THEOREMS COMPARISON

I nternat. J. Math. Math. Sei.

Vol. (1978) 31-40

³¹

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.

I. Introduction. The Sturmian comparison theorems for ordinary differential equations have been extended extensively to partial differ- ential 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 quasilinear differential systems.

However,

for partial differential

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. (aij ^(y)u

⁺

^p(y)u

⁰

i 3

Yi Yj

(Bij(X)Vx.)x. (bij(y)Vy)

⁺

^q(y)v

⁰

1 J ⁱ

Yj

and the pair of singular ultrahyperbolic equations

x

(aij

+--u

)- (y) Uy) (Ux.

^X X. X.

I i 1 1

iYj

(v _x.x.1

⁺ ___1 v (b

(y)Vy)

1

x.1 x.1

^ij i

Yj

+

p(y)u

0

+ q(y)v 0

in a domain D H

G,

where H and G are bounded regular domains in Eⁿ and E

m,

respectively. For brevity, we let x

(x

I ,x

n)

^denote

EQUATIONS

33 a point in

H

and

Y (Yl ,ym

^{a point in G.}

^Moreover,

^we

^adopt

Einstein’s summation convention concerning

repeated

indices.

The matrices

(Aij), (aij), (Bij)

^and

(bij)

are all assumed to

be symmetric and positive definite with continuously differentiable elements in their respective domains of definition. The coefficients

p

and q are continuous in G, while the

a.’s

are real parameters,

1

i= l,...,n.

We associate with the above equations the following eigenvalue

problems

(i.I)

and

-(aijyi )yj

^{+ pC}

^X

^{in G}

+

r(y)

0 on G

(bijyi)y

_j ⁺

^q

^{in G}

-

^{3nb +}

^s(y)

^{0 on G}

where r and s are continuous functions on G, and

---

^a

^{j, --}

^b

^Cyi

na

^ij

Yi ⁿ

b ij j

(Vl ’Vm)

^{being the outward} unit normal vector on 3G.

2. The Normal Case. We consider the following boundary value problems

(2.1)

(AijUx)x (aiju

i

Yi Yj

+ pu 0 in D

u

+ ru 0 on H ^x 3G

n

a

and

(2.2)

{BijVx.)x.- {bijv _{Yi Yj}

⁺

^qv

^{0 in D}

v

_{+ sv 0 on}

H DG.

n

_b

C²

Theorem 2.1. Let u

(D)

be a solution of

{2.1)

such that u > 0 in

D,

u 0 on

H

G. If

{2.3) {Bij)

^<

{Aij), {aij)

^<

{bij), ^p

^<

^q,

^{r _< s,}

where at least one strict inequality holds, then

every

solution v e

C2{D)

of

{2.2)

has a zero in D.

Proof.

Suppose

v is a solution of

{2.2).

Let

0

^and

0

^{be the}

positive eigenfunctions [12] corresponding to the first eigenvalues X 0 and

0

^of

^(1.1)

^and

^{1.2),

respectively. Define

U(x) I u(x’Y)00(Y)dY

G

V(x) I v(x’Y)qo(Y)dY"

G

Since u and v satisfy

(2.1)

and

(2.2),

respectively, and

0

^and

0

are eigenfunctions of

(1.1)

and

(1.2),

respectively, it follows by the divergence theorem that

(2.4)

(Aij Ux.)x.

^{+ X U}0 ^{0 in}

^H

U 0 in 3H

EQUATIONS

35 and

x. ⁺

u0

^{V 0 in H.}

(2 5) (BijVx.)

1

By

the variational characterization of the eigenvalues of (i.i) and

(1.2),

it follows from the assumptions

(2.3)

that

0

^<

0" ^Hence,

^{by Theorem i0}

of

[4],

V has a zero in

H,

say at x x

0.

^{Then, since}

40

^{> 0 in}

G, the equation

V(Xo) I V(Xo’Y)qo(Y)dY

⁰

G

implies that v must vanish at some point Y

Y0

^{in G. Thus}

(x0,Y 0)

is a zero of v in

D,

and the theorem is

proved.

We remark that if the solution u of

(2.1)

is required to satisfy the boundary condition

A..u t. ⁺

R(x)u

0 ij x

i ^j

on

H

^x

G,

instead of u 0, the conclusion of Theorem 2.1 remains valid for solutions v of

(2.2)

satisfying the additional condition

B..v t. ⁺

S(x)v

0 i x

i

on

H

^x

G,

where

R(x)

^<_

S(x)

and

(tl,...,t)

^{is the} outward unit n

normal vector on H. This is a

consequence

of Theorem 2 of [3].

3. The Singular Case. We consider the singular boundary value probiems

C.

O +

_!u _(a

i

Uy

^{+ pu 0 in D}

x.x.

x. x. j y:

1 1 I 1 i

u

+ ru 0 on

H

^x G

n

a

and

(3.2)

+--v (b ^v

VX’X’I

¹

X.1 X.1

^ij

Yi Yj

+ qv 0 in D

--+

v

sv 0 on H ^x 3G

3nb

where now H is the domain

E^{n <} a., 1 ^<_ i ^<_ n}.

H

{x

I0

^{< x}_{i 1}

Theorem 3.1. Let a.

<-

-i, i l,...,n, and assume that

1

(3.3) (aij)

^<

(bij), ^p

^<

^q,

^{r < s in G.}

C²

If u e

(D)

is a solution of

(3.1)

which is positive in D and satisfies the conditions

(3.4)

fa I x=[u(x’y)[2

0 G

dydx

^<

u(x,y) 0 on x

i

ai(l ^-<

ⁱ

^{-< n)}

C²

then every solution v e

(D)

of

(3.2)

satisfying

(3.5) x=lv(x,y) [’2dydx

^<

0 G has a zero in D.

Here

denotes the integral

al I ^an I

^x

^al

^{I ...x}

^{an [v[2dydx}

0 0 G n

where as usual dx dx

l...dxn

^{and dy dy}

l...dym.

Proof. As in the

proof

^{of Theorem} 2.1, we let

0

^and

0

^{be the}

positive, normalized, eigenfunctions of

(1.1)

and

(1.2)

corresponding ^to the first eigenvalues

XO

^and

UO"

^Then the functions U and

V,

defined as in the

proof

of Theorem 2.1, satisfy the equations

(XaUx.)x.

⁺

_xO ^xaU

^{0 in H}

1 1

(3.6) xa[u(x) 12dx

^<

0

U(x)

0 on

x.

a., (i ^< i ^<

n)

1

and

(3.7)

(xaV

^x_i

)xi

⁺

oXaV

^{0 in H}

fa

⁰

^{xalV(x) 12dx}

^<

^.

It can be shown, [13], that for a.

<-

-1, i 1 ,n, the only solutions

1

of

(3.6)

and

(3.7)

are given by n

(1-ai)/2

U(x) il=

^{x J}

_{(l_ai)}

_/2

(.].

^xi

and

n

V(x) il

^{xi J}

_(l-ei)/2 ^xi)

X0 +...+ X and

U0 i

^{+’’" +}

Un’

where

X1

ⁿ

the roots of the equation

the

X.’s

being

J(l_ai)/2( ^{Xia i)}

^{0, (i}

^l,...,n)

’s

are arbitrary constants. Here J

(t)

denotes the Bessel and the

i p

function of the first kind of order

p.

From condition

(3.3),

it follows that

X0

^<

u0"

Hence there exists an integer j such that X. ^<

u..

^This implies that V vanishes along the line x.

/X-./.

_{3 3} a.₃ ^<

a..

₃ Since

0(y)

^{> 0 in G we conclude by}

the same argument as in the proof of Theorem 2.1 that V has a zero in D.

Theorem 3.2. Let a. >-i, i 1 n, and assume that condition

(3.3)

holds. If u is a solution of

(3.1)

which is positive in

D

and satisfies the conditions

lim x.u

X.-J i X.

1 1

0, (i 1

n)

u(x,y) 0, on x. a., (i

l,...,n).

I 1

then every solution v of

(3.2)

satisfying

x .i 0

^xi

Vx.

1

0, (i l,...,n)

has a zero in D.

The proof of this theorem is similar to that of Theorem 3.1 and is therefore omitted.

ULTRAHYPERBOLIC EQUATIONS

39 We conclude this

paper

with a theorem that is valid for all values of the parameters

..

Theorem 3.3. Let <

.

_i ^<

^,

^{i 1 n, and assume that}

^(3.3)

holds. If u is a solution of

(3.1)

which is positive in D and satisfies the conditions

u(x,y) 0 on x. a., (i i n)

1 i

lu(0,y)

^{< for} y ^e G

then

every

solution v of

(3.2)

satisfying

]v(O,y)[

^<

,

y has a zero in D.

REFERENCES

i. Kuks, L. M.

Sturm’s

Theorem and Oscillation of Solutions of Strongly Elliptic Systems, Kokl. Aka. Nauk. SSSR 142

(1962)

32-35.

2.

Swanson,

C. A. Comparison and Oscillation

Theory

of Linear Differen- tial

Equa.tlo.n.s,

^Academic

Press,

New

York,

1968.

3. Swanson, C. A. A Generalization of

Sturm’s

Comparison Theorem, J.

Math. Anal. Appl. 15

(1966)

512-519.

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.

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4__1 ⁽¹⁹⁷²⁾

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Accad. Sci. Fis. Mat.

Napoli

36

(1969)

406-410.

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Math. Soc. 22

(1969)

277-281.

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(1974)

139-151.

i0. Travis, C. C. Sturmian Theorems for a Class of Singular Hyperbolic Equations, to appear.

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(1975)

383-390.

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Krasnosel’skii,

M. A. Positive Solutions of

Operator Equations,

Fitzmatgiz,

Moscow,

1962; English Transl., Noordhoff, Gronlngen, 1964.

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Young.

Uniqueness of Solutions to Singular Boundary Value Problems, SlAM J. Math. Anal. 8

(1977)

111-117.