A nonexistence result for travelling front solutions in a cylindrical domain

Nonlinear Analysis, Theory, Methods & Applications, Vol. 26, No. 3, pp. 525-538, 1996 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/96 $9.50 + .00

Pergamon 0362-546X(94)00296-7 A NONEXISTENCE

RESULT FOR TRAVELLING FRONT IN A CYLINDRICAL DOMAIN

SOLUTIONS

SONIA OMRANI Laboratoire d'Analyse Numerique, tour 55-65, Universit6 Pierre et Marie Curie, 4 place Jussieu, 75252 Paris, Cedex 05, France (Received II March 1994; received for publication 19 October 1994) Key words and phrases: Nonexistence, travelling front, removable singularities. 1. INTRODUCTION In this work, we construct a counter-example which shows that in some cases, there m a y not exist travelling fronts for a reaction-diffusion equation of the type -Au

au + v(Y)~xl = g(u)

(1.1)

in a infinite cylindrical domain E = I(xl, Y) e ~ x o91, where o9 is a smooth and bounded domain in R z, and with appropriate boundary conditions. This type of equation arises in some models in combustion theory (see for instance [1]), it describes the propagation of a curved premixed flame in the infinite tube E. The unknown u is a normalized temperature, v(y) is a velocity distribution and g(u) represents a reaction term. More precisely, the problem we consider is the following -Au

Ou + cc~(y)-~x~ = g(u)

au --=0 av u ( - o o , y ) = o,

in E

on0E = ~x0o9 u(+oo,y) = 1

(1.2) (1.3)

for y e o9.

(1.4)

In (1.3), v is the outward unit normal on 0E, c~ is a given function in C°(69, ~+) and g is a Lipshitz continuous function from [0, 1] into ~+ satisfying some conditions depending on the model under discussion. The unknowns are the parameter c which represents a propagation velocity and the function u which represents a "travelling f r o n t " of the heat equation c~(y) ~

- Av := g(v).

Berestycki and Larrouturou [1] establish the existence of solution (u, c) for the problem (1.2)-(1.4) under the following hypotheses 3 0 e (0, 1), g > 0

g -= 0

on(0, 1)

and 525

on [0, 0]; g(l) = 0.

(1.5) (1.6)

526

S. OMRANI

Moreover, they assume c~ > 0

on (5.

(1.7)

The hypotheses (1.5), (1.6) correspond to the assumption of flame propagation with an ignition temperature 0. Their proof is based on a topological degree method to show the existence of a solution (u~, ca) in a bounded domain IEa = l - a , a[ x co. Then using an auxiliary normalization condition on u , , they obtain a solution for the problem in I2 by passing to the limit as a --* +oo. The p r o o f here uses strongly the assumption (1.7). In the same way, when (1.5), (1.6) are replaced by the hypotheses describing the (Z.F.K.) model, g(0) = g ( 1 ) = 0

and

g> 0

on(0,1)

(1.8)

Berestycki and Nirenberg [2], prove (using (1.7)) the existence of c* > 0 such that the problem (1.2)-(1.4) has a solution (u, c) if and only if c _~ c*. To do this they truncate g near the origin approximating it by functions go satisfying (1.5), (1.6) with 0 tending to zero. Then they show that the Co corresponding converge to c* as 0 tends to zero. Finally, they show that for every c _> c* the problem admits solutions while solutions do not exist for c < c*. Here, we are concerned with the study of this existence problem when the condition (1.7) is relaxed to c~ _> 0

on d)

and

c~ ~e 0.

(1.9)

In fact, in this paper we construct a counter-example which shows that the two existence results mentioned above no longer hold for some particular choice of domains co when o~is allowed to vanish in a portion of co.

Remark. When the term co4y) is replaced by c - ~(y) in the equation (1.2), the assumption (1.7) is no more needed to get the existence result [3]. In Section 2, we construct the counter-example, state the main results and describe the main steps in the proof. Lastly, in Sections 3 and 4 we prove some technical results used in Section 2. In the p r o o f of the main result we need some results on removable singularities of solutions of elliptic equations. For this purpose, we present and generalize in Section 4 some results of Serrin [4, 5] dealing with this subject. These results may be of independent interest. The results presented here have been announced in [6]. 2. CONSTRUCTION OF THE COUNTER-EXAMPLE Let eo be a small positive number, for each e e (0, ao), we define ¢0~ as a smooth and bounded domain in ~2 composed by two discs DL = B(0, 1) and D 2 a translate of D 1 smoothly connected by a strip of width e (Fig. 1). As e goes to zero, co~ decreases and we have lim co~ = N co~ = D1 U D 2 1.3 L, -~0 ~>0 where L is a straight segment of ~z. Let A be the point of R z defined by [A] = / ) 1 7 / L = L = O (0/31 - aco~).

A nonexistence result

527

@ 0 4

Fig. 1.

We define a function o~ e C°(tb~, E+) such that c~ = 0 o n D 1 , ote = 1 on D2 and 0 _ a~ ___ 1 in the rest o f the d o m a i n . The main result o f this paper is the following theorem. THEOREM 2.1. The p r o b l e m (1.2)-(1.4) with a = c~ and g a Lipschitz function satisfying either the hypotheses (1.5), (1.6) or the hypothesis (1.8) has no solution in R × in,, for e small.

Remark. It is an open question to know if o~(y) > 0 in ~ (possibly vanishing on 0o9) is enough to get the existence of a solution. F u r t h e r m o r e , we conjecture the existence o f solutions o f the p r o b l e m (1.2)-(1.4), with the hypothesis (1.9), when 09 is a convex d o m a i n . This question is open even in dimension N - 1 when ~o is an interval. The p r o o f of t h e o r e m 2.1 is obtained by a contradiction a r g u m e n t . We suppose the existence of solution (u~, c~) such that

cgue -Au~ + c ~ ( y ) ~ x l = g(u~) 3u~ Ov

0

u~(-oo, .) = 0;

in ~ x oJ~

on fir x 0~o~ u~(+oo, .) = 1.

(2.1)

(2.2) (2.3)

Since the p r o b l e m is translation invariant, we can add to equations (2.1)-(2.3) the condition u~(0, P ) = r/,

(2.4)

where P is a point fixed in D1 and q a n u m b e r fixed in (0, 1) or in (0, 1) for conditions (1.5), (1.6). This additional condition (2.4) is a normalization condition which refers to the translation invariance o f the solutions o f (1.2)-(1.4) along the Xl-direction. Indeed, as (ue, ce) is solution o f the p r o b l e m (1.2)-(1.4) then (u,(x~ + a, y), e,) is also solution of the same p r o b l e m for any a~. By setting g -= 0 on ] - ~ ,

O[ U ]1, + ~ [ , we get f r o m the m a x i m u m principle,

0<<_ u e < 1

in ~?xo9 e.

Let us also observe that from the classical elliptic estimates [7], we have, u~ ~ WlZof(~ x o9~)

for all p ~ (1, +oo).

(2.5)

528

S. OMRAN1

The p r o o f is divided into several steps: the first step concerns the properties of convergence when e ~ 0, this is the aim of proposition 2.2. The second step deals with removing the singularity of the limit function u, which will be clone using proposition 2.4 and theorem 2.5. Then in the next step we prove that I~×D, Au = 0. Finally, we conclude by reaching a contradiction. PROPOSITION 2.2. U~ converges in W~2~p (p > 3) to a function u satisfying U E Wl2ocP(~ × D1) O CI([R ) < / ~ I \ I A I )

- A u = g(u) Ou Ov

--

= 0

0<_

in [~ x Dl

on ~ ×

u_<

for all p > 3

1

(2.6) (2.7)

(OD1L[A})

in~xD1

(2.8)

(2.9)

u(0, P) = r/.

(2.10)

Proof o f proposition 2.2. For any a > 0 and any p > 0 small, we denote a

~)p = ] - a , a[ ~ (DIkBp(A)). To prove proposition 2.2, we use the following lemma. LEMMA 2.3. There exists a positive constant K = K(a, p, p) independent of e such that for any a > 0 and any p > 0 small, the solution u~ satisfies the estimate

NltC.II w2,P(]_a,a[)..(DI\Bo(A))) <~ K

(2.11)

for all p ~ (1, +oo). Then, since the injection W 2 " p ( ~ ) ~ C L ( ~ ) is compact for all p > 3, it follows f r o m the estimate (2.11) that we can extract a subsequence (still noted u~) such that u~ converges to a function u strongly in C~(~-~p) and weakly in WZ'P(ff2~p) for all a > 0 and all p > 0 small. Obviously, the limiting function u satisfies u ~ WIZo~P(E× D 0 (3 Cl(~? :,~ DI\[A})

-•u Ou Ov

--

= g(u)

-- 0

on

in I1~× D l ~ x

(OD1\[A])

and from (2.4), (2.5) we have also 0<_u-
in ~ × D 1

u(0, P) = ~. This completes the p r o o f of proposition 2.2.

for all p > 3

•

A nonexistence result

Proof of lemma

529

2.3. For any a > 0 and p > 0 small, we consider

R. = ] - a , a[ × D~,

f~g = l - a , a[ × (D~\Bp(A)) and

T. = l - a , a[ × (ODI\Bp(A)).

We have f~, CC R,+I U T~+ 1 and u~ satisfies

-Au~ = g(u~) Ou~ = On

-By the local

Lp estimates

0

in R~+ 1

on T~+ 1 for e small.

near the b o u n d a r y ([7, 8]), we have for all p e (1, +co)

Ilu~llw~.,,~,~-<

Cp(llg(uPll~,~°+,> +

Ilu, llL~ko+,p,

where Cp is a constant independent o f e but which in principle depends on p and a. Therefore, since 0 _ u~ < 1 and g is b o u n d e d , we get the estimate (2.11). II Thus, u is constructed with a possible singularity on the line F = ~ × {A] o f the b o u n d a r y . We will show that this singularity is removable. That is, u m a y be defined on F so that the resulting functions is in C~([~ × D~) and so satisfies the N e u m a n n condition on all o f R × 0D~. To this end, we first extend the function u in such a way that it is defined on a connected and open n e i g h b o u r h o o d o f F, let us say the cylinder o f ~73, ~ = ~ × B~/z(A), where B~/2(A) is the disc o f ~2 with center A and radius ½. In Section 3, we prove the following proposition. PROPOSITION 2.4. The function u can be extended to a function fi defined on £2\F and satisfying ~ W~ZcP(92\F) 3

02t~

~

t~l ~ i.j=lE ai

for all p > 3 3 _

Off

+ '=IEb,(x)-~x = g(ft)

in~\F

0 ___ fi <__ 1,

(2.12) (2.13)

where L is a uniformly elliptic operator with coefficients d~.j Lipschitz continuous on 92 and bi b o u n d e d . Then we use a variant o f a theorem o f Serrin [4] on removable singularities o f strong solutions o f elliptic equations in ~" ; n > 2 o f the type Zu

32u Ou ai,j(x)~ + ~ bi(X)~xi + c(x)u

= i,j=t

.

= 0

(2.14)

i=]

with the hypotheses

col~l 2 <_

i,j= 1

ai,j(x)g~gj <- Co ~

v g ~ ~", Co, Co > 0

(2.15)

the ai,j are Lipschitz continuous

(2.16)

the b i and c are b o u n d e d .

(2.17)

We, therefore, derive the following generalization.

530

S. O M R A N I

TrIWOREM 2.5. Let u ~ WlZo~P(D\Q), (p _ n) be a strong solution o f (2.14) in DXQ, where D is open, Q is a c o m p a c t set and the coefficients o f L satisfy the hypotheses (2.15)-(2.17). If Q is a set o f o r d i n a r y capacity zero and u is b o u n d e d . Then u m a y be defined on Q so that the resulting extended function is a strong solution o f (2.14) in all o f D. The p r o o f o f theorem 2.5 is given in Section 4. Now, taking D = ] - a , a [ x B 1 / 2 ( A ) , theorem 2.5 and the relation (2.13)

Q=

[-a,a]xlA}

ft ~ W12o'cP(~.x BI/z(A))

for any a > 0 ,

for all p > 3

in ~ x Bl/2(A ).

Lft = g(ft)

we have using (2.18) (2.19)

F r o m (2.18) we deduce that ft e CI(IR x B~/2(A)) and with (2.8), we get

Oft Ov

Ou - 0 Ov

-

on ~ x OD1.

(2.20)

Lastly, our original function u is o f class C a and satisfies

- A u = g(u) Ou Ov

-

-

=

in ~ x D1

on all o f ~ x 0 D ~ .

0

Let us now turn to the p r o o f o f t h e o r e m 2.1. The function u satisfies

- A u = g(u) Ou = 0 Ov

- -

0_

in II~ x D1 on R x OD~

(2.21) (2.22)

u_< 1

(2.23)

u(0, P ) --: q.

(2.24)

Then we need the following lemma. LEMMA 2.6. The function u satisfies Au = 0.

'

(2.25)

~'xD 1

Proof o f lemma 2.6. F r o m (2.22), we get when applying the G r e e n ' s first identity on the domains R m -- ] - m , m [ x 0 1 , Ou ~xl (m, y) dy -

Au = Rm

D1

Ou ~xl ( - m, y) dy.

(2.26)

DI

Let us show that the two integrals in the right-hand side o f (2.26) converge to zero as m ~ +oo. For this, we first prove that the integrals f~×o, g(u) and f~×D, Ivul 2 are convergents.

A nonexistence result

531

For z > 0, we denote Rz+ = [(xl, y) e ~ × D1 ; 0 < xl < z],

and

R [ = {(X 1 , y) E ~ X D 1 ; - Z < x l < 0}.

Integrating the equation (2.21) in Rz+, we get

i'n~ g(u) = -A(z) + A(O),

(2.27)

z

where

A(z) : t' ~.. Ouo.1x(z, y) dy : -d-zd i ,.Ol Since g _> 0, the limit of IR~+ g(u) as z --' + ~ lim A ( z ) = - ~ .

exists. If this limit is infinite, then by (2.27),

Z~+Oo

Writing

A(z)

(d/dz)V(z) with V(z) = Jo, u(z,y)dy, we ___ V(z) <_ m e s ( D 0 for all z > 0. We prove

as

is absurd since 0

then get,

lim

V'(z)

= -~.

This

Z--+ + ~

in the same way that

Ii'n:g(u) < +~.

lira z ~+¢~

~

Hence, we get that

"

g(u)

< +oo.

(2.28)

t ~xDI

To prove the convergence o f the second integral, we multiply the equation (2.21) by u and we integer in R Z+ thus we get I

[Vul 2 =

W(Z) - W(O) +

R +

g(u)u, R+

where w(z) =

t'DI u(z, y) ~v"~l- O['(z, l y)

dy :

21d i'ot uZ(z' y)

We have, by (2.23) and (2.28),

n:g

g(u)u <_

R~

g(u) < +~

Vz >0.

Therefore, if one assumes that, t

lim Z ~ +oo

~

I v u l ~ --- + ~

~R +

then (2.29) leads to lim Z ~+e~

this is absurd since 0 _< u ___ 1.

d t" OZZ

Ol

uZ(z,y)dy

+~,

(2.29)

532

S. OMRANI

We prove using the same arguments that

I'R:IvulZ

lim

< +~"

Hence, we get that IVul 2 < +oo. S

(2.30)

IRxD I

Now, by considering the sequence of functions: u+(x~, y) = U(X1 -'1-n, y) for (Xl, Y) in the fixed domain R~, we derive from the classical elliptic estimates that u + ~ v -+

(2.31)

in CI(/~I).

Since i'~xo, IVu[2 <

+oo,

t,

we find that necessarily, Vv -+ = 0 and thus by (2.31),

i

• 'D! ~01d x l ( m , y ) --" 0

a s m --, +

oo.

Au = 0 .

•

It follows from (2.26) that t

Au=

lim

[RxDI

Z~+

°°

S

R m

Conclusion o f the p r o o f o f theorem 2.1. From (2.21), (2.25) we deduce that

i '

g(u) = 0

, ~XOl

and using the fact that g _> 0, we infer that g(u) = 0

in ~ x D~.

(2.32)

Using the conditions (1.5), (1.6) or (1.8) on g, (2.32) leads to u ( x , , y ) ~ [0, 01U{1}

V (Xl, y ) E [~.~× D 1

i f g s a t i s f i e s (1.5), (1.6)

V (xl, Y) e ~' × DI

if g satisfies (1.8).

or

u ( x l , y ) ~ 10, 1}

This obviously contradicts the normalization condition (2.24) on u. We have, therefore, reached a contradiction which shows that there are no solutions when e > 0 is small. •

A nonexistence result 3. C O N S T R U C T I O N

533

OF THE EXTENSION

t2

In this section, we prove proposition 2.4. We first make some notations precise. We recall that f2 = ~ × Bt/z(A) and for 0 < p < ½ we denote ~p = ~ × ( B 1 / 2 ( A ) \ B v ( A ) ) ,

Sv = ~p f') ( ~ x OD1)

Sp is a regular hypersurface of Op which separate it into the two domains ~1 and ~pZ. Then we consider the d i f f e o m o r p h i s m q) from ~ 3 \ ~ x [0} into R 3 \ ~ x [0} defined by 0(XI,X2,X3)

= ( O I ( X I , X z , X 3 ) , q~2(X'l,X 2 , x 3 ) , ( ~ 3 ( X l , X 2 , X 3 ) )

(

2x2

2x3

xl

For any p e (0, ½) we have ~(~0z) C f2~. We recall that u satisfies the equations (2.6)-(2.10), then from (2.6) we deduce that 2 p (f~p) 1 for all p 6 ( 0 , ~-) 1 and all p > 3 and then the function u2 defined on ~p2 by u ~ W~oc uz = u o ~ satisfies 2 u z ~ Wlo2 ~p (f~p)

for all p > 3 and all p E (0, ½)

3

O2uz

i,j= l

OAi UAj

S" b

Luz = Z a~.~(Y)~.--Tq-S~..+ ,-,

i=1

0U2

~(Y)-z2-.. = g(u2)

(3.1)

in Opz,

where

ai.j(Y) =

r=l

-~x

(~b-l(Y)) ~Xr

1 < i , j <. 3

O2~i

bi(y) =

- 0~-z (~-~(y))

1 _< i _< 3.

r=l

Dl O ' ~

Fig. 2.

x3

(3.2)

534

S. OMRANI

The c o m p u t a t i o n of the coefficients of L leads to al,l(y ) = -1,

a2'2(Y) = -

(

al,2(Y) = al,3(Y) = 0

2x22)3/2 1

(x~ +

x~ + x2)3/2 3

bl(Y) = O,

)

x~ + x~) 1/2

1

4x2 x3 (x~ + x~) 3/2

1

2X2 (x~ + x~) 3/2'

b2(Y) -

xZ3)3/2

(, l

a2,3(Y) = a3,a(Y) - (x~ + xZ) 3 3/z

a3"3(Y) = -

(x2z +

2x3 (x 2 + x2) 3/2

b3(y) =

with x = (xl, x2, x3) = ¢ l(y) = ¢(y). Furthermore, u 2 satisfies on S a,

01d2 191d2 0U 2 (0/d Ov - -ax 2xx + -ox3X3 = ~ 3 0 20x2 -

(~/,/ 003~X 2 ( O~x2 302 + -0x3 + 3x 3 a-Z l

Ou 0 0 3 ) x 3

----+--Tx., 3x3

- -

Here v is the outward unit normal on 0f2~. F r o m the condition (2.8), we have Ou Ou 3u 3v - 3x2 x2 + ~x3 .*3 = 0

on Sp

and a straightforward computation yields 3u2 #v

-

o

onSp.

Consequently, u and u2 are strong solutions (in W ~2 op~ , P > 3 )

of

- A u = g(u)

in f2p1

(3.3)

Lu2 = g(u2)

in Op2

(3.4)

and satisfy the following interface conditions u = u2 3u 3v We deduce that fi = u ' x e . strong solution of

-

Ou2 - 3v

on Sp

(3.5)

on Sp.

(3.6)

+ u 2 " z e ~ (Z~ denotes the characteristic function o f fa) is a

Lfi = g(fi)

in f2p

for all p ~ (0, ½),

(3.7)

where/~ is the elliptic operator whose coefficients are those o f - A in f2~ and those o f L in f ~ for all p e (0, ½) (see [9]).

A nonexistence result

535

Writing 3

L =

Z

02

i,j=l

a~,j(x)

~

3

O

+ E b,(x) ~x~ ' i=l

we have ai,y, bi ~ L~(~).

(3.8)

The coefficients ~ , j are Lipschitz continuous in ~ and satisfy 3

Co1¢12 _< Y~ ~ , j ( x ) ~ j _< Col¢l 2

v x ~ f~, v ( ~ ~3, Co, Co > 0

(3.9)

i,j= 1

b2 and/3 3 are discontinuous on OS = OD~ f) f~. F r o m (3.7), we deduce that fi is a strong solution (in W~o~ 2,p , p > /.t~ = g(t~)

3) o f

in ff~\F.

(3.10)

satisfies also, 0 _ fi _< 1.

(3.11)

The function t~, therefore, satisfies all the properties of the proposition 2.4.

•

4. REMOVABLE S I N G U L A R I T I E S OF SOLUTIONS OF E L L I P T I C EQUATIONS

In this section, we briefly describe some results concerning r e m o v a b l e singularities o f solutions of the elliptic partial differential equation in R n, n > 2

Lu =- i,j=l ~ ai,j(x) ~

OZu

Ou + l El.= bi(X) oxii + c(x)u = O

(c <-- O)"

(4.1)

The first result we mention is due to Serrin, it concerns r e m o v a b l e singularities o f classical solutions o f (4.1). It assumes that the coefficients ai,o~, bi and c are H61der continuous functions and satisfy ~1~12 ~

~ ai,j(X)~i~ i < Af~12, i,j=l

(4.2)

where 2 and A are positive constants and ~ is an arbitrary vector in ~n. THEOREM 4.1 [4]. Under these assumptions, let u be a classical solution o f (4.1) in a d o m a i n

D \ Q , where D is open and Q is a c o m p a c t subset of D. If Q is a set o f o r d i n a r y capacity zero and u is b o u n d e d . Then u m a y be defined on Q so that the resulting extended function is a solution o f (4.1) in all o f D. We recall here the definition o f the ordinary or 2-capacity of a regular c o m p a c t set K o f ~n. Let u be the h a r m o n i c function (often called the conductor potential) defined in the c o m p l e m e n t o f K and satisfying the b o u n d a r y conditions u = 1 on OK and u = 0 at infinity.

536

S. OMRANI

Then the quantity capK=

j

3u

i V u ] 2 = _ , t' ~n',K

OK OV

where v denotes the outer normal, is defined to be the capacity of K (see [8, 9]). Using a slight modification of the p r o o f of theorem 4.1, we can easily show that the result in theorem 4.1 is true if u ~ W~P(DkQ), (p >_ n) is a strong solution of (4.1). We assume here that the a~,j are o~-H61der continuous (0 < ~ _< 1) and satisfy (4.2), and that the b~ and c are bounded. Since the p r o o f will be useful later in order to further extend this result, we present it and make the arguments complete.

Proof. We shall assume n > 2 for simplicity. Firstly, we observe that there is no loss of generality in assuming the domain D to be bounded and to have a smooth boundary. In the same way, it is clear that the solution u in question can be be assumed continuous on the boundary of D. Then consider u* ~ l,VlZo'~V(D)fq C°(D) the unique solution of the Dirichlet problem I Lu* = 0 u*

u

in D on OD,

see [8]. The function v = u - u* satisfies, v e wIE~p(D\Q), p > n IL;=0

inD\Q 0

on OD.

The theorem will then be proved if we can show that v = 0 in D\Q. Since the ai,j are o~-H61der continuous and b~, c are bounded, there exists a function G(x, y) which for fixed y in D satisfies

LG ~ 0

x ~ D\{y]

c2[x-yl2-n<-G(x,y)<.c31x-y[

2-n

c2,c3 > 0 ,

see [10, p. 317]. (In fact we can choose G(x,y) := Ix - yt2-n(1 + g l x - yl~).) Let T~, Tz . . . . . T~... denote a nested sequence of closed sets with smooth boundary, converging to Q. Q c ... T~... T~ C D. It is known that there exist positive surface densities a~(Y) on OT~ such that the functions ~(X)

= i

Ix - y l 2 - n g ~ ( y )

da(y)

or, are harmonic in the exterior of T~ and identically one on T~. Since Q has capacity zero and Q is the limit of the nested sequence I T~]~ we have L

capQ = lim cap T~ = v~ +oo

lim ~

v~ +~ ,] ~n\T ~

Iv~12 = 0

A nonexistence result

537

and, therefore, by the Sobolev inequality, for a subsequence • ~(x)

for x ~ D\Q.

, 0

We define

f LP~(x) = I G(x, y)l~,(y) dry(y). J aL Obviously, Lq~ < 0

in D - T~

and

c:~Ax) - °eAx) __: c3~Ax). Denoting by M t h e upper bound of v: Iv[ -< M in D\Q, then f r o m the m a x i m u m principle for strong solutions, we get that, M v -< --~P~

in D \ T~,

C2

whence letting v --* + oo, we find v _< 0 on D\Q. Similarily, v _> 0 on D\Q. •

Remark 4.2. We remark that an inhomogeneous term could be allowed on the right-hand side of equation (4.1) without altering the results, in fact by subtracting f r o m u a particular solution of the inhomogeneous equation this case reduces at once to the one already treated.

Remark 4.3. The condition that Q is a subset of D is not essential for the above result. To state this observation more precisely, we use a theorem of Serrin [5] on the removable singularities which touch the boundary for weak solutions of the equation

i,j= l ~

ai'j(x)

q i=l

ci(X) ~xi + d(x)u = O,

(4.3)

where ai,j, ci and d are bounded and measurable and the ai,j satisfy condition (4.2). THEOREM 4.4 [5]. Let Q be a compact set of 2-capacity zero and let D be a domain in IR". If u is a bounded and continuous solution of (4.3) in the set D\Q, then u can be defined on the set Q so that the resulting extended function is a continuous solution of (4.3) in all of D.

Proof o f theorem 2.5. We obtain the p r o o f of theorem 2.5 by recalling that any strong solution (in W~o 2,p c , p ___ n) of (4.1) in D \ Q with L satisfying the hypotheses of theorem 2.5, is a continuous weak solution of an equation of type (4.3) in D \ Q and thus by theorem 4.4 it can be defined on Q so that the resulting function is a continuous solution of (4.3) in all of D. Then we use again the p r o o f given above to complete. •

538

S. OMRANI REFERENCES

1. BERESTYCKI H. & LARROUTUROU B., A semilinear elliptic equation in a strip arising in a two-dimensional flame propagation model, J. reine angew. Math. 396, 14-40 (1989). 2. BERESTYCKI H. & NIRENBERG L., Travelling fronts in cylinders, Ann. Inst. H. PoincardAnalyse non Lindaire 9(5), 497-572 (1992). 3. BERESTYCKI H., LARROUTUROU, B & LIONS P.-L., Multidimensional traveling wave solutions of a flame propagation model, Archs ration. Mech. Analysis 111, 33-49 (1990). 4. SERRIN J., Removable singularities of solutions of elliptic equations, Archs ration. Mech. Analysis 17, 67-78 (1964). 5. SERRIN J., Local behavior of solutions of quasilinear equations, Acta. Math. sin. 111,247-302 (1964). 6. OMRANI S., Non existence de solutions pour certains probl6mes d'ondes progressives darts un domaine cylindrique, C. r. Acad. Sci. Paris 315(I), 1165-1170 (1992). 7. AGMON S., DOUGLIS A. & NIRENBERG L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conitions 1, Communs pure appl. Math. 12, 623-727 (1959). 8. GILBARG D. & TRUDINGER N., Elliptic Partial Differential Equations o f Second Order, 2nd edition. Springer, Berlin (1983). 9. DAUTRAY R. & LIONS J. L., Analyse Mathematique et Calcul Numerique, Vol. 2. Masson, Paris (1987). 10. GILBARG D. & SERRIN J., On isolated singularities of solutions of second order elliptic differential equations, J. Analyse Math. 4, 309-340 (1955/1956).