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34
Вычислительные технологии
Том 5, № 5, 2000
ON THE WEAK SOLUTIONS OF THE EQUATION RELATED TO THE DIAMOND OPERATOR *
A. KANANTHAI, S. SUANTAI, V. LÜNGANI Chiangmai University, Department of Mathematics, Thailand e-mail: malamnka@science.cmu.ac.th
Рассматривается функция Грина оператора ®k, определенного следующим обра-
зом:
®k =
p+q ö
A", dx2
\3=P+1 J
4-1
где р + ц = п — размерность пространства Сп векторов х = (Х\,Х2, ..., хп) с п комплексными компонентами х^, к — целое неотрицательное число. Выполнено исследование функции Грина, которая затем применяется для построения слабого решения уравнения К(х), такого что
фк К (х) = / (х),
где / — обобщенная функция.
1. Introduction
The operator ©fc can be factorized in the following form
ek
p д2 \ 2 i p+q d2 4 2
S dx2 J — dx2
j=i
V — + г V —
dx2 dx2
. i=1 ' j=p+1 J.
v —-i p+q —
dx2 dx2
. i=1 ' j=P+1 J.
(1.1)
where i = y/—1 and p + q
n.
P d 2 V / P+q d 2
The operator | ^^ -hj j — у =+ -x^ ) has first been
i=1
introduced by A. Kananthai [4] and is named the Diamond operator which is denoted by
Let us denote the operators
Lk
v —Y- f V —
к — Vj=P+I dj
v—+i p+q —
dx2 dx2
,i= 1 ' j=p+1 J.
(1.2)
(1.3)
*The authors are responsible for possible misprints and the quality of translation. © A. Kananthai, S. Suantai, V. Longani, 2000.
к
4
к
к
к
к
2
к
and
Lk
p+q
.... ^ dxf
i=1 ' j=p+1 J_
P
dx2
E--1
(1.4)
Thus the operator ©k, iterated k-times defined by (1.1) can be written in the form
©k = ♦ Lk L2. (1.5)
In this work, we obtain the Green function of the operator ©k, i.e. ©kG(x) = 6 where 6 is the Dirac-delta distribution and G(x) is the Green function and x £ Rn. Moreover, we find the weak solution of the equation
©k K(x) = f (x)
(1.6)
where f is a given generalized function and K(x) is an unknown and x £ Rn
k
2. Preliminary
Definition 2.1. Let x = (x1, x2,..., xn) £ Rn Let us denote by
P P+q
u = E x2 "E x2 (2.1)
i=1 j=P+1
the nondegenerated quadratic form, whereas p + q = n is the dimension of Rn. Let r+ = {x £ Rn : x1 > 0 and u > 0} and r+ denotes its closure. For any complex number a, we define the function
(a —n)
u 2
RH(u) = ^ Kn(0) for x £ Г+, (2.2)
0 for x £ r+, where the constant Kn(a) is given by the formula
• K^nl ) ™
r ^ 2 + a - P^ r /P - a
The function RH is called The Ultra-Hyperbolic Kernel of Marcel Riesz and was introduced by Y. Nozaki (see [3], p. 72).
It is well known that RH is an ordinary function if Re (a) > n and is a distribution of a if Re (a) < n. Let us supp RH (u) denote the support of RH (u). Assume RH (u) C r+. Definition 2.2. Let x = (x1,x2, ..., xn) be a point of the Euclidean space Rn and
n
v = £ x2. (2.3)
i= 1
Define the function
a — n
V 2
K(v) = -^T-:, (2.4)
where a is any complex number and the constant Hn(a) is given by the formula
n 12*V( a
Hn(a) = —,-^f. (2.5)
r / n - a
Now the function R' (v) is called the Elliptic Kernel of Marcel Riesz. Definition 2.3. Let x — (x1,x2, ..., xn) be a point of the Cn and let
W — xx + + ••• + xp i(xp+i + x^+2 + + xp+q), (2-6)
where i — \J—\ and p + q — n is the dimension of Rn. Define the function
a — n
W 2
Sa(w) = WT^ ' (2-7
Hn(a)
where a is any complex number and Hn(a) is defined as the formula (2.5). Definition 2.4. Define the function
a — n
z 2
Ta (z) — -—-, (2.8)
Hn(a)
where
z — xi + x 2 + ... + xp + i(xp+1 + xp+2 + '" + xp+q) (2*9)
and i — \J —1, p + q — n and Hn(a) is defined as (2.5).
Lemma 2.1. The convolution RHk (u)*(—1)k R'k (v) is an elementary solution of the operator remove off ♦ where ♦ is defined by (1.2) and RHk (u) and R'k(v) are defined by (2.2) and (2.4) respectively with a — 2k.
Proof. The elementary solution of ♦ is the solution of the equation ♦ K(x) — 5 where 5 is the Dirac-delta distribution, K(x) is an unknown and x £ Rn. Now we need to prove that
K(x) — RHk(u) * (—l)kR2k(v).
To prove this , see ([4], p. 33).
Lemma 2.2. (i) The function K(x) — S2(w) is the solution of the equation L1K(x) — 0 where L1 is defined by (1.3) and S2(w) is defined by (2.7) with a — 2.
(ii) The function K(x) — (—1)k(—i) 2 S2k(w) is an elementary solution of the operator L'k, where L'k is the operator iterated k times defined by (1.3) and S2k (w) is defined by (2.7) with a — 2k.
p d2 p+q d2 proof. « Now L1 — £ ^ + i £ ^.
i=1 i j=p+1 j We need to show that L1S2(w) — 0. Now if a is real, we have for 1 < r < p
r\ r\ / a — n\, .. a — n —2 a — n —2
d d w 2 \ (a — n) w 2 w 2
-Sa(w) — -7— u f s —-~--u , s 2xr — (a — n)xr
dxr dxr \ Hn(a) I 2 Hn(a) Hn(a)
r\2 a — n— 2
d2 ^ , s , ,w 2 a — n, a—n—4 2
Sa(w) = (a — n) ^ (a — n — 2)w 2 X2
dx2 H„(a) H„(a) r'
Thus
2
p n2 p V"^ d2 a — n a —n —2 a — n a — n — 4
Sa(w) = Pu f \w 2 + U ( M — n — 2)w 2 .
dx2 Hn(a) Hn(a)
Similarly
p+q o2 / n p+q
d2 q(a — n) a—n—2 a — n a—n—4 2
i / tttSa(w) = TT . , w 2 — 7 (a — n — 2)w 2 > x,-.
^ dx2 ^ Hn(a) Hn(a)v 7 ^ j
j=p+i j nV 7 nV 7 j=p+i
Thus
(p + q), \ a—n—2 (a — n)(a — n — 2) a—n—4 /^ 2 . 2
Lisa(w) = Hn(ay<a—n)w 2 +——nw—w 2 ^x2 — !5>22
\i=1 j=p+1
(\ / \ / \ a — n — 2
a — n) a—n—2 (a — n)(a — n — 2) a—n—2 , ,, , ,w 2 /„.„n
= H ( / w^ + --ht^-^^ = (a — 2)(a — n) —-—. (2.10)
Hn(a) —n(a) —n(a)
For a = 2, we have L1S2 = 0. That is K(x) = S2(w) is a solution of the homogeneous equation L1K(x) = 0.
(n) To show that K(x) = (—1)k(—7) 2 S2k(w) is an elelmentary solution of Lk, that is Li(—1)k(—i)2S2k(w) = 5. At first we need to show that Lk(—1)kSa(w) = Sa-2k(w) and S-2fc (w) = (—1)k (i) 2 Ll5.
Now, from (2.10) and (2.5)
a—n—2 , v , v a—n—2
w 2 (a — 2)(a — n)w 2
L1Sa(w) = (a — 2)(a — n)- —
Hn(a) » 2ar(f)
' ' 1 \ / n — a \
2
By direct calculation with the property of Gamma function we obtain
w 2 w 2
LiSUfw) =----TT-— =--;-- = — Sa-2(w).
1 ^ n 0a-2 r() Hn(a - 2) a 21 ;
n2 '2 rr
r( ^^^)
By keeping on operating the operator L1 k-times to the function Sa(w), we obtain
LkSa(w) = (-1)kSa-2k (w)
or
ii
Li (—1)1 Sa(w) = Sa-2fc (w). (2.11)
Then we show that S—2k = (—1)k(i)2 5.
Now
S-2k (w) = lim Sa(w) = lim
a^—2k a^—2k
a —n
w 2
Hn(a)
lim
a^-2k
a —n
w 2
v i n — a
■ n 2 ■ lim 2 ar
a^-2k
lim |T(a )! a^-2k V 2
a^-2kl 2 J
(2.12)
a—n—2
a—n—2
Now consider liinjw 2 ]. We have w = xf +x2 + ...+xp — ¿(x^+i+xp+2 + ). By changing
the variables, let xi = yf, x2 = y2, • ••, xp = yp and xp+i = —+= , xp+2 = —=, ...,xp+9 = -p+9
—i y—i V —
Thus w = y2 + + ... + yp + + ... + , where y^i = 1, 2,..., n) is real and p + q = n. Let r2
w
y2 + yi + ••• + y« and consider the distribution wA, where A is a complex parameter. Since < wA,Q >= f wAQ(x)dx, where Q(x) is the element of the space D of the
Q
infinitely differentiable functions with compact supports and x £ Rn, dx = dxidx2...dxn. Thus
A r\ I 2A d(Xi
< wA,Q >= / r2
Rn
d (yi ,y2,...,yn)
Qdyidy2...dy„
(—i)
- , r2AQdyidy2...dy„
< r2A,Q > .
(—i)
2
Thus
res <wA,Q>= —— _ A=—n (—i)2 A=
res < r2A, Q >
— n 2
1 2nn -q—^ < ^(x),Q >
(—i)2r(n) ( ),Q
or
A 1 2n2 , res w = -—— _ /__^ o(x).
\ _ —n
—n
Now, by Gelfand and Shilov (see [1], p. 271), < wA,Q > hase simple poles at A = —--k
and for k = 0 the residue of r2A at A = —^ is given by res r2A = —y— $(x).
2 A=—2n r( n)
(2.13)
(-i)2 r (2)
Now we find res wA for k is nonnegative integer by, Gelfand and Shilov (see [1], p. 272) we A= -f -k
have
1
wA =
4k (A + 1)(A + 2)...(A + k)(A + n )(A + n + 1)...(A + n + k — 1)
fc„,,A+fc
Lk w
Thus
2
A=r—ns-k wA = A=—n LiwA' 4k (A + 1)...(A + k)(A + n )...(A + n + k — 1)
2
A= —n-k
by (2.12) we have
Thus
res w A= —n-k
A
2n 2
(—i)2 4kk!r(n + k)
Lk ¿(x).
lim [w 2 ] = lim wA
a^-2k
—n-k
(2.14)
Now from (2.12), we have
n
, . a — n
lim (a + 2k)w 2 S—2k (w) = -—^n 22k W - + k
lim (a + 2k)r (f) V2
a — n
res w 2
a=—2k — A k~
a^-2k
,-Y 4k r - + k res r (f) V 2 y
a=-2k V2/
1
2
1
Now
-a \ 2(-1)k
res r . . ,, a=-2k \2J k!
Thus by (2.14), we obtain
(-1)k 2n2n-r k!4kr (f + k) k
S-2k(w) = --V-——, v 2 , ' Lk5(x) =
v ; (-i)2 2 ■ 4kk!r (f + k) 1 v ;
= Lk 5(x) = (-1)k (i)2 L1 5(x). (-i) 2
Thus
So(w) = (i) 2 5(x). (2.15)
From (2.11) and (2.15), we obtain
Ll (-1)k S2k (w) = S2k-2k (w) = So(w) = (i) 2 5(x)
or
Lk (-1)k (-i)2 S2k (w) = 5.
It follows that K(x) = (—1)k(-i)2S2k(w) is an elementary solution of the operator Lk. Similary K(x) = (—1)k(i)2T2k(z) is an elementary solution of the operator L^k where z is defined by (2.9) and T2k is defined by (2.8) with a = 2k.
3. Main results
Theorem 3.1. Given the equation
ek k (x) = 5 (3.1)
where ©k is the operator iterated k-times defined by (1.1), 5 is the Dirac-delta distribution, x = (x1,x2,...,xn) £ Rf and k is a nonnegative integer. Then the convolution
K(x) = RHk(u) * (-1)kReek(v) * (-1)k(-i)2S2k(w) * (-1)k(i)2T2k(z) (3.2)
is an elementary solution or the Green function of the equation (3.1) where R^ (u),R2k (v),S2k (w) and T2k (z) are defined by (2.2), (2.4), (2.7) and (2.8) respectively with a = 2k. Proof. By (1.5) the equation (3.1) can be written as
0kK (x) = ♦ Lk Lk K (x) = 5. (3.3)
Since the function R^.(u),R^k(v),S2k(w) and T2k(z) are tempered distributions (see [5], p. 34, Lemma 2.1) and the convolution of functions in (3.2) exists and is a tempered distribution (see [5], p. 35, Lemma 2.2 and [2], pp. 156-159). Now convolving both sides of (3.3) by R^k,(u) * (-1)kR2k(v) * (-1)k(-i)2S2k(w) * (-1)k(i)2T2k(z) we obtain
♦ k [RHk (U) * (-1)kR22k (V)] * Lk [(-1)k(-i) 2 S2k (w)] * Lk [(-1)k(i) 2T2k(z)] * K(x) =
= [Rfk(u) * (-1)kR^k(v) * (-1)k(-i)2S2k(w) * (-1)k(i)2T2k(z)] * 5.
By Lemma 2.1 and Lemma 2.2 (ii), we obtain (3.2) as required, we call the solution K(x) in (3.2) the Green function of the operator ©k we denote the Green function
G(x) = RHk(u) * (-1)kR2k(v) * (-1)k(-i)2S2k(w) * (-1)k(i)2T2k(z). (3.4)
Theorem 3.2. Given the equation
©k K (x) = f (x) (3.5)
where ©k is defined by (1.1) and f (x) is a generalized function, then K(x) = G(x) * f (x) is a weak solution for (3.5) where G(x) is a Green function of ©k defined by (3.4). Proof. Convolving both sides of (3.5) by G(x) defined by (3.4) we obtain
G(x) * ©kK(x) = G(x) * f (x)
©kG(x) * K(x) = G(x) * f (x). 6 * K(x) = G(x) * f (x) K(x) = G(x) * f(x)
or
By Theorem 3.1, we have
or
as required.
The author would like to thank the Thailand Research Fund for financial support.
References
[1] Gelfand I. M., Shilov G. E. Generalized Functions. 1, Academic Press. N. Y., 1964.
[2] Donoghue W. F. Distributions and Fourier Transforms. Academic Press, 1969.
[3] Nozaki Y. On Riemann-Liouville integral of ultra-hyperbolic type. Kodai Mathematical Seminar Report, 6(2), 1964, 69-87.
[4] Kananthai A. On the solutions of the n Dimensional Diamond operator. Appl. Math. and Comp., 1997, 88:27-37.
[5] Kananthai A. On the convolution equation related to the Diamond Kernel of Marcel Riesz. J. Comp. Appl. Math., 100, 1998, 33-39.
Received for publication April 26, 2000
