Internat. J. Math. & Math. Sci.
VOL. 12 NO 2 (I989) 385-390
385
KELVIN PRINCIPLE FOR A CLASS OF SINGULAR EQUATIONS
ABDULLAH ALTIN
Faculty of Science
University of Ankara
Besevler, -Ankara, Turkey
EUTIQUIO C.
YOUNG
Department of Mathematics
Florida State University
Tallahassee, Florida 32306
(Received September 7, 1988)
ABSTRACT.
The classical Kelvin principle concerns invariance of solutions
of
the
By employing a hyperbolic-
Laplace equation with respect to inversion in a sphere.
polar coordinate system, the principle is extended to cover a class
of singular
equations, which include the ultrahyperbolic equation.
Kelvin principle, Laplace equation, ultrahyperbolic equation,
KEY WORDS AND PHRASES.
Lorentzian distance.
1980 AMS SUBJECT CLASSIFICATION CODES.
i.
35LI0, 35C99.
INTRODUCTION.
As is well known the classical Kelvin principle introduced in 1847 (Thomson [i])
concerns solutions of the Laplace equation. For solutions of some class of elliptic
differential equations and their iterated forths in n independent variables, n z 2
the extension of Kelvin principle is usually
proved
using
rectangular coordinates
[2], Germain and Bader [3], Huber [4], Weinstein [5] ).
(Diaz and Martin
In
a generalization of Kelvin principle was established by Weinstein [5] for
equation
2
n
r.
i=l
using polar coordinates.
k
u +
x.1
i
x.
1
u
k.l
0,
x.1
Following Weinstein method we shall give in this
Kelvin principle for solutions
of
1960
the
the
class of
<
paper a new formulation
singular
partial
of
differential
equations
n
L(u)
where
.
(i
E
i=l
(2,,v_ + ui u)
2
x.1
i S n) and
1
x.l
B.
m
Z
x.1
j=l
2
8j 8u
( +---- +yj yj yj r2
s m)
(i
u
i
are
real
P(u)
0
parameters, r is the
(i.])
],orentzia
metric defined by
2
r
n
F.
i=l
2
x.
1
m
I
j=l
2
y_
(1.2)
A. ALTIN AND E.C. YOUNG
386
and
P
is a general linear operator of arbitrary
vanishing for u
Zp
L
The domain of the operator
of class
C2(D)nCq(),
n,
denote points in R
u
2.
in R
n+m
order q in the
variables Zl, z 2,
0.
where x
m
R and R
is the set of all real valued functions u(x,y,z)
(Xl,.
p,
x
n
),
,ym
p
D x R is a regularity domain of
(Yl
respectively, and
(Zl,...,z)
and z
p
x R
HYPERBOLIC-POLAR COORDINATE SYSTEM FOR EQUATION (i. i).
First let us consider the n+m-dimensional Laplace operator
2
n
Au
2
3u
7.
+
x.
i=l
2
m
u
7
j=l
1
(2.1)
2
Xn+j
and introduce the polar coordinates
x
x
x
x
x
x
r
I
r cos8 cos8 ...cos8
cos8 cos8
...cos8
sin0
1
2
m+l
m-i
m
m+n-2
m+n-i
2
m
3
cosOlcos82 .cOSSm_ 1 cos0 m cos8 m+l ...sin8 m+n-2
r cos0 cos0 ...cos0
cose sin0
1
2
m-i
m
m+l
n
r cosO
n+l
e.
i
cose 2 ...cos0
r
sine
m
sin01
n for
N
m-i
(2.2)
r cos8 sine
2
i
n+m-i
Xn+m
where 0
cosolcoso2...cOSSm_iCOSSmCOSOm+ 1...cosSm+n_2cosom+n_ 1
l,
r
n+m-2, 0 N 8
[x
2
I
n+m-i
+ x2
n+m
+
2
N
and
)%
(2.3)
Under this change of variables, the polar form of the Laplace operator is given by
n+m
Au
2
2
u
---Z
---Z
D u
1
where the operator
I
u +
n+m-I
+
Dr
r
Dr
i=l Dx.
2
r
depends only on the variables
1
(u)
(2.4)
@I,
n+m-i
is the
and r
Euclidean distance given by (2.3).
Now in (2.1) and (2.2) let
i
/-i and
cos(iCj)
8m+.] @.] for
Cj and sin(iOj
i
let
ch
iy4
Xn+
i sh
.,
and
n-l.
e icj
Since
for
(D/DXn+ j)
i
2
m
with
2
u
(D/Dyj)
u,
the operator (2.1) reduces to the ultrahy-
perbolic operator
2
[h=
7.
i:l
u
---T
x.
2
Z Du2
j:l
On the other hand, the polar coordinate system (2.2) takes the form
(2.5)
KELVIN PRINCIPLE FOR A CLASS OF SINGULAR EQUATIONS
x
x
x
x
I
2
3
n
Yl
Y2
r
chich2"’’ChCm-lChCmCSl’’’cSn-2CS@n-i
r
chich2...Chm_iChmCOSl...cOSn_2Sin@n_l
r
chich2...Ch#m_iChmCOS@l...sinn_ 2
r
chich 2.. "ChCm-lChmsinl
r
chich2...Chm_iShm
chich#2...ShCm_ 1
r
r
Ym_l
(2.6)
chish 2
r she
Ym
38F
I
where r is the Lorentzian distance given by (1.2).
We
refer
this
to
as
"polar-
hyperbolic transformation". In the polar-hyperbolic coordinate system, the operator
(2.5) assumes the form
n
Z
[]u
i=l
where
2
2
2
m
Du
---,/-
Dx.I
2
Du
E
j=l
Du +
[
2
r
Dr
yj
u +
n+m-i
depends only on the variables
Dr
1
+ -u
r
u
2
+
Au= u
rr
+
r is given in (1.2).
2, 3 are given by
1
--_Uo0
r
u
r
r
(2.7)
’n-i and
i’’’" ’m’l
For example, the polar forms of Au for n
Au
1
--z2(u)
r
(u
+
r
OO
0
+ cos
sin O
1
Uo
sin2 0
u
The corresponding forms for the hyperbolic equations are given by
[
rl
[h
where
u
xx
+ u
yy
r
--Ur
r
2
U
r
2
+
zz
1
--U
r
Urr
u
1
+
u
1
r
r
x
2
+ Y
2
Uee
--z
+
2
sh0
1
--Ucho O
ch2o
and
r
x
u
ch0cos, y
r
chOsin
z
r sh0
3.
A FUNDAMENTAL THEOREM AND MAIN RESULT.
In [5]
using his main three recursion formulas, Weinstein gave the followir,
theorem which will be used to establish our main result.
THEOREM I.
Let v
v(r,l,...,n_l)
2
D v
---2
D r
+
k Dv
1
r Dr
r
where k is a real or complex number and
for v
satisfy the differential equation
2
(v)
(3.1)
is a linear, differential operator vanishing
l-k
Then
w satisfies the same
0 which is independent of the variable r.
A. ALTIN AND E.C. YOUNG
388
equation (3.1) in the variables
w(p,
v(i/P,l
n_l
1
P’
where
i’’’’’ n-i
i/r
p
and
n_l
Using Theorem 1 we can now establish an extension of Kelvin principle to ultrahyperbolic equations
THEOREM 2.
i. 1 ).
u(x l,...,xn,yl,...,ym
Let u
,
Then
w
r
-I
Xl
ur
x
Yl
r
r
’zl’’’’’zp)
Ym
-,
r
z
be a solution of the equation
,z
I
(3.2)
P
is also a solution of the same equation (i.i), where
n
I
n + m
2 +
m
Z u. + Z 8.
i
i=l
j=l
(3.3)
and r is the Lorentzian distance defined by (1.2).
Let us consider the polar-hyperbolic transformation (2.6) which can be
written in the following form
PROOF.
x
r
i
functions of
i
l,...,n
(3.#)
gj(,),
fi(,), gj(,) or without
’m’l"" "’n-i
r
yj
where the notations
i
1
m
subscripts f(,), g(,) denote
We note that from (3.#) we have
x.
r x.
Yk Yk
Dr
3yj
where
6jk
+
Dr
7.
m
+
yj
is the Kronecker delta.
__r
xj
x.]
Dr
x
i=l
(i <
7.
i=l
+
7.
i=l
Yk
6jk
x]
6jk
i YJ
From (1.2) we have
_< n)
and
YJ
__Dr
y.
and we may express the partial derivatives
(i _< j <_ m)
r
8i/8xj, 8i/Sx4j
and
8i/8yj
the denominator of which is the Jacobian of the transformation (2.6).
as a quotient
It can
shown that
(x
I,..
3(r, 1
,Xn,Y 1
m,i
Ym
n+m- I
r
h ,)
Cn_l
The numerator of this quotient contains obviously only a factor r
n+m-2
hence
be
KELVIN PRINCIPLE FOR A CLASS OF SINGULAR EQUATIONS
On the other hand, since
x.
Du
Dx.
m
Du
Dr
r
Du
Yj
Du
Dyj
r
Dr
m
1
1 Du
1 Du
xj Dxj
r
z
+
Dr
yj Dyj
and
2
l(U),
r
1 Du
+
r Dr
1 Du
i
Fji(
n-i
Du
i=l
D.
#
1 < j < n
2(u)’
r
1
Since
+
n
n
+ m
r
l-(n+m-i +
w
1 +
m
r
_k
x
the substitution P
2
xi/r
(i)
+
Du
7.
j=l
8j)
x
i "’m’@l "’’’n-i
(i.i), then
our equation (i.i)
1
+
(u)
(3.6)
0
r
by Theorem 1
k
,..,- ,-Yl ,..,-
I
u(r
Y
r
r
r
,Zl,..,zP
Here we note that, since
n
2
and
I
Xi=le i + 7.j= i 8j
satisfies the equation (i.i).
by
m
X e.
i:l
n
-< m
depend only on the ariables
becomes
2
(3.5)
-<
If we substitute the expressions (2.7) and (3.5) into
D u
D r
3x
r i=l D# ix
i
we see that
where the operators
_Du
r i=l
+
H..(,)
(,),
=--G
--Fji(,),
389
(xi/r 2)
i=l
Z
2
m
2
1
(yj/r2)
:j=l
7.
r
i/r in the solution u means replacing the variables x_. and y_.
yj/r 2,
]
respectively.
We note that, since the r defined by (1.2) is not real for
the solutions of (i.i) is valid only in the domain D x
n
D
x D
n
m
D
{(x,y)
n+m.
x e D
n
y
D
m
7.
i=l
2
x.
1
,
7.n.1=i X.l2
where
m
>
<
Z
i-=-
y
2
J
is a hyperconoidal domain in R
Here D and D are the spherical domains centered
m
n
n
p is the regularity domain of
R
at the origins of R and R
respectively, and
m,
u with respect to the variable z.
A. ALTIN AND E.C. YOUNG
390
(ii)
In equation (i.i) if we have addition instead of subtraction of the two
summations, then Theorem 2 remains valid, where
r
n
2
m
2
2
Z x. + 7.
1
i=l
j=l
yj
This includes Weinstein’s [5] and Altin’s [2] results.
7u where Y= const, the formula (3.2) gives the
In the special case Pu
result obtained in [2]
(iii)
(iv)
If we multiply both sides of the equation (i.i) by -I, we get the equation
2
m
Z
j:l
-n(u)
where
r
(
u
+
2
yj
zmj=l Yj2 Enj=l x.]2
-r
2
2
n
j u
yj yj
j u
u
Z (-----2 +
x. x.
j:l
This shows that if
Ym
Xl
-r -r -r -r
0
r
U(Xl,..,Xn,Yl,..,Ym,Z l,..,zp)
is a solution of the equation (i.i), then by Theorem 2
yi
x
n
-X
r U(
1
Wl
(u)
+
x.
Zp)
zl
where X is given by (3.3) It is clear
is also a solution of the same equation (i.i),
2
2
that this solution is valid in a different domain where r I > 0 that is, r < 0
(v) It is clear that by a simple linear transformation, Theorem 2 also holds
for a more general equation of the form
2
n
a2(
i
Z
where a
n
o
i
bi,e i
o
o
"(nl’’’’’m
w
8
e.
u
i
1
+
m
u
Z
?
8
are real parameters (a
are fixed points in
r
-X
2
b.32(
u(
alr
where
2
anr
nz (i-i). 2
o
r
2
i=l
a.
n
2
’ 0 u)
+
q
0, b.
0),
1
Dn and Dm,
n-n
i-
8.
u
nm-nm2
blr
m
Z
j=l
2
bmr
jb.
0
r
o (,..., on
respectively
nl-n
(u)
+
and
Here
’Zl’’’’z P
o
J)2
]
REFERENCES
I.
THOMSON, W. Extraits de deus Lettres Addressees a M. Liouville, J. Math. Pures
AppI. 12(1847), 256.
2.
ALTIN, A.
3.
GERMAIN, P. and BADER, R.
Some Expansion Formulas for a Class of Singular Partial Differential
Equations, Proc. Amer. Math. Soc. 85(1982), 42-46.
Sur le Problems de Tricomi, Rend. Circ. Mat. Palermo
2(1953), 53-70.
4.
HUBER, A.
5.
WEINSTEIN, A.
Some Results on Generalized Axially Symmetric Potential Theory, Proc.
Conf. Partial Diff. Eqs., University of Maryland, 1956, 147-155.
359-365.
6.
On a Singular Differential Operator, Ann. Mat. Pura Appl.
49(1960),
DIAZ, J. B. and MARTIN, M. H. Riemann’s Method and the Problem of Cauchy, II.
The Wave Equation in n Dimension, Proc. Amer. Math. Soc. 3(1952), 476-483.