I nternat. J. Math.
Vol.
Math. Sei.
31
(1978) 31-40
COMPARISON THEOREMS FOR ULTRAHYPERBOLIC EQUATIONS
C. C. TRAVIS
Department of Mathematics
University of Tennessee
Knoxville, Tennessee 37916
E. C. YOUNG
Department of Mathematics
Florida State University
Tallahassee, Florida
32306
(Received July 2, 1977)
ABSTRACT.
Sturmian comparison theorems are obtained for solutions of a
class of normal and singular ultrahyperbolic partial differential equations.
In the singular case, solutions are considered which satisfy non-standard
boundary conditions.
KEY WORDS AND PHRASES. Compon thrzm, hyprbolie
singular boundary value pblems, EDP equatn.
AMS(MOS) SUBJECT CLASSIFICATIONS (1970).
34B05, 35L20.
32
C. C. TRAVIS & E. C. YOUNG
I.
Introduction.
The Sturmian comparison theorems for ordinary
differential equations have been extended extensively to partial differential equations of the elliptic type.
For example, see Kuks [13,
Swanson [2], [3], Diaz and McLaughlin [4], and Kreith and Travis [5],
to mention only a few.
By employing Swanson’s technique, Dunninger [63
obtained a comparison theorem for parabolic partial differential equations.
His result was recently generalized by Chan and Young [7] to time-dependent
However, for partial differential
quasilinear differential systems.
equations of the hyperbolic type, very little is known.
In fact, as far
as these authors know, the only comparison results known for hyperbolic
equations were obtained just recently by Kreith [8], Travis [9], [i0],
and Young [ii].
In this paper, we shall present some comparison theorems for the pair
of normal ultrahyperbolic equations
(Aij (X)Ux)x.
i 3
(aij (y)uYi Yj
+
p(y)u
0
(Bij(X)Vx.)x.J
(bij(y)Vy)i Yj
+
q(y)v
0
1
and the pair of singular ultrahyperbolic equations
(Ux.I X i
(v
in a domain
E
n
and
E
m,
x.x.11
D
x
+--u
X.
1
X.
1
)-
+ ___1 v
(b
x.1 x.1
H
G,
(aij (y) Uy)iYj
where
respectively.
ij
H
(y)Vy)i Yj
and
G
p(y)u
+
+
0
q(y)v
0
are bounded regular domains in
For brevity, we let
x
(x I
,xn)
denote
ULTRAHYPERBOLI C EQUATIONS
H
a point in
and
,ym
(Yl
Y
33
Moreover, we adopt
G.
a point in
Einstein’s summation convention concerning repeated indices.
The matrices
and
(Aij), (aij), (Bij)
(bij)
are all assumed to
be symmetric and positive definite with continuously differentiable elements
in their respective domains of definition.
are continuous in
G,
a.’s
1
while the
The coefficients
p
and
q
are real parameters,
i= l,...,n.
We associate with the above equations the following eigenvalue problems
-(aijyi )yj
X
+ pC
G
in
(i.I)
+
r(y)
0
G
on
and
-
(bijyi)yj
3n
where
r
and
+
’Vm)
2.
q
s(y)
G
in
0
G
on
b
- s
are continuous functions on
a
na
(Vl
+
ij
b
Yi j, nb
ij
Cyi
G,
j
being the outward unit normal vector on
The Normal Case.
(AijUx)x
i
+ ru
(aijuYi Yj
0
3G.
We consider the following boundary value problems
+
pu
(2.1)
u
na
and
on
H
x
3G
0
in
D
C. C. TRAVIS & E. C. YOUNG
34
and
{BijVx.)x.- {bijvYi Yj
qv
+
0
in
D
(2.2)
in
v
nb
+ sv
Theorem 2.1.
Let
D, u
{2.3)
on
0
{Bij)
<
0
C 2 (D)
u
H
H
on
G.
DG.
be a solution of {2.1) such that
If
{Aij), {aij)
<
{bij),
p < q, r
_<
s,
where at least one strict inequality holds, then every solution
of {2.2) has a zero in
Proof.
u > 0
C2{D)
v e
D.
Suppose v
is a solution of {2.2).
Let
0
and
be the
0
positive eigenfunctions [12] corresponding to the first eigenvalues
and
0
of (1.1) and {1.2), respectively.
U(x)
V(x)
Since
u
and
v
I
G
I
G
X
u(x’Y)00(Y)dY
v(x’Y)qo(Y)dY"
satisfy (2.1) and (2.2), respectively, and
0
and
are eigenfunctions of (1.1) and (1.2), respectively, it follows by the
divergence theorem that
(Aij Ux.)x.
0
Define
+
X U
0
in
H
U
0
in
3H
0
(2.4)
0
ULTRAHYPERBOLIC EQUATIONS
35
and
(2 5)
(BijVx.) x.
u0V
+
1
H.
in
0
By the variational characterization of the eigenvalues of (i.i) and (1.2),
(2.3) that
it follows from the assumptions
of [4],
G,
V
H,
has a zero in
say at
x
x
Hence, by Theorem i0
0 < 0"
0.
Then, since
Y
Y0
40
> 0
in
the equation
I V(Xo’Y)qo(Y)dY
V(Xo)
0
G
implies that
v
must vanish at some point
is a zero of
v
in
D,
in
G.
Thus
(x0,Y 0)
and the theorem is proved.
We remark that if the solution
u
of (2.1) is required to satisfy
the boundary condition
A..u
ij
on
H
x
G,
x
t. + R(x)u
instead of
B..v
i xi
H
x
G,
where
normal vector on
3.
prob i ems
u
t. + S(x)v
R(x)
H.
the conclusion of Theorem 2.1 remains
0,
of (2.2) satisfying the additional condition
v
valid for solutions
on
0
i j
<_
0
S(x)
and
(tl,...,t)
n
is the outward unit
This is a consequence of Theorem 2 of [3].
The Singular Case.
We consider the singular boundary value
C. C. TRAVIS & E. C. YOUNG
36
C.
O
x.x.
_!u
+
x.
I
1 1
x.
(a i
j
u
na
0
on
1
+ ru
Uy i
y:
H
+
pu
0
in
D
0
in
D
G
x
and
VX’X’I
(b
+--v
X.1 X.1
1
ij
v
+
qv
Yi Yj
(3.2)
v
--+ sv
3n
H
where now
H
If
x
3G
is the domain
{x
E
n
<
(aij)
2
u e C (D)
I0
x i < a., 1
<
<_
i <_ n}.
1
Let
a.
(bij),
p
Theorem 3.1.
(3.3)
H
on
0
b
<- -i, i
1
<
< s
q, r
l,...,n,
in
and assume that
G.
(3.1) which is positive in
is a solution of
satisfies the conditions
fa I
0
x=[u(x’y)[2 dydx
<
G
(3.4)
u(x,y)
then every solution
0
on
x
i
2
v e C (D)
x=lv(x,y) [’2dydx
(3.5)
0
has a zero in
G
D.
ai(l
-< i -< n)
of (3.2) satisfying
<
D
and
ULTRAHYPERBOLIC EQUATIONS
37
Here
denotes the integral
Ian I
al
0
0
G
dx
where as usual
al ...xan [v[2dydx
x
dx
I
n
l...dxn
dy l...dy
dy
and
m.
As in the proof of Theorem 2.1, we let
Proof.
0
and
0
be the
positive, normalized, eigenfunctions of (1.1) and (1.2) corresponding to
XO
the first eigenvalues
and
Then the functions
UO"
U
and
V,
defined as in the proof of Theorem 2.1, satisfy the equations
(XaUx.)x. xO xaU
+
(3.6)
0
xa[u(x) 12dx
U(x)
0
in
H
1
1
on
0
<
x.
a., (i
< i <
1
n)
and
(xaV x
(3.7)
)xi oXaV
+
i
0
in
H
fa xalV(x) 12dx .
<
0
It can be shown, [13], that for
a.
1
<- -1, i
of (3.6) and (3.7) are given by
n
U(x)
il=
x
(1-ai)/2 J
(l_ai) / 2 (.].
x
i
1
,n,
the only solutions
C. C. TRAVIS & E. C. YOUNG
38
and
n
V(x)
X
where
X1
0
x
il
J
i
+...+ X
and
n
xi)
(l-ei)/2
U0
+’’" +
i
Un’
the
X.’s being
the roots of the equation
J(l_ai)/2( Xia i)
i ’s
and the
l,...,n)
0, (i
are arbitrary constants.
function of the first kind of order
Here
X.
such that
denotes the Bessel
X0 < u0"
Hence there exists
p.
From condition (3.3), it follows that
an integer
J (t)
p
V
vanishes along
> 0 in G
a. < a.. Since
3
3
the same argument as in the proof of Theorem 2.1 that V
we conclude by
j
This implies that
/X-./.
x.
the line
u..
<
Let
Theorem 3.2.
(3.3) holds.
0(y)
3
3
If
u
a. >-i, i
n,
1
has a zero in
and assume that condition
is a solution of (3.1) which is positive in
D
and
satisfies the conditions
lim
X.-J
x.u
i
1
u(x,y)
X.
0,
then every solution
x .i 0
x
1
has a zero in
i
0, (i
n)
1
1
Vx.
on
v
x.
I
a., (i
1
l,...,n).
of (3.2) satisfying
0, (i
l,...,n)
D.
The proof of this theorem is similar to that of Theorem 3.1 and is
therefore omitted.
D.
ULTRAHYPERBOLIC EQUATIONS
We conclude this paper with a theorem that is valid for all values
of the parameters
..
Theorem 3.3.
Let
<
.
<
i
holds.
If
u
,
i
n,
1
and assume that (3.3)
(3.1) which is positive in
is a solution of
D
and satisfies
the conditions
u(x,y)
0
lu(0,y)
has a zero in
1
for
<
then every solution
]v(O,y)[
x.
on
v
<
a., (i
i
i
y
39
n)
e G
of (3.2) satisfying
,y
D.
REFERENCES
Sturm’s Theorem and Oscillation of Solutions of Strongly
Elliptic Systems, Kokl. Aka. Nauk. SSSR 142 (1962) 32-35.
i.
Kuks, L. M.
2.
Swanson, C. A. Comparison and Oscillation Theory of Linear Differential Equa.tlo.n.s, Academic Press, New York, 1968.
3.
Swanson, C. A.
4.
Diaz, J. B. and J. R. McLaughlin. Sturm Separation and Comparison
Theorems for Ordinary and Partial Differential Equations, Atti.
Accad. Naz. Lincei Mem. CI. Sci. Fis. Mat. Nat. 9 (1969) 135-194.
5.
Kreith, K. and C. C. Travis. Oscillation Criteria for Selfadjoint
Elliptic Equations, Pacific J. Math. 4__1 (1972) 743-753.
6.
Dunninger, D. R. Sturmian Theorems for Parabolic Inequalities, Rend.
Accad. Sci. Fis. Mat. Napoli 36 (1969) 406-410.
7.
Chan, C. Y. and E. C. Young.
A Generalization of Sturm’s Comparison Theorem, J.
Math. Anal. Appl. 15 (1966) 512-519.
Unboundedness of Solutions and Comparison
Theorems for Time-Dependent Quasilinear Differential Matrix Inequalities, J. Diff. Eqs. 14 (1973) 195-201.
C. C. TRAVIS & E. C. YOUNG
40
8.
Krieth, K. Sturmian Theorems for Hyperbolic Equations, Proc. Amer.
Math. Soc. 22 (1969) 277-281.
9.
Travis, C. C. Comparison and Oscillation Theorems for Hyperbolic
Equations, Utilitas Mat. 6 (1974) 139-151.
Sturmian Theorems for a Class of Singular Hyperbolic
Equations, to appear.
i0.
Travis, C. C.
ii.
Young, E. C.
12.
Krasnosel’skii, M. A. Positive Solutions of Operator Equations,
Fitzmatgiz, Moscow, 1962; English Transl., Noordhoff, Gronlngen,
Comparison and Oscillation Theorems for Singular
Hyperbolic Equations, Accademia Nazlonale Del Lincel, LIX (1975)
383-390.
1964.
13.
Travis, C. C. and E. C. Young. Uniqueness of Solutions to Singular
Boundary Value Problems, SlAM J. Math. Anal. 8 (1977) 111-117.