1 x m 1 x m 6 t can aso be consdered negatve. Athough reaton 2.11 was derved by assumng we are movng n the postve tme drecton, ceary t s equvaenty vad for when movng n the negatve tme drecton e.g. for when constructng the souton on [ α+t 0,t 0 ] wth δt = α/2 N > 0 ). 9
has a unque Lpschtz contnuous 7 or C r souton respectvey, f : B P 2 for V B P 1, B contanng a neghbourhood of V nt, wth V nt x P 1 x m = x 0m, x x 0 < ā} and f reducng to the nta condton functon I on V, fu) = Iu) for u V. Proof. In a smar approach as the aternatve proof of the Pcard-Lndeof theorem presented n the prevous Secton we assume a souton exsts and fnd bounds for ths souton by dvdng the doman aong the x m drecton nto equa parttons and ater show that these bounds approach each other as the number of parttons goes to nfnty. Frst we defne a standard doman to construct the souton on. Let M D be a bound for on P and ᾱ = mna,b M I y0 )/M D }. Let M C and m C denote an upper and ower bound for for = 1,...,n on P, respectvey. We defne the pus standard doman S + x P 1 0 x m x 0m α, ā+m C x m x 0m ) x x 0 ā+m C x m x 0m )} 3.2) wth α > 0 chosen suffcenty sma as to satsfy the foowng condtons: ) a ocaty crtera the frst reaton of 3.47) to be derved n Subsecton 3.2, ) α ᾱ, ) to ensure the nequates for x n the defnton of 3.2 are satsfed. Smary an S doman can be defned for beow the hyperpane V 8. The standard doman S + s defned n a way as to ensure the foowng two propertes. If a souton f to 3.1 on S + exsts satsfyng the nta condton then: ) fs + ) P 2. ) Each characterstc curve x ) of ths souton es wthn S + and connects wth a pont n the nta condton doman V. Inwhat foowswew constructaunque soutonto 3.1on S + thatsatsfesthe nta condton. We w be usng ots of notatons and defntons. For a p S + after ntegratng 3.1 based on an assumed souton f on S + that satsfes the nta condton we obtan the foowng ntegra and characterstc equatons: pm f p) = f p ) 0 )+ x ) t),fx ) t)))dt, x ) x 0m ) = p ) 0 V, x ) p m ) = p x 0m dx ) j t) = C j x ) t),fx ) t))), C m 1, j = 1,...,m dt 3.3) note that the parameter of the characterstc equatons t, s the same as the x m coordnate. Next we dvde S + aong the x m drecton nto 2 N for N = 0,1,... equa parttons and fnd upper and ower bounds for the vaue of the assumed souton f at the ntersecton of these parttons n S +, we have f N,k,m x) fn,k 1,m,V res. x)+m N,k x) α 2 N f x),v res. x)+m N,k x) α 2 N fn,k x) 3.4) x V N,k,V N,k z S + z m = x 0m +kα/2 N }, V N,0 V, k = 1,...,2 N 7 By Lpschtz contnuous souton we mean a Lpschtz contnuous functon that soves the system of PDE 3.1 at ts dfferentabe ponts. By Rademacher theorem a Lpschtz contnuous functon s dfferentabe amost everywhere. 8 A st of equvaent defntons for when constructng the souton on the S doman can be found n Appendx A. 10
f N,0 z) = fn,0,m z) I z) for z V. f N,k,fN,k,m : V N,k R form upper and ower bounds for the vaue of the souton on V N,k.,V res. x),,m,v res. x), M N,k x) and m N,k x) for x V N,k w be defned beow. The bounds of reaton 3.4 can be understood better n terms of the frst reaton of 3.3. Wrtng ths reaton for x V N,k as the fna pont and x V N,k 1 as the nta pont we have x0m+kα/2 N f x) = f x )+ x ) t),fx ) t)))dt, x ) x m ) = x, x ) x m ) = x 3.5) x 0m+k 1)α/2 N note that x m = x 0m + kα/2 N for x V N,k and x m = x 0m + k 1)α/2 N for x V N,k 1. The bounds of reaton 3.4 are such that,m,v res. x) f x ),V res. x) and m N,k x) x ) t),fx ) t))) M N,k x) for k 1)α/2 N t x 0m kα/2 N. Next we gve precse defntons for these bounds. We frst defne,v x), fn,k 1,m,V x), MN,k x) and m N,k x) for x V N,k : wth S N,k,V x) max,m,v x) mn M N,k x) max z z) S N,k V N,k 1}, z,m z) S N,k z,y) z,y) P N,k and PN,k for x V N,k gven by VN,k 1}, },m N,k x) mn } z,y) z,y) P N,k 3.6) S N,k z S + α/2 N z m x m 0, M C z m x m ) z x m C z m x m )} P N,k z z,y) S N,k, fn,k 1,m,V x) M α D 2 N y,v x)+m D α 2N, = 1,...,n} 3.7) for when the characterstc curves x ) t) of the assumed souton f pass through a x V N,k for = 1,...,n,.e. x ) x m ) = x, S N,k s defned n a way as to ensure that x) t) S N,k for α/2 N t x m 0 and P N,k s defned n a way as to ensure that x ) t),fx ) t)) ) P N,k for α/2 N t x m 0.,V res. x) and,m,v res. x) for x V N,k are gven by,v res. x) max,m,v res. x) mn,m } z z) V N,k 1 res.,,x } z z) V N,k 1 res.,,x V N,k 1 res.,,x z S N,k M VN,k 1 N,k x)α/2 N z x m N,k x)α/2 N} wth M N,k x) and m N,k x) havng smar defntons as M N,k x) and m N,k x) n reaton 3.6 respectvey wth repaced by } } M N,k x) max z,y) z,y) P N,k, m N,k x) mn z,y) z,y) P N,k From the defntons above t can be verfed that the bounds of reaton 3.4 for the assumed souton are correct. For exampe the maxmum of f at a pont x V N,k conssts of the maxmum 3.8) 3.9) 11
vaue of f n a regon of V N,k 1 whch the characterstc curve of f passng through ths regon has the possbty of passng through x, ths regon of V N,k 1 s gven by V N,k 1 res.,,x defned n 3.8 and the maxmum vaue s gven by,v res. x), pus the maxmum vaue whch f can change when ts characterstc curve passng through S N,k reaches x, ths s gven by M N,k x)α/2 N. Aso t can be verfed that the bounds for the souton at the N +1 step of the parttonng e wthn the bounds of the N step. Ths s dscussed n deta n Appendx B, therefore wth defnng P N + 2N k=1 x V N,kPN,k ) we have PN + PN+1 +.... From the defntons and reatons above t s cear that the graph of the assumed souton on S + es wthn the set P N + at the N step of parttonng, x,fx)) P+ N for x S +, therefore n order to show that P+ N converges to the graph of a unque functon for the souton Ufs) we ony need to show that f N,k x) fn,k,m x) 0 as N. For ths we w try to fnd a smar recurson reaton as n 2.11 for f N,k f N,k x) fn,k,m x), x V N,k. Startng from 3.4 we have } x) fn,k,m x) = fn,k 1,V res. x),m,v res. x)+ M N,k x) m N,k x) α/2 N 3.10) f N,k an upper bound for M N,k x) m N,k x) s gven by M N,k x) m N,k x) +,V x) fn,k 1 M C m C ) α 2 N + α 2 N α } 2 N 3.11) }L D,m,V x)+2m D wth L D a Lpschtz constant for the functons and the expresson n brackets corresponds to an upperbound for the dstance p 1 p 2 1 between any two ponts p 1,p 2 P N,k defned n 3.7. We aso need to fnd an upper bound for,v x) fn,k 1,m,V x) n 3.11. For ths we w assume the functons f N,k x) and fn,k,m x) are Lpschtz wth Lpschtz constant LN,k. We w show ths to be true and derve a recurson reaton for the Lpschtz constants L N,k n Subsecton 3.1. We have,v x) fn,k 1,m,V x) = fn,k 1 x max) L N,k 1 M C m C )α/2 N + x mn)+ x mn ) fn,k 1 x mn),m x mn),m x mn ) ) 3.12) ) wth x max and x mn denotng the ponts n SN,k V N,k 1 whch and,m assume ther maxmum and mnmum vaues n S N,k VN,k 1, respectvey. Combnng 3.11 and 3.12 we have M N,k x) m N,k x) n +L N,k 1 n + 2nM D + M C m C ) α ) α M C m C )+1 2 N 2 N }L D 3.13) 12
wth an upper bound for the foowng quantty z) fn,k 1,m z) fn,k 1, z V N,k 1 3.14) note that based on 3.14, we can take f N,0 = 0 snce we defned f N,0 = fn,0,m = I. Smary we can obtan a bound for,v res. x),m,v res. x) n 3.10,V res. x),m,v res. x) = z max ) fn,k 1 z mn )+ L N,k 1 ) M N,k x) m N,k α x) 2 N + wth zmax and zmn and mnmum vaues n V N,k 1 res.,,x m N,k x). We have denotng the ponts n V N,k 1 res.,,x whch fn,k 1 z mn ) fn,k 1 z mn ) fn,k 1 and,m,m z mn ) ),m z mn ) ) 3.15) assume ther maxmum, respectvey. Smar to 3.13 we can obtan a bound for MN,k x) M N,k x) m N,k x) n +L N,k 1 n + 2nM D + M C m C ) α ) α M C m C )+1 2 N 2 N }L C 3.16) wth L C a Lpschtz constant for the functons. Now usng 3.13, 3.15 and 3.16 we can fnd a bound for 3.10, we have f N,k + L C m 1)L N,k 1 +L D ) α 2nM D + 2 N n +L N,k 1 n M C m C ) α 2 N+ ) } α M C m C )+1 2 N f N,k x) fn,k,m x), x V N,k 3.17) InSubsecton3.1wewdervearecursonreatonfortheLpschtzconstantsL N,k andnsubsecton 3.2 we w show that they are ocay.e. for a suffcenty sma α) bounded. Wth knowng ths we can wrte 3.17 as f N,k = 1+C 1 α/2 N) +C 2 α/2 N ) 2 3.18) wth C 1 and C 2 beng constants whch bound the foowng quanttes C 1 nm 1)L C L N,k 1 +nl D 3.19) C 2 m 1)L C L N,k 1 ) +L D L N,k 1 n M C m C )+2nM D + } M C m C )+1 3.18 s the recurson reaton smar to 2.11 we were ookng for. For competeness we ncude the recurson reaton for the Lpschtz constants to be derved n Subsecton 3.1, the ocaty crtera for α and a bound for the Lpschtz constants L N,k, to be derved n Subsecton 3.2, and a Lpschtz 13
constant for the unque functon for the souton Ufs) to Theorem 3.1 to be derved beow, here f N,k = α ) α ) 2 1+C 1 2 N +C 2 2 N L N,k = L N,k 1 α 1+m 1)L C 2 N +nl α ) α D 2 N +nm 1)L C L N,k 1 ) 2 α +LD 2 N 2 N 1 α < expθc 1 )c 1 α)nm 1)L C L I +1/n), c 1 = nl D m 1)L C 3.20) L L N,2N I +1/n)expc 1 α) 1 nm 1)L C αl I +1/n)expθc 1 )c 1 α) 1/n L f L Ufs = max } L f,m D +L f m 1)M C Reatons 3.20 consttute the man reatons of Theorem 3.1. L I refers to the Lpschtz constant of the nta condton functon I and θc 1 ) the step functon. Wth knowng that the Lpschtz constants L N,k are ocay bounded we can use the frst reaton n 3.20 to show that f N,k 0 as N smar to the steps n reaton 2.12 f N,k = C 2 α/2 N ) 2 1+1+C 1 α/2 N )+...+1+C 1 α/2 N ) k 1 } = C 2 /C 1 α/2 N 1+C 1 α/2 N ) k 1} 3.21) = f N,k C 2 /C 1 expc 1 αk/2 N ) 1}α/2 N from reaton 3.21 t s cear that f N,k 0 as N, hence P N + converges to the graph of a unque functon for the souton Ufs) to Theorem 3.1. We w prove n Subsecton 3.3 that Ufs ndeed soves the PDE of Theorem 3.1 subject to the nta condton. Before movng on to the next Subsecton we show that Ufs s aso Lpschtz n the x m drecton. L f n reaton 3.20 can be consdered as the Lpschtz constant of Ufs aong the hyperpanes x m = const. n S + for x 0m const. x 0m +α. Consder V N,kN and V N,k N for q = kn /2 N and q = k N /2N hed fxed as N and x m = q q for q > q. It can be easy seen that a bound for the dfference f N,k N x+ê m x m ) f N,kN x) for x V N,kN and x+ê m x m V N,k N s M D x m +L f m 1)M C x m, wth M C beng a bound for on P and ê m the unt vector n the x m drecton, hence M D +L f M C m 1) can be consdered as a Lpschtz constant for Ufs n the x m drecton. Therefore L Ufs = maxl f,m D +L f m 1)M C } 3.22) s a Lpschtz constant for Ufs on S + or S ). Note that reatons of 3.20 are equvaenty vad for when constructng the souton on the S doman wth α > 0 beng the extent whch, n genera, the souton can be constructed beow the nta condton hyperpane V. A st of the equvaent of the defntons used n ths Secton for when constructng the souton on the S doman s gven n Appendx A. 14
3.1. A recurson reaton for the Lpschtz constants L N,k In ths Subsecton we w obtan a recurson reaton for the Lpschtz constants L N,k of the functons f N,k x). A smar resut w be reached f we work wth the functons fn,k,m x). Let L N,0 = L I, wth L I beng the Lpschtz constant of the nta condton functons I. Take two separate ponts p 1,p 2 V N,k. Wth assumng L N,k 1 s known we woud ke to fnd an expresson for L N,k f N,k p 1) f N,k p 2) L N,k For ths we w make use of the foowng Lemma: p 1 p 2, = 1,...,m 1 3.23) Lemma 1. Let g : W R n R be a Lpschtz contnuous functon wth Lpschtz constant L g for the 1-norm. W 1,W 2 W be compact sets and consder d wth the foowng characterstcs: w 1 W 1, w 2 W 2 : w 1 w 2 1 d, and vce versa: w 2 W 2, w 1 W 1 : w 1 w 2 1 d then we have the foowng reatons: M g W 1 ) M g W 2 ) L g d and m g W 1 ) m g W 2 ) L g d. Where M g W r ) and m g W r ) denote the maxmum and mnmum vaues of g n W r for r = 1,2, respectvey. Proof. By the assumpton ofcompactnessof W r and contnuty ofg there exsts w r W r such that gw r ) = M g W r ) for r = 1,2. By assumpton ofthe emma there s a y 2 W 2 such that w 1 y 2 1 d so we have gw 1 ) gy 2 ) L g d and snce gy 2 ) gw 2 ) we have : gw 2 ) + L g d gw 1 ) and smarytcanbeconcudedgw 1 )+L g d gw 2 )whchproves M g W 1 ) M g W 2 ) L g d. Smary t can be concuded that m g W 1 ) m g W 2 ) L g d. Note: Consder B 1 n h=1 [a h,b h ],B 2 n h=1 [c h,d h ] R n. Then d = n h=1 max a h c h, b h d h } has the characterstcs of the dstance d n Lemma 1 wth respect to the subsets B 1 and B 2. From 3.4 ) f N,k p 1) f N,k p 2) =,V res. p 1 ),V res. p 2 )+ M N,k p 1 ) M N,k α p 2 ) 2 N 3.24) assumng d 1 has the characterstcs of the dstance d n Lemma 1 for the two sets V N,k 1 res.,,p 1 and V N,k 1 res.,,p 2 we have,v res. p 1 ),V res. p 2 ) L N,k 1 d 1 3.25) based on the defntons of V N,k 1 res.,,p r for r = 1,2 n 3.8 and the Note after Lemma 1 we can fnd an expresson for d 1 d 1 ) p1 max p 2 + M N,k p 2 ) M N,k α ), p1 p 1 ) 2 N p 2 + m N,k p 2 ) m N,k α } p 1 ) 2 N 3.26) 15
a bound for M N,k p 1 ) M N,k p 2 ) or m N,k p 1 ) m N,k p 2 ) s gven by M N,k p 1 ) M N,k p 2 ) L C d 2, m N,k p 1 ) m N,k p 2 ) L C d 2 3.27) wth d 2 havng the characterstcs of the dstance d n Lemma 1 for the two sets P N,k +,p 1 and P N,k +,p 2. Based on the defntons of P N,k +,p r for r = 1,2 n 3.7 and the Note after Lemma 1 we can fnd an expresson for d 2 d 2 p 1 p 2 + } max,v p 1),V p 2),,m,V p 1),m,V p 2) 3.28) a bound for,v p 1),V p 2) or,v p 1),V p 2),m,V p 1),m,V p 2) s gven by L N,k 1,m,V p 1),m,V p 2) L N,k 1 p 1 p 2 p 1 p 2 3.29) from the defntons of 3.6 and 3.7 t can be verfed that p 1 p 2 has the characterstcs of the dstance d n Lemma 1 for the two sets S N,k +,p 1 V N,k 1 and S N,k +,p 2 V N,k 1. From 3.28 and 3.29, d 2 s gven by d 2 = p 1 p 2 1+nL N,k 1) 3.30) and from 3.26 and 3.27 d 1 s gven by d 1 = p 1 p 2 +m 1)L C α 2 Nd 2 3.31) Smary a bound for M N,k p 1 ) M N,k p 2 ) s gven by M N,k p 1 ) M N,k p 2 ) L D d 2 3.32) From 3.25, 3.30, 3.31 and 3.32 we obtan a bound for 3.24 f N,k p 1) f N,k p 2) L N,k 1 α 1+m 1)L C 2 N +m 1)L C α +L D 1+nL N,k 1 )} 2 N p 1 p 2 α 2 NnLN,k 1) 3.33) Comparng 3.23 and 3.33 we fnd an expresson for L N,k L N,k =L N,k 1 α 1+m 1)L C 2 N +nl α ) α D 2 N +nm 1)L C L N,k 1 ) 2 α +LD 2 N 2 N 3.34) 16
3.2. Loca boundedness of L N,k The nonnear term L N,k 1) 2 n 3.34 s the term that can ead to an unbounded ncrease of the Lpschtz constants L N,k, but f the coeffcent of ths term nm 1)L C α) s sma enough we expect to be abe to show that the Lpschtz constants are bounded. We frst rewrte 3.34 n a smper form b k = b k 1 γ +b k 1 ) 2, b k c 2 α 2 NLN,k +1/n), k = 1,...,2 N γ = 1+c 1 α/2 N), c 1 = nl D m 1)L C, c 2 = nm 1)L C 3.35) For convenence we have suppressed the ndex N n b k. Note that for c 1 > 0, γ > 1 but for c 1 < 0 and a suffcenty arge N, 0 < γ < 1 wth γ 1 as N. The frst few terms of the sequence b k read b 1 = b 0 γ +b 0 2 b 2 = γ 2 b 0 +γ +γ 2 )b 0 2 +2γb 0 3 +b 0 4 3.36) From 3.35 and 3.36 t s cear that b k s a poynoma of degree 2 k n b 0 b k = Ck h γ)bh 0, Ch k = 0 for h > or h < 1, k N 0} 3.37) 2k 2 k h=1 wth C h k γ) a poynoma n γ. To show the oca boundedness of LN,k we need to fnd a bound for the coeffcents C h k. For ths nsert b k 1 from reaton 3.37 nto reaton 3.35 to obtan 2 b k = γck 1 h bh 0 + Ck 1 h ) k 2 bh 0 = γc h k 1 b h 0 + h 1 1 h 1=2h 2=1 summaton over h s mpct. From 3.38 a recurson reaton for the coeffcents C h k C h1 h2 k 1 C h2 k 1 bh1 0 3.38) 2, Ck h = γch k 1 +2Ch 1 k 1 C1 k 1 +...+2Ch/2+1 k 1 C h/2 1 k 1 + Ck 1) h/2 f h s even C h k = γc h k 1 +2C h 1 k 1 C1 k 1 +...+2C h+1)/2 k 1 C h 1)/2 k 1, f h s odd can be derved 3.39) In what foows we w show that the coeffcents C h k are bounded by the nequates beow C h k kh 1 γ kh, γ 1 C h k k h 1 γ k 1, 1/2 γ < 1 3.40) one mght be abe to mprove the bounds n 3.40 and accordngy mprove the bounds of reaton 3.47 by a more carefu study of the coeffcents Ck h. But these bounds suffce to capture the man features of a ocaty condton for α. From reaton 3.35 and 3.36 t can be verfed that Ck 1 = γk and Ck 2 = 2k 2 h=k 1 γh, satsfyng the nequates of 3.40. So assumng Ck h k h 1 γ kh hods for 1 h < h ets try to prove 17
C h k kh 1 γ kh for γ 1 and for h 3. Appyng ths to 3.39 we have C h k γc h k 1 +2h/2 1)k 1) h 2 γ hk 1) +k 1) h 2 γ hk 1), f h s even Ck h γck 1 h +2 h 1) k 1) h 2 γ hk 1), f h s odd 2 3.41) so n both cases we obtan C h k γc h k 1 +h 1)k 1) h 2 γ hk 1) 3.42) appyng ths nequaty to Ck 1 h,ch k 2,... we obtan C h k γk C h 0 +h 1)0h 2) γ 0+k 1 +h 1)1 h 2) γ h+k 2 +...+h 1)k 2) h 2 γ hk 2)+1 k +h 1)k 1) h 2 γ hk 1) 0+γ kh h 1)x h 2 dx = k h 1 γ kh 3.43) 0 note that C h 0 = 0 for h 3. We aso used the fact that γkh γ hk 1 r)+r for γ 1, r = 0,...,k 1, k = 1,...,2 N and h 3 n the above reaton. Smary f we assume C h k k h 1 γ k 1 hods for 1 h < h, t s possbe to prove that C h k kh 1 γ k 1 for 1/2 γ < 1 and h 3. Appyng ths to 3.39 for both even and odd cases we obtan C h k γc h k 1 +h 1)k 1) h 2 γ 2k 2) 3.44) appyng ths nequaty to Ck 1 h,ch k 2,... we have Ck h γk C0 h +h 1)0h 2) γ 2+k 1 +1 h 2) γ 0+k 2 +2 h 2) γ 2+k 3 +...+k 2) h 2 γ 2k 3)+1 } +h 1)k 1) h 2 γ 2k 2) 0+γ k 1 2 k } h 1) x h 2 dx+ x h 2 dx =k h 1 γ k 1 3.45) 0 we used the fact that γ k 1 γ 2 k 2 r)+r,r = 0,..., k 3, k = 3,...,2 N and 2 0 xh 2 dx 1/γ for 1/2 γ < 1 n the above reaton. Hence the nequates of 3.40 are proven. Appyng 3.40 to 3.37 for k = 2 N we fnd 2 c 2 αl N,2N +1/n) = 2 N b 2 N 2 N b 0 γ 2N ) h < c 2αL I +1/n)expc 1 α) 1 c 2 αl I +1/n)expc 1 α), c 1 0 2 2N c 2 αl N,2N +1/n)=2 N b 2 N h=1 2 2N h=1 γ 2N γ 2N b 0 ) h < 1 γ expc 1 α)c 2 αl I +1/n),c 1 <0,1/2 γ < 1 1 c 2 αl I +1/n) 3.46) wth L N,0 = L I the Lpschtz constant of the nta condton functon I. We used γ 2N = 1 + αc 1 )/2 N ) 2N expc 1 α) n the above reatons and assumed 2 N b 0 expc 1 α) < 1 n the frst reaton and 2 N b 0 < 1 n the second reaton of 3.46. From these assumptons and 3.46 we can fnd a ocaty 18
condton for α and a bound for the Lpschtz constants L N,k L N,2N 9 1 α <, c 1 = nl D m 1)L C expθc 1 )c 1 α)nm 1)L C L I +1/n) L L N,2N I +1/n)expc 1 α) 1 nm 1)L C αl I +1/n)expθc 1 )c 1 α) 1/n L f 3.47) wth θc! ) the step functon. 3.3. Unque functon for souton Ufs) soves Theorem 3.1 In ths Subsecton we w show that the Ufs obtaned n the prevous Subsectons s the souton of the system of PDE of Theorem 3.1. Wth the Lpschtz condton for the nta condton and the coeffcents and, Ufs s Lpschtz. Due to Radamechar theorem t s dfferentabe amost everywhere. Here we w show that Ufs soves the system of PDE at ts dfferentabe ponts. Consder two hyperpanes n S + : V β = z S + z m = β,x 0m β x 0m + α} and V β+δβ = z S + z m = β + δβ,x 0m β + δβ x 0m + α} for δβ > 0. Defne the functon g for x ê m δβ V β and x V β+δβ g x) Ufs x C x ν,ufsx ν ))δβ)+ x ν,ufsx ν ))δβ, 3.48) C = C 1,...,C m 1,1), x ν = x νδβ, wth ν = m C +M C )ê /2+ê m ê j s the unt m-vector n the x j drecton. Smar to before we can take V β as the nta condton hyperpaneand V β+δβ asthe fna hyperpane, but we w not partton the spacen between, nstead we take the mt δβ 0. Based on how g x) s defned t can be seen to e wthn the upper and ower bounds for the souton 10 : f 0,1,m x) g x) f 0,1 x). Usng the frst reaton of 3.20) wth N = 0, k = 1, α = δβ and notng that f 0,0 = 0, we have f 0,1 = C 2 δβ) 2 3.49) 9 We have dropped the 1/γ factor on the rghthand sde of the second reaton of 3.46 as γ 1 for N. But now snce L N,2N s an ncreasng functon of N and the second reaton of 3.47 s true n the mt of N then t must be true for a N 0} N. To see how L N,2N s an ncreasng functon of N consder L N,k = L N,k 1 1+e 1 /2 N) +e 2 /2 N ) L N,k 1) 2 +e3 /2 N from 3.34 wth e 1,e 2,e 3 0. It suffces to show L N+1,2k L N,k for k = 1,...,2 N. Note that L N+1,0 = L N,0 = L I, therefore ets assume L N+1,2k 1) L N,k 1 and try to prove L N+1,2k L N,k. We have L N+1,2k 1 = L N+1,2k 2 1+e 1 /2 N+1) + e 2 /2 N+1 ) L N+1,2k 2) 2 + e3 /2 N+1 and L N+1,2k = L N+1,2k 1 1+e 1 /2 N+1) +e 2 /2 N+1 ) L N+1,2k 1) 2 +e3 /2 N+1 L N+1,2k = L N+1,2k 2 1+e 1 /2 N) + e 2 /2 N ) L N+1,2k 2) 2 +e3 /2 N +terms greater than or equa to zero. Ths proves L N+1,2k L N,k. 10 e.g. t can be verfed that x ν S 0,1, xν,ufsx ν )) P 0,1 therefore m0,1 x) x ν,ufsx ν )) M 0,1 x) and M 0,1 x) x ν,ufsx ν )) m 0,1 x) hence x C x ν,ufsx ν ))δβ) V 0,0 res.,,x Ufs x C x ν,ufsx ν ))δβ) f 0,0,V res. x). and f0,0,m,v res. x) 19
snce Ufs x) aso es wthn the upper and ower bounds for the souton f 0,1,m x) Ufs x) f 0,1 x), based on reaton 3.49 we have g x) Ufs x) = Oδβ 2 ). Therefore Ufs x) Ufs x ê m δβ) = Ufs x) g x)+g x) Ufs x ê m δβ)= Oδβ 2 )+Ufs x ê m δβ) x Ufs x ê m δβ) x ν,ufsx ν ))δβ+ Rδβ)+ x ν,ufsx ν ))δβ Ufs x ê m δβ) 3.50) wth Rδβ)/δβ 0 as δβ 0 and we used the fact that Ufs s dfferentabe at x ê m δβ. Note that x ê m δβ V β s a fxed pont and x V β+δβ s vared as δβ 0. Another pont to consder here s that we ony used the fact that Ufs s dfferentabe on V β and dd not need to assume t s dfferentabe n the x m drecton n 3.50. Dvdng reaton 3.50 by δβ and takng the mt δβ 0 we fnd x,ufsx)) x Ufs x)+ x m Ufs x) = x,ufsx)) 3.51) Ths shows that Ufs soves the PDE of reaton 3.1 at ts dfferentabe ponts subject to the nta condton 11. Next we w show that f the nta condton and the coeffcents and are C 1 then Ufs s C 1. We frst show that Ufsx) s C 1 on V N,k. We w make use of the foowng two theorems n mathematca anayss [2]: 1. Arzea-Ascotheorem: AnyboundedequcontnuoussequenceoffunctonsnC 0 d h=1 [a h,b h ],R) has a unformy convergent subsequence. 2. Theorem: The unform mt of a sequence of functons n C 1 d h=1 [a h,b h ],R) s C 1 provded that the sequence of ts parta dervatves aso converges unformy and the parta dervatve of the unform mt functon s the same as the unform mt of the parta dervatve. Consder the coecton of functons f N,k f N,k f N,0 : V N,k R defned recursvey as foows x) = x C x ν, x ν ))α/2 N )+ x ν, x ν ))α/2 N I, x V N,k, x ν = x ν α 2 N, ν = m C +M C )ê /2+ê m 3.52) from the way the functons f N,k are defned t can be seen 12 f N,k,m x) fn,k x) f N,k x), x V N,k 3.53) we consder a fxed V N,kN for 1 k N 2 N ) at x m = x 0m + qα wth q = k N /2 N hed fxed as 11 Athough the constructon of Ufs was done by movng n the postve x m drecton t s cear that wth smar methods t s possbe to start from an nta condton hyperpane and construct the souton n the negatve x m drecton c.f. Appendx A). Therefore the dscusson here s equvaenty vad for when makng the repacement δβ δβ for δβ > 0 and evauatng the dervatve of Ufs n the negatve x m drecton. 12 A smar reasonng as the footnote of the prevous page hods here: x ν S N,k, xν,ufsx ν )) P N,k C x) hence therefore m N,k x) x ν,ufsx ν )) M N,k x) and M N,k x) x ν,ufsx ν )) m N,k x C x ν,ufsx ν ))α/2 N ) V N,k 1 res.,,x and fn,k 1,m,V res. x) Ufs x C x ν,ufsx ν ))α/2 N ),V res. x). 20
N. To show that Ufs s dfferentabe on V N,kN we have to show the foowng: 1. Unform convergence of the sequence of functons f N,kN on V N,kN. 2. Unform convergence of the sequence or at east a subsequence) of the parta dervatves f N,kN / x on V N,kN. To show the unform convergence of a subsequence of the parta dervatves t suffces to show the foowng: 2.1 Boundedness of the sequence of parta dervatves f N,kN / x. 2.2 Equcontnuty of the sequence of parta dervatves f N,kN / x. The frst statement foows from reaton 3.53 and the fact that as N the upper and ower bounds approach each other unformy on V N,kN as shown n 3.21. To show statements 2.1 and 2.2 we take the parta dervatve of 3.52 13 f N, k x x) = f N, k 1,z z)+ C h,x p ν )α/2 N C h,ys p ν )f N, k 1 s,x x ν )α/2 N} f N, k 1 z) p ν ) α 2 N +,y s p ν )fs,x N, k 1 x ν ) α 2 N,1 h m 1,1 s n,1 k k N 3.54) wth p ν = x ν, x ν )) and z = x C p ν )α/2 N n the above reaton. Summaton over h and s s mpct. To show the boundedness of the sequence of dervatves we assume a bound L N, k 1 f f N, k 1 x )/ x s known for the parta dervatves of fn, k 1 x ) for x V N, k 1 and ook for L N, k f f N, k x)/ x. From 3.54 we can fnd such recurson reaton L N, k f L N, k 1 α f 1+m 1)L C 2 N +nl α ) D 2 N +nm 1)L C L N, k 1 f ) 2α 2 N +L D α,z h 2 N fn, k,x x) 3.55) where L C and L D are Lpschtz constants for x,y) and x,y) whch bound,x x,y), C,ys x,y) and,x x,y),,ys x,y) respectvey, for x,y) P. Reaton 3.55 s exacty smar to reaton 3.34 obtaned prevousy for the Lpschtz constants L N,k. Ths proves 2.1 that the sequence f N,kN,x f N,kN,x s bounded ocay n α). To prove 2.2 we have to show that the sequence s equcontnuous. The C 1 assumpton for the nta condton and the coeffcents x,y) and x,y) mpes that f N,kN,x s contnuous and snce they are defned on a compact set they are unformy contnuous, therefore we ony have to show that for a ǫ > 0 there s a common δ > 0, ndependent of N, such that f x x 1 < δ f N,kN,x x) f N,kN,x x) < ǫ, for x, x V N,kN. Takng the functons f N, k 1 x ) as known, for an ǫ N, k 1 > 0 choose δ N, k 1 > 0 such that f x x 1 < δ N, k 1 for x, x V N, k 1 and p p 1 < δ N, k 1 1+L f ) for p, p P wth L f gven by 3.20, then f N, k 1,x x ) f N, k 1 x ) < ǫ N, k 1, x,y s p ),y s p ) < ǫ N, k 1,,x p ),x p ) < ǫ N, k 1,ys p ),ys p ) < ǫ N, k 1,,x p ),x p ) < ǫ N, k 1 3.56) 13 For brevty we have used the symbo H,x H/ x. 21
= 1,...,m 1. For these ǫ N, k 1 and δ N, k 1 ets see whch ǫ N, k and δ N, k we w obtan for f N, k,x. For ths ets evauate f N, k,x x) f N, k,x x) usng the rght hand sde of 3.54 for x, x V N, k and x x 1 < δ N, k. Note that the dfference of the product of any number of terms can be wrtten n terms of the dfference of each of the terms mutped by other terms, for exampe A 1 A 2...A t Ã1Ã2...Ãt = δa 1 A 2...A t +Ã1δA 2 A 3...A t +...+Ã1Ã2...Ãt 1δA t 3.57) for δa h A h Ãh, 1 h t. Therefore the dfference of the rght hand sde of 3.54 can be wrtten n terms of the dfference of each of the terms at ther correspondng two dstnct ponts mutped by other terms whch are bounded. Ther two dstnct ponts are ether x ν = x να/2 N and x ν x να/2 N or p ν = x ν,f N, k 1 x ν )) and p ν x ν,f N, k 1 x ν )) or z = x C p ν )α/2 N and z x C p ν )α/2 N. A bound for the dfference between these ponts are p ν p ν 1 < δ N, k1+l f ) or x ν x ν 1 < δ N, k or z z 1 < δ N, k1+l C 1+L f )α/2 N ). Assumngδ N, k1+l C 1+L f )α/2 N ) = δ N, k 1 note that wth ths assumpton δ N, k δ N, k 1 and δ N, k1+l f ) δ N, k 1 1+L f )) and usng 3.56 we can fnd a bound for f N, k,x x) f N, k,x x) f N, k,x x) f N, k,x x) < ǫ N, k 1 +ǫ N, k 1 Gα/2 N = ǫ N, k 3.58) wth G 0 a bounded constant. Therefore the δ N, k δ N, k 1 ) and ǫ N, k ǫ N, k 1 ) obtaned for f N, k,x 14 n terms of δ N, k 1 and ǫ N, k 1 and eventuay n terms of δ N,0 and ǫ N,0 are as foows ǫ N, k = ǫ N, k 1 1+Gα/2 N ) = ǫ N,0 1+Gα/2 N ) k δ N, k = δ N, k 1 /1+L C 1+L f )α/2 N ) = δ N,0 /1+L C 1+L f )α/2 N ) k 3.59) for k = k N = q2 N we have ǫ N, q2n = ǫ 0 1+Gα/2 N ) q2n < ǫ 0 expgqα) = ǫ 3.60) δ N,q2N = δ 0 /1+L C 1+L f )α/2 N ) q2n > δ 0 /expl C 1+L f )qα) = δ 3.61) where ǫ 0 = ǫ N,0, δ 0 = δ N,0. Therefore for a ǫ > 0, we can choose ǫ 0 sma enough such that 3.60 s satsfed: ǫ 0 expgqα) = ǫ. For ths δ 0 has to be chosen such that z z 1 < δ 0 I,x z) I,x z) < ǫ 0 z, z V p p 1 < δ 0 1+L f ),y s p),y s p) < ǫ 0, 3.62),x p),x p) < ǫ 0,,ys p),ys p) < ǫ 0,,x p),x p) < ǫ 0, p, p P for the δ 0 of 3.62 the N ndependent δ s gven by 3.61: δ = δ 0 /expl C 1+L f )qα). Ths shows that the sequence f N,kN,x s equcontnuous and therefore statement 2.2 s proven. Therefore there exsts 14 Note that for the δ N, k and ǫ N, k obtaned, reaton 3.56 for the dervatves of and s aso satsfed: p, p P, p p 1 < δ N, k1 + L f ),y s p),y s p) < ǫ N, k,,x p),x p) < ǫ N, k,,ys p),ys p) < ǫ N, k,,x p),x p) < ǫ N, k snce δ N, k δ N, k 1 and ǫ N, k ǫ N, k 1. 22
a subsequence of f N,kN,x for = 1,...,m 1 that converges unformy and snce the sequence of f N,kN converges unformy to Ufs on V N,kN ths shows that Ufs,x x) exsts and s contnuous n the drecton of the varabes x for = 1,...,m 1 on V N,kN. Snce the hyperpanes V N,kN are dense n S + ths easygenerazesto a hyperpanesparaeto the nta condton hyperpanen S + e.g. by varyng α). Next we show that Ufs,x x) s contnuous n the x m drecton. Consder V β and V β+δβ for δβ > 0, defned at the begnnng of Subsecton 3.3, as the nta condton and fna hyperpane, respectvey. We dscretze the space n between aong the x m drecton smar to before. Consder 3.54 wth α repaced by δβ and V N,0 and V N,2N correspondng to V β and V β+δβ, respectvey, wth notng that a the terms have a bounded behavour as N the recurson reaton can be wrtten as f N,2N,x x) = f N,2N 1,x x ê m δβ/2 N + ê O δβ)/2n ) + O δβ)/2 N wth O δβ) and O δβ) terms of order δβ, therefore upon sovng ths reaton for f N,2N,x x) x V β+δβ ) n terms of f N,0,x x ) = Ufs,x x ) x V β ), we fnd f N,2N,x x) = Ufs,x x ê m δβ +ê O N δβ)) +O N δβ), wth x ê m δβ + ê O N δβ) V β, O N δβ) and O N δβ) terms of order δβ. From f N,2N,x there s a subsequence e.g. f an,2an,x ) that converges unformy to Ufs,x x), therefore 15 Ufs,x x) Ufs,x x ê m δβ) = m n fan,2an,x x)} Ufs,x x ê m δβ) = Ufs,x x ê m δβ +ê O δβ))+oδβ) Ufs,x x ê m δβ) 3.63) we aready proved that Ufs,x s contnuous n the drecton of the varabes x on V β, therefore upon takng the mt δβ 0 n 3.63 note that for x V β+δβ, x ê m δβ V β s a fxed pont) t can be concuded that Ufs,x s contnuous n the x m drecton 16. From 3.50 and 3.51 t foows that Ufs x) soves the system of PDE of 3.1 subject to the nta condton for a x S + and that Ufs,xm x) exsts and s contnuous. Smary wth assumng that the nta condton and the coeffcents and are C r+1 for r 1 we can show that the souton s C r+1. For ths consder the r+1 parta dervatves of 3.52, by smar methods t can be shown that the sequence of a r +1 parta dervatve of f N,kN s bounded and equcontnuous and wth a subsequence of ts ower r dervatve convergng unformy, t can be concuded that the r+1 parta dervatve of Ufs n the x drectons exsts and s contnuous n the x drectons for 1 m 1, aso smar to the argument above t can be concuded that the r +1 parta dervatve n the x drectons s contnuous n the x m drecton. Then usng 3.51 t can be shown that a r+1 parta dervatves n the x j drecton for j = 1,...,m exst and are contnuous. Note that wth the Lpschtz or C r assumpton on the coeffcents and the nta condton we obtan a Lpschtz or C r souton, respectvey but the characterstc curves and the souton aong 15 m n ê O an δβ) ê O δβ) and m n O an δβ) Oδβ), these mts are we defned. To see ths consder f an,2an,x x) = Ufs,x x ê mδβ +ê O an δβ))+o an δβ), as noted f an,2an,x x) converges to Ufs,x x). x ê mδβ + ê O an δβ) V β converges to the pont n V β whch the characterstc curve of the souton f passng through x V β+δβ passes through n V β, therefore the O an δβ) term aso has a we defned mt as n. 16 As prevousy noted athough the constructon of Ufs was done by movng n the postve x m drecton t s cear that wth smar methods t s possbe to start from an nta condton hyperpane and construct the souton n the negatve x m drecton c.f. Appendx A). Therefore the dscusson n ths page s equvaenty vad for when makng the repacement δβ δβ for δβ > 0 and showng the contnuty of Ufs,x x) n the negatve x m drecton. 23