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1 Math-Nt.Ru All Russian mathmatical portal Mihail S. Kildyushov, Valry A. Niishin, Asymptotic bhavior at infinity of th Dirichlt problm solution of th ordr quation in a layr, J. Sib. Fd. Univ. Math. Phys., 14, Volum 7, Issu 3, Us of th all-russian mathmatical portal Math-Nt.Ru implis that you hav rad and agrd to ths trms of us Download dtails: IP: Sptmbr 3, 18, :34:3

2 Journal of Sibrian Fdral Univrsity. Mathmatics & Physics 14, 73, УДК Asymptotic Bhavior at Infinity of th Dirichlt Problm Solution of th Ordr Equation in a Layr Mihail S. Kildyushov Valry A. Niishin Institut of Computr Tchnology, Moscow Stat Univrsity of Economics, Statistics and Informatics, Nzhinsaya, 7, Moscow, Russia Rcivd 1..14, rcivd in rvisd form , accptd.4.14 For th oprator ux + ν ux with x R n n, an xplicit fundamntal solution is obtaind, and for th quation ux + ν ux = fx for f C R n with compact support th lading trm of an asymptotic xpansion at infinity of a solution is computd. Th sam rsult is obtaind for th solution of th Dirichlt problm in a layr in R n+1. Kywords: asymptotic bhavior, lliptic quation, fundamntal solution, stimation of solution, G-Myr function. A fundamntal solution for th oprator ux + ν ux, x R n n,, is obtaind in [1, Sction 3,.8.3]. Th lading trm of an asymptotic xpansion at infinity xponntially dcrass and dos not contain a rapidly oscillating factor. In [] th gnral form of all solutions of th quation U + a 1 1 U a U =, in a domain is dducd. Hr a 1,a,...,a ar complx constants. In [3] a fundamntal solution for th oprator ux + µux, satisfying th radiation condition, is considrd. Lt hξ b th Fourir transform of th function hx L 1 R n hξ = F[h]ξ = πıxξ hxdx. R n If g L 1 R n, thn th invrs Fourir transform is F 1 [g]x = πıxξ gξdξ. R n Lmma. Th quation Ex + ν Ex = δx x R n n, ν > 1 has a radially symmtric solution Er of th form n 4/ νr Er = π n/ ν G+1, rn, n, n + 1 n,..., + 1, 1,,..., 1, hr r = x 1 + x x n, G is th G-Myr function in th notation of [4]. ridan111@gmail.com vniishin@msi.ru c Sibrian Fdral Univrsity. All rights rsrvd, 311

3 Mihail S.Kildyushov, Valry A.Niishin Asymptotic Bhavior at Infinity of th Dirichlt Problm... Proof. Th Fourir transform Ẽξ of th fundamntal solution Ex of th quation 1 is a solution of th quation πρ + ν Ẽξ = 1. Hnc, whr ρ = ξ 1 + ξ ξ n. So Ẽξ Ex = Eρ 1 = πρ + ν, Rn πıxξ πρ dξ. + ν Using th formulas for th invrs Fourir transform of th radially symmtric function Ψρ s [5,.113,.114], w hav: for n = p for n = p Ex = π p 1 lim ε + r 1 Ex = π p 1 lim ε + r r r p 1 [ π p 1 [ r + + ] ερ ΨρρJ πrρdρ, 3 ] ερ Ψρρsinπrρdρ. 4 Notic that in our cas for ε = th intgrals in squar bracts in formulas 3, 4 convrg. Thrfor, w can considr formally th following xprssions, although it is not yt provd that thy dfin a solution of 1: for n = p for n = p + 1 Ex = π p 1 1 r Ex = π p 1 1 r p 1 [ + ] ρ π r πρ + ν J πrρdρ, 5 p 1 [ + ] ρ r r πρ sinπrρdρ. 6 + ν Considr th vn dimnsion cas. W calculat th intgral on th right hand sid of 5 at = 1,,3,4,..., using Mathmatica th licnc L , and construct for it th following xprssion for any : + ρ πρ + ν J πrρdρ = 1 ν r 8π G+1, ν, 1,,...,, 1,,..., 1,. 7 Substituting it in 5 and using th formula for diffrntiation of th G-Myr function s [6, 8...3] w obtain th function on th right hand sid of th xprssion. Lt us prov that this is th xprssion for th fundamntal solution. Using th wll-nown asymptotic xpansion of th G-Myr function, w put G m,,q z Gm, z b1,b.,...,b q From Thorm s [7, 5.9.] it follows that for m q 1,arg z =,z + : G m,,q,q z Am, q H p,q z ıπq m + Ām, q H p,q z ıπq m. 31

4 Mihail S.Kildyushov, Valry A.Niishin Asymptotic Bhavior at Infinity of th Dirichlt Problm... In th cas undr considration m = + 1,p =,q =. H, ζ = 1 π 1 xp ζ 1 1 ζ 4 j= M j ζ j, whr M = 1 [7, ], A +1, = 1 1 πı 1 xp ıπ [7, 5.8..]. Finally, for th G-Myr function in 7 w obtain G +1, ν r, 1,,...,, 1,,..., 1, j=+1 π νr xp νr sin π b j 3π 1 cos νr cos π. 4 Th xponntial dcras nsurs th convrgnc of th intgrals and th applicability to th right hand sid of of th formula [5,.18] for th Fourir transform of th radially symmtric function: In our cas it loos li Ψρ = Ψρ = π ρ n / + n πρ n 1 ν Φrr n/ J n / πρrdr. 8 + νr G +1,, n, n + 1 n,..., r 1 n J n πρr + 1, 1,,..., 1, dr. This intgral s [6,.4.4.1] is xprssd via th G-Myr function and simplifid to th form 1 ν 1 Ψρ = πρ G1,1 ν. πρ Using th intgral rprsntation of th G-Myr function [6, Dfinition 8..1], whr th intgration contour L is th straight lin ı, + ı and formula [8, ], w find 1 G 1,1 1,1 z = z 1 + z. From hr it follows that 1 πρ G1,1 ν 1 ν πρ 1 1 = 1 1 πρ. 9 + ν Finally, th statmnt is provd for vn n. Lt us prov th lmma for odd n. First, for n = 3 w substitut 9 into th invrs Fourir transform formula formula 8, whr ρ and r ar intrchangd. W obtain th intgral s [6,.4.4.1] and substitut th rsult in 6. Now w procd similarly to th cas of vn n. Th lmma is provd. Lt us spcify th lading trm of th asymptotic xpansion of ux at infinity. 313

5 Mihail S.Kildyushov, Valry A.Niishin Asymptotic Bhavior at Infinity of th Dirichlt Problm... Thorm 1. Lt x R n n, ν >, fx b a smooth function with compact support. Lt th solution ux of th quation ux + ν ux = fx 1 xponntially dcras at infinity. Thn th following rprsntation holds ux = r 1 n/ sin νr cos π νr sin π Φ1 θ 1,...,θ n 1 + +r 1 n/ cos νr cos π νr sin π Φ θ 1,...,θ n 1 + Or n+1/ νr sin π, 11 whr Φ 1 θ 1,...,θ n 1,Φ θ,...,θ n 1 ar diffrntiabl functions on th unit sphr. Proof. Lt fx hav its support in a ball Q R of radius R. Thn for th solution ux of th quation 1 w hav th following rprsntation ux = E x y fydy. R n Introduc th following notation x = x 1,x,...,x n. Suppos that x runs along a ray, w can turn th coordinat systm so that this ray coincids with x 1 >, x =... = x n =. Thn ux 1,,..., = x 1 y 1 + y y n fydy = R n E = E x 1 y 1 fydy + E x 1 y 1 + y y n E x 1 y 1 fydy. Q R Q R Dnot th intgrals on th right hand sid of th this quality by J 1 and J. For Er w mploy th asymptotic xpansion Er n+1 4 ν πr n 1 νr sin π cos πn νr cos π and obtain, as x 1 +, J 1 = x 1 y 1 1 n/ sin ν x 1 y 1 cos π ν x 1 y 1 sin π c1 + Q R + x 1 y 1 1 n/ cos ν x 1 y 1 cos π ν x 1 y 1 sin π c + r +, 1 +O x 1 y 1 n+1/ ν x1 y1 sin π fydy = = x 1 n/ 1 sin νx 1 cos π νx 1 sin π c3 + x 1 n/ 1 cos νx 1 cos π νx 1 sin π c4 + +Ox n+1/ 1 νx1 sin π, whr c 1, c, c 3, c 4 ar constants. Turn now to th stimation of J. Using 1 and th man valu thorm w arriv at th following inqualitis for som < Θ < 1: E x 1 y 1 + y y n E x 1 y 1 = = x E 1 y 1 + Θ x 1 y 1 + y y n x 1 y 1 314

6 Mihail S.Kildyushov, Valry A.Niishin Asymptotic Bhavior at Infinity of th Dirichlt Problm... x 1 y 1 + y y n x 1 y 1 C ν x1 y1 sin π x1 y 1 n+1 C 1 νx1 sin π x n+1 1 if y R, x 1 R. Thrfor J C νx1 sin π x n+1 1, whr C,C 1,C ar constants. Going bac to initial coordinats, w gt th rprsntation 11. W shall apply th obtaind rsults to th Dirichlt problm in a layr. Dnot Π = {x,x n+1 R n+1 x R n,x n+1 a,b}, < a < b < +, n. Considr th problm 1 n + v + αv = h, x,xn+1 Π, x n+1 j v x j xn+1=a n+1 x j=1 j = j v x j xn+1=b n+1 =, j =,..., Lt < λ 1 < λ <... b th ignvalus and ϕ l,l = 1,,... b th ignfunctions of th problm { y t λ yt =, t a,b, 14 y j a = y j b =, j =,..., 1. Put µ l = α + λ l l = 1,,... In[9] Thorm 6 th solvability of th problm 13 and uniqunss of th solution v was provd for hx,x n+1 C Π with compact support and α + λ l >, as wll as th stimat vx,x n+1 C µ1 sin π ε x, x,x n+1 Π, 15 hr ε > is sufficintly small. Study th proof of th stimat 15 mor closly. Dnot by ṽ ṽξ 1,...,ξ n,x n+1 th Fourir transform with rspct to x of th function vx,x n+1. Thn ṽ is a solution of th on-dimnsional boundary valu problm in x n+1 on [a,b] with th paramtrs ξ 1,...,ξ n : { 1ṽ + αṽ + π ξ ξ n ṽ = F[h], x n+1 a,b, Th singular sts in this cas ar givn by th conditions Put ṽ j a = ṽ j b =, j =,..., 1. µ l = π ξ ξ n, l = 1,,3, ζ j = R ξ j, τ j = Im ξ j j = 1,...,n, ζ = ζ 1,...,ζ n, τ = τ 1,...,τ n. To apply Thorm 6 of [9], th intrsction of th cylindr π τ = γ with th singular sts 16 must b mpty. W shall find γ, for which this condition is fulfilld, that is th following systm has no solutions: { π τ = γ, π ξ ξ n = µ l l = 1,,3,

7 Mihail S.Kildyushov, Valry A.Niishin Asymptotic Bhavior at Infinity of th Dirichlt Problm... This systm splits into systms of th form { π τ = γ, π ξ ξ n = µ l cos 1+sπ whr s =,1,..., 1. For ach s th systm consists of th ral quations π τ = γ, π ζ τ = µ l π ζ,τ = µ l + ı sin 1+sπ cos 1+sπ l = 1,,3,..., sin 1+sπ l = 1,,3,... l = 1,,3,..., Th Cauchy-Schwarz inquality implis that th contradiction is achivd if th following condition is fulfilld: 4γ µ 1 + sπ l cos + γ < µ sπ l sin. 1 + sπ Solving this inquality for γ, w obtain γ < µ l sin. Thus, if γ < µ 1 sin π, thn th systm 17 is inconsistnt. Thorm. Lt hx,x n+1 C Π in th problm 13 hav compact support, th constant α satisfy th condition α+λ 1 >, whr λ 1 is th first ignvalu, and ϕ 1 b th corrsponding ignfunction of th problm 14. Lt th solution vx,x n+1 of th problm 13 xponntially dcras at infinity. Thn vx,x n+1 = sin α + λ 1 r cos π Φ1 θ 1,...,θ n cos α + λ 1 r cos π Φ θ 1,...,θ n 1 r 1 n/ α+λ 1 r sin π ϕ1 x n Or n+1/ α+λ 1 r sin π, whr Φ 1 θ 1,...,θ n 1,Φ θ,...,θ n 1 ar diffrntiabl functions on th unit sphr. Proof. Put h 1 x = v 1 x = b a b a hx,x n+1 ϕ 1 x n+1 dx n+1, vx,x n+1 ϕ 1 x n+1 dx n+1. Not that h 1 x has compact support and h 1 x C R n. Th function v 1 x is a solution of v 1 x + µ 1 v 1 x = h 1 x x R n. Thn for th solution vx,x n+1 of 13 w hav th rprsntation vx,x n+1 = v 1 xϕ 1 x n+1 + ˆvx,x n+1, 18 and for ˆvx,x n+1, by Thorm 6 of [9], w hav th stimat ˆvx,x n+1 C µ sin π ε x, ε > is sufficintly small. Th asymptotic xpansion for vx,x n+1 follows from 18 and Thorm

8 Mihail S.Kildyushov, Valry A.Niishin Asymptotic Bhavior at Infinity of th Dirichlt Problm... Rfrncs [1] I.M.Glfand, G.E.Shilov, Distributions and actions ovr thm, Moscow, Dobrosvt, in Russian. [] I.N.Vua, About mtaharmonic functions, Tr. Tbil. Mat. Inst., 11943, in Russian. [3] A.V.Filinovsy, About asymptotic bhavior of solutions of on non-stationary mixd problm. Dif. Uravn, 11985, no. 3, in Russian. [4] G.S.Mijr, On th G-function, Ndrl. Aad. Wtnsch. Proc. Sr. A, , 7 37, , , , , , , [5] S.Mizohata, Th thory of partial diffrntial quations, Cambridg Univrsity Prss, Nw Yor, [6] A.P.Prudniov, J.A.Brychov, O.I.Marichv, Intgrals and sris. Supplmntary chaptrs, Moscow, Naua, 1986 in Russian. [7] Y.L.Lu, Th Spcial Functions and Thir Approximations, Nw Yor, Acadmic Prss, vol. I II, [8] S.Gradshtyn, I.M.Ryzhi, Tabl of intgrals, sris, and products, Fourth dition prpard by Ju. V. Gronimus and M. Ju. Citlin. Translatd from th Russian by Scripta Tchnica, Inc. Translation ditd by Alan Jffry, Acadmic Prss, Nw Yor, MR #595 [9] V.A.Niishin, On stimats of solutions to boundary-valu problms for lliptic systms in a layr, Funct. An. and Its Appl., 4511, no., Об асимптотике решения задачи Дирихле для уравнения порядка в слое Михаил С.Кильдюшов Валерий А.Никишкин Для оператора ux + ν ux в R n n, получен явный вид фундаментального решения, а для уравнения ux + ν ux = fx с финитной бесконечно дифференцируемой функцией f первый член асимптотики решения на бесконечности. Изучается также задача Дирихле в слое из R n+1. Ключевые слова: асимптотика, эллиптическое уравнение, фундаментальное решение, оценки решений, G-функция Мейера, слой. 317

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