The Cauchy problem for hyperbolic operators with variable multiple characteristics-猫眼课题宝

具有可变多重特征的双曲算子的柯西问题

DOI：

10.1090/s0002-9939-1978-0503542-1

发表时间：

1978

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作者：

Kazuhiro Yamamoto

设P（t, x, D ‘ Dx）是一个双曲微分算子，其主符号为pm（t, x, t, q）。我们设pm表示为IÇ_ i(t W&flYi* \) ** D <t\ \ x ’。x,Q + 0iî （i,j）¥■（*,m N + k） (k = 1，…s),电视= 2 j_ z«,和t \ (z, x,问e C°°Q0, t] x Rn (R " \ 0))。在E. E. Levi的广义条件下，我们证明了柯西问题Pu = f in [0, T] x R '， I>/»|/-o = % U ~ 1»•••> m ~ 1)是适定的。当m × 1 （J■1）j)，我们的结果与Ohya和Petkov的结果一致。结果的声明。我们将考虑以下微分算子：P(t,x,D ' Dx)= S aa(t,x)Dr*DÏ， \a\<m where (i, jc) E [0, t] x R ' ' (0 < t < 0)， Z),a°Z>/ = a ‘ 0^",’■•••a ？和Z), = Z '8/3i, Djj。= id / dxj。我们假设<zm00(i, x) = 1, aa（/, x）属于$（[0,T] x R”），它由所有这些任意导数有界于[0，T] x R”的函数组成。设（t，£）为（r, x）的协变量。然后定义如下符号：pk(t,x,j,i)= 2aa (t,x)ra°ia' （/c = 0，…，m）。首先，我们将在Pm上施加以下假设：s m - N+s Pm(t,x,rA)=W(t- Xj)mj IT (t- x,), (A.1) j=\ j=s+\其中mx> m2>•••> ms, N = 2*_i wy和所有函数Xj（t, x，£）（j= 1,. .…）， m N + s)是关于£的1次实数且正齐次的，并且当£E s ' -1时属于< s （[0, T] x Rn x Sn~x）。这里是R的单位球面。(a)对任何夫妇(i, j) = h (k、m N + k) (k = 1,……， s)我们假设如下：|(\ Xj)（t, x，€）| > C, (r, x, |) E [0, t] x Ä“x s”-其中C是一个正常数。在本注释中，伪微分算子的符号是<$（[0,T] X R“ X 5“）或$([0,T] X 7T X S”-1) if （T，£）E 5”或由编辑于1977年5月13日和1977年8月8日修订的<$（[0,T] X 7T X S"-1）的元素。AMS （MOS）学科分类（1970年）。主要35个翻译;二次35给。©美国数学学会1978 109许可或版权限制可能适用于再分发；参见http://www.ams.org/journal-terms-of-use 110 KAZUHIRO YAMAMOTO£G S“1-1”。此外，我们使用以下符号。对于P的低阶项我们做如下假设：（A.3）对于任意k （k = 1，…）我们可以用以下形式表示P(t, x,D ' Dx): mk P=IlQkJ(t,x,Dl,Dx)（Ak(t,x,D ' Dx)）。这里Ak是由符号t （A^f, x，£）和Qkl（1 =0，…）定义的伪微分算子， mk)是m阶伪微分算子，其主符号qkj（t, x, t，£）具有以下性质：？M^e0 modXk Xm_N+k。（1.2）显然，如果Xk = Am_Ar+fc，则上述条件（A.3）是E. E. Levi的条件。在本注的最后一部分，当mk = 1或2时，我们用pk的条件表示（A.3）。对于非负整数k和s E R，函数空间C*（[0, T]； HS(R")）由满足D{u(T) (j = 0，…， k)作为Hs_j（R”）的一个元素存在，并且在Hs_j（R”）的拓扑上连续。我们使用以下规范：ikoiift * = 2 l|£/«(0ll？-y> 7 = 0其中||•\\s_j是Hs_j（R”）的通常规范。现在我们可以陈述定理了。定理。设P（t, x, D ' Dx）是m阶微分算子。如果P满足假设（a . 1），（a .2）和（a .3），则柯西问题Pu = fin [0, t] x R ', D/«|/=,0 = gj(J = 0，…，m I)是适定的，即，对于f 6 C* - + ' '+,aft T)；HS -m+mi + x(Rn))和gj E Hs_j+m（R»）存在一个唯一解u(t, x) G C*([0, t]； HS(R '))，使得(m-\ IIKOIIU < C 2 llg,IL-,+mi + ll/(0)|IL-m+mi,A_m+mi I 7=0•ill /tollL-m+m +m。+密歇根大学+ m。+ l ^ l。wAere k > m m, 1, |||/(0)|||J_m+m,>_1 = 0 am/。G [0, t]。当m， = 1 （J = 1,. .）时，[3]和[4]的定理相同,年代)。在另一种情况下，在b[1]中，他们考虑三重情况下的柯西问题。许可或版权限制可能适用于再分发；参见http://www.ams.org/journal-terms-of-use双曲算子的柯西问题111 2。条件改革（A3）。在本节中，我们陈述（A.3）的等价条件。考虑根X × j = 1的多重性，…,年代),我们可以表示点(t, x, t,£)通过(t \“_N”+ s)■■- (r-Xs + xWr■■&、>,d >”(f, x t E) (v = 1,…， u)是关于t的s次多项式，等于II^。我(t Xj)。这里ms × x =••••- m, （v - 1，…）u)。注意sx < s2 <■■■< s^ = s, mx = 2；=1/z”，用svnp表示N”。我们引入一个积伪微分算子$ " (/,x, Dt, Dx) = (Aj•••a,)（/, x, Dt, Dx）。然后记为阶为j （j = 0，…）的Ay（t, x, D ' Dx）， m)由A， 1， A, A "…,,,…， àN Of * *?'，…， A ' +, = A1+t•••AJ+1A^，…, = Am_”+我•••AJ + 1个,,,=像•••,W_j•••美元;如果y = (Nx + • • • + N”值)+ os”+ 8 (o = 0 . .。， n, - 1,8 = 0，…sy_x)。然后我们有下面的，命题2.1。设P（t, x, D ' Dx）是满足条件（a .1）和（a .2）的微分算子。那么条件（A.3）等价于下面的语句。我们可以表示771年P, P (t, x, D”Dx) = 2 Q, IU x, D”Dx) /((>(2.1)(= 0 =«M + • • • + w”+ 1 + o (1 < < n”),然后我(我)= Nx + • • • + 第四“年代”o和Q¡(i = 0,。， m,)是一个Mi = m - i - i (i)阶的伪微分算子和t的微分算子，并且q的主符号q′（t, x, t，£）满足下列条件：(7,1^=0 mod a * - k -s+k ifk<s”。（2.2）由于任意阶/ (< m)的伪微分算子是t的微分算子，用a…表示。先生，我们有以下内容：提案2.2。设P是满足条件（A.l），（A.2），（2.1），（2.2）的微分算子。则P表示为771，M′′=2 2*z,,('>*>¿>,)A/0)+,,（2-3）其中R′是M′- j阶伪微分算子，其主符号为R′（t, x，£），end rtJ具有以下性质：k-\ 2 rtJAj■•••A…^ 0 mod A* x”，_N+k, k < s”（2.4）y-o if i = zzM +••••+ rt”+1 + o （1 < A < zz”）。3. 还原为一阶系统及定理的证明。由于定理的证明是通过类比一个简单的情况推断出来的，我们假设j = 2, ttz, = 2, m2 = 1。因此，根据第2.3条，我们考虑的运营商许可或版权限制可能适用于再分发；见http://www.ams.org/journal-terms-of-use 112 KAZUHIRO YAMAMOTO（1.1）中的P（t, x,D " Dx）表示为2m - i P(t, x,D " Dx) = Am + '2 2*r‘ a > í-1 j=0，其中Rf"~’是m - i - j阶。条件（2.4）表明r¿-l(t,x,Dx) =0, (3.1) ri ' ' 1 （/, x，¿）= r0-2 (/, x, |) =0 mod x, \ ' _2, (3.2) (ri ' -1 + r2 - ' (X2 Xx))(t, x,^) =0 mod X2 x ',_x。（3.3）我们用项a表示符号为|£|的伪微分算子，并定义列向量U ='(Am- 3v > Am ' 3A, ', Am- 4a2w, Am- 4a3m，…，AAm_2M,Am_lM)（3.4）和F '(0，…0 /)。由式（3.1），方程Pu = /变成如下一阶系统：MU = （D，¿（/,x， DX））U + B(t, x， DX)U = 77，其中5为0阶，A为一阶伪微分算子，符号为

Let P(t, x, D„ Dx) be a hyperbolic differential operator with the principal symbol pm(t, x, t, Q. We assume that Pm is denoted by IÇ_ i(t W&flYi* \) **d <t\ \X'. x,Q + 0iî (i,j) ¥■ (*, m N + k) (k = 1, . .. , s), where tV = 2j_,z«, and t\,(Z, x, Q e C°°Q0, T]X Rn X (R"\0)). Under a generalized condition of E. E. Levi, we shall show that the Cauchy problem Pu = f in [0, T] x R", I>/»|/-o = % U ~ 1» • • • > m ~ 1) is well posed. When m¡ — 1 (J ■ 1.j), our result coincides those of Ohya and Petkov. 1. Statement of the result. We shall consider the following differential operator: P(t,x,D„Dx)= S aa(t,x)Dr*DÏ, \a\<m where (i, jc) E [0, T] X R" (0 < T < oo), Z),a°Z>/ = A"0^",' ■ • • A? and Z), = z'8/3i, Djj. = id/dxj. We assume that <zm00(i, x) = 1 and aa(/, x) belongs to $([0, T] X R"), which consists of all functions having that these arbitrary derivations are bounded in [0, T] X R". Let (t, £) be the covariable of (r, x). Then we define the following symbols: pk(t,x,j,i)= 2 aa(t, x)ra°ia' (/c = 0,...,m). First we shall impose the following assumptions onpm: s m — N+s Pm(t,x,rA)=W(T-XJ)mj IT (t-X,), (A.1) j=\ j=s+\ where mx> m2> • • • > ms, N = 2*_i wy and all functions Xj(t, x, £) (J = 1,. .., m N + s) are real and positively homogeneous of degree 1 with respect to £ and belong to <S([0, T] x Rn x Sn~x) if £ E S"-1. Here 5"1"1 is the unit sphere of R". (A.2) For any couple (i,j) =h (k, m N + k) (k = 1,..., s) we suppose the following: |(\ Xj)(t, x, €)| > C, (r, x, |) E [0, T] X Ä" X S"-\ where C is a positive constant. Throughout this note, the symbols of pseudo-differential operators are elements of <$([0, T] X R" X 5") or $([0, T] X 7T X S"-1) if (t, £) E 5" or Received by the editors May 13, 1977 and, in revised form, August 8, 1977. AMS (MOS) subject classifications (1970). Primary 35L30; Secondary 35L45. © American Mathematical Society 1978 109 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 110 KAZUHIRO YAMAMOTO £ G S"1-1. Moreover we use the following notation. For symbols a(t, x, |), b(t, x, £) and A(t, x, t, £), A\T=a = 0 mod b means that there exists a symbol c(t, x, £) such that A(t, x, a(t, x, £), Í) = c(t, x, £)b(t, x, £). For the lower order terms of P we assume the following: (A.3) For any k (k = 1,.. . , s) we can denote P(t, x, D„ Dx) by the following form: mk P=IlQkJ(t,x,Dl,Dx)(Ak(t,x,D„Dx)). (1.1) 1=0 Here Ak is the pseudo-differential operator defined by the symbol t A^f, x, £) and Qkl (1 = 0,..., mk) is a pseudo-differential operator of order m mk, whose principal symbol qkj(t, x, t, £) has the following property: ?M^e0 modXk Xm_N+k. (1.2) Clearly if Xk = Am_Ar+fc, then the above condition (A.3) is that of E. E. Levi. In the final part of this note we denote (A.3) by the condition with respect to pk, when mk = 1 or 2. For a nonnegative integer k and s E R the function space C*([0, T]; HS(R")) consists of functions such that D{u(t) (j = 0, ..., k) exists as an element of Hs_j(R") and is continuous on the topology of Hs_j(R"). We use the following norm: IIKOIIft* = 2 l|£/«(0ll?-y> 7 = 0 where || • \\s_j is the usual norm of Hs_j(R"). Now we can state our theorem. Theorem. Let P(t, x, D„ Dx) be a differential operator of order m. If P satisfies the assumptions (A.l), (A.2) and (A.3), then the Cauchy problem Pu = fin [0, T]X R", D/«|/=,0 = gj(J = 0, . . .,m I) is well posed, i.e., for f 6 C*—+"'+,aft T); Hs-m+mi + x(Rn))andgj E Hs_j+m(R») there exists a unique solution u(t, x) G C*([0, T]; HS(R")) such that ( m-\ IIKOIIU < c 2 llg,IL-,+mi + lll/(0)|IL-m+mi,A_m+mi I 7=0 • illl/tollL-m+m. + U-m+m. + l^l. wAere k > m m, 1, |||/(0)|||J_m+m,>_1 = 0 am/. G [0, T]. This Theorem is the same as those of [3] and [4] when m, = 1 (J = 1,. .., s). Under a different situation, in [1] they consider the Cauchy problem of a triple case. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE CAUCHY PROBLEM FOR HYPERBOLIC OPERATORS 111 2. Reform of the condition (A3). In this section, we state an equivalent condition of (A.3). Taking care of multiplicities of the roots X¡ (j = 1,. . ., s), we can denote pm(t, x, t, £) by (t \„_N+S) ■ ■ -(r-Xs+xWr ■■&,>, where d>„(f, x,t,E)(v = 1, ..., u) is a polynomial of degree s with respect to t and equal to II^.i(t Xj). Here ms¡ + x = • • • — m, (v — 1, . . . , u). Remark that sx < s2 < ■ ■ ■ < s^ = s, mx = 2¡;=1/z„ and denote N„ by svnp. We introduce a product pseudo-differential operator $„(/, x, Dt, Dx) = (Aj • • • A,)(/, x, Dt, Dx). Then we denote Ay(t, x, D„ Dx) of order j (J = 0,..., m) by A, 1, A, A„ ..., A,,..., àN Of• • *?',..., A„+, = A1+t • • • AJ+1A^, . . . , Am = Am_„+i • • • AJ+1AN, where A, = As • • • A,W_j •••$;' if y = (Nx + • • • + N„_x) + os„ + 8 (o = 0,.. ., n, — 1,8 = 0,..., sy_x). Then we have the following: Proposition 2.1. Let P(t, x, D„ Dx) be a differential operator which satisfies the conditions (A.1) and (A.2). Then the condition (A.3) is equivalent to the following statement. We can denote P by 771, P (t, x, D„ Dx ) = 2 Q, ÍU x, D„ Dx )A/((> (2.1) ( = 0 where if i = «M + • • • + w„+1 + o (1 < a < n„), then I(i) = Nx + • • • + iV„ — s„o and Q¡ (i = 0, . . ., m,) is a pseudo-differential operator of order Mi = m — i — I (i) and differential operator of t. Moreover the principal symbol q¡(t, x, t, £) of Q¡ satisfies the following condition: (7,1^=0 mod A* -K-s+k ifk<s„. (2.2) Since any pseudo-differential operator of order / (< m) which is a differential operator of t is represented by A,,, . . . , Am, we have the following: Proposition 2.2. Let P be a differential operator which satisfies the conditions (A.l), (A.2), (2.1) and (2.2). Then P is denoted by 771, M¡ ^=2 2*z,,('>*>¿>,)A/0)+,, (2-3) where R¡j is a pseudo-differential operator of order M¡ — j whose principal symbol is r¡j(t, x, £), end rtJ have the following property: k-\ 2 rtJAj ■ • • A,,,^ 0 mod A* X„,_N+k, k < s„ (2.4) y-o if i = zzM + • • • + rt„+1 + o (1 < a < zz„). 3. Reduction to a first order system and the proof of the Theorem. Since the proof of the Theorem is inferred on the analogy of a simple case, we assume that j = 2, ttz, = 2 and m2 = 1. Thus by (2.3) our considered operator License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 112 KAZUHIRO YAMAMOTO P(t, x, D„ Dx) in (1.1) is denoted by 2 m — i p(t,x,D„Dx) = Am + '2 2*r'A> í-1 j=0 where Rf"~' is of order m — i — j. The condition (2.4) says that R¿-l(t,x,Dx) = 0, (3.1) ri""1 (/, x, ¿) = r0—2(/, x, |) = 0 mod X, \„_2, (3.2) (ri"-1 + r2—' (X2 Xx))(t, x,^) = 0 mod X2 X„,_x. (3.3) We denote a pseudo-differential operator with the symbol |£| by the term A and define a column vector U ='(Am-3V> Am"3A,«, Am-4A2w, Am-4A3M,...,AAm_2M,Am_lM) (3.4) and F '(0,..., 0,/). Then by (3.1) the equation Pu = / becomes the following first order system: MU = (D, ¿(/, x, DX))U + B(t, x, DX)U = 77, where 5 is of order 0 and A is a first order pseudo-differential operator with the symbol

The Cauchy problem for hyperbolic operators with variable multiple characteristics

具有可变多重特征的双曲算子的柯西问题

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