Liouville theorems for nondiagonal elliptic systems in arbitrary dimensions

Math Z 176, 123- 133 (1981) Mathematische Zeitschrift

�9 Springer-Verlag 1981

Liouville Theorems for Nondiagonal Elliptic Systems in Arbitrary Dimensions

Michael Meier

Mathematisches Institut der Universit/it Bonn, Wegelerstr. 10, D-5300 Bonn 1, Federal Republic of Germany

Introduction

Let ~ aik(x ) be bounded measurable coefficients on lR"(e, f l = l , . . . , n ; i,k = 1, ..., N), and set

A(u, ~)= S ~ aik (x)D~ vi D~ u k dx, (1)

where u, v~H~(IR ~, 1RN). Moreover, assume that the bilinear form A is coercive in the sense that there exists a constant 2 > 0 such that the inequality

A(q~, ~b)> 2 ~ I Vq~ 12 dx (2)

holds for every 0eH~(IR", IR ~) with bounded support. By an entire solution of the system

D~(a~(x)D~uk)=O ( i=1, . . . ,N) (3)

we mean a vector function ueH~,]oo(lR", IR N) which satisfies the equation A(u, 4)) =0 for all test vectors qSEC~(IR",IRN).

In the first part of this note we consider the question under which conditions one may conclude that the only bounded entire solutions of (3) are constant vectors (Liouville theorem).

While a Liouville theorem holds without further assumptions in the case n = 2 (cf. [3], Theorem A1, and [10], Theorem 4 for a generalization), a corresponding result ceases to be true for higher dimensions. As an example of Giusti and Miranda shows, the function u(x)= x/Ix[ is a bounded entire solution of the system (3), where n = N > 3, and

0025-5874/81/0176/0123/$02.20

124 M. Meier

(cf. [5]). On the other hand, a Liouville theorem has been proved by Frehse [3], Theorem A3, for arbitrary ~dimensions n>2, provided that the coefficients aik are constants.

Our first theorem generalizes this result to the case where the coefficients aik (x) differ from constants only by a sufficiently small quantity for large values of the argument x.

In the subsequent sections, we present several variants and related results for homogeneous systems (3) as well as for systems of the form

D~(a~(x)D~uk)=fi(x ,u, Vu) ( i=1 . . . . . g),

the right hand side f = ( f ~ . . . . ,fN) of which grows at most quadratically in the derivatives Vu.

Our proofs are based upon the method of Campanato (cf. [1]) which originally was used to show regularity of weak solutions of elliptic equations and systems. The close connection between Liouville type theorems and the problem of obtaining C%estimates for the solutions has already been observed by several authors (cf. [3, 7, 8, 10]): not only the assumptions in the theorems but also the methods of proof are often quite similar. This can be seen once more by comparing our Liouville results (see Theorems 1, 3, 4) with recent works of Giaquinta and Modica (cf. [4], in particular Theorem 4.1), Ivert (cf. [6], Theorems IV.l, IV.2), and Sperner [11]. Note, however, that the theorems we state do not follow immediately from the corresponding a priori bounds by the argument given in [7]. The reason is that in our case certain conditions of the regularity theorems are allowed to be violated on a bounded subset of IR ~. Therefore, we carry out the proofs using only estimates for the Dirichlet integral of the solutions rather than estimates for the modulus of continuity.

In the sequel, I ] denotes the Euclidean norm of a vector in IR", IR N, and IR ~N, respectively. Moreover, for an arbitrary matrix ~' (Cik)i,k~l ..... N;~,~=I ........ we set

tl(c,k)l[ - s u p ~ ~ ~ 2 .

Repeated Latin indices i, k are to be summed from 1 to N, Greek indices ~, fl from 1 to n. By B R w e denote the open ball in IR" with center 0 and radius R > 0.

A Liouville Theorem for Homogeneous Systems

Theorem 1. Let ~ " blk (t, k = 1 . . . . . N; c~, fi = 1, ..., n) be constant coefficients satisfying the inequalities

b~ . i glk ik', (~(~>;~olgl l2 l ( I 2 for all glEN N and all (6 lR ~, (4)

Ih (b~)II </~o, (5)

where )~o and i~o are positive constants.

Liouville Theorems for Nondiagonal Elliptic Systems 125

Then there exists a number c~=d n, ~(0, 1) with the following property:

I f ~P aik (x) are bounded measurable coefficients on IR", the associated bilinear form A of which is coercive, and if there is a ball B o c lR ~ such that

]l(aik (x)--bik)] ] <620 for a.e. xff]Rn-BRo, (6)

then every bounded entire solution of the system

D~(a~(x)D~uk)=O (i=1, . . . ,N) (3)

is a constant vector.

a~ (x) Remarks. i) Condition (6) is in particular fulfilled for a certain R o > 0 if the ~ converge to bi~ as I x ] ~ oo.

ii) By virtue of (4), the coerciveness of the bilinear form A follows if there exists a C~o~(0, 1) such that

]](aik(x)--bik)]]<6o2o for a.e. xelR".

The proof of Theorem 1 rests upon the following three lemmata.

Lemma 1. Let b~[ be constant coefficients with the properties (4) and (5). Then

( / ~ ) > 1 such that every weak solution there exists a constant C o = C o n, veH12(BR, ]R N) of the system

D~(b~D~vk)=O (i=1 . . . . . N) (7)

satisfies the inequality

B o B R

for arbitrary pc(O, R).

Lemma 1 is proven with the same technique as Lemma7.I in [1] and Lemma 3 in [5]. We note that for the proof it is not necessary to assume the estimate ~ i k bik ~ / ~ 2 o I{I 2 for all ~EIR "N. In fact, in the case of constant coefficients, the weaker Legendre-Hadamard condition (4) implies that

S b~ r~ ,~i r~ ,~k dx >=2o S [ V d~i2 dx (9) i k ~ tU ~ f l '4" N~ N~

holds for all ~b~H~(IR",IR N) with bounded support; this suffices to derive (8).

Lemma 2. Let ~b=~b(p) be a nonnegative, monotone nondecreasing function of p~(Rl ,~ ) , RI>=O , and suppose that there exist constants e~(0,1), ao>0, and H >= 1, such that

4a(p)<= H +~ ~(R) foral l p ,R with R ~ < p < R . (10)

126 M. Meier

Moreover, assume that for some c~e(O, %)

Then the inequality

e< c%H (11)

~(p)N c%-H ~o-, qS(R) (12)

holds for all p, R with R I < p < R.

The proof of a modified version can be found in [9], Lemma 3.I. In fact, Lemma 2 follows immediately from this result (note that we may always assume R 1 = 0 - by setting 4~(p)=0 for pe(0,R1] - and that (11) implies condition (3.3) of [9]).

Lemma 3. Let ~ aik (X) be bounded measurable coefficients on IR ~, and suppose that the associated bilinear form A is coercive (i.e. satisfies (2)). I f u~H~,loc(lR n, IR N) is an entire solution of the system (3), and if ~ IVul2 dx < oo, then u is a constant vector. ~'~

Proof. We fix an arbitrary R>0 , and set T2R=B2R--BR. Choose a "friend" tl~C~(B2R), 0 < r / < l , U--1 on B R, I Vr/I<2/R. Let co= ~ udx be the mean value

T2R

o f u on T2R, and w=u-co . As a test vector for (3) we take ~b =wt/2; this yields

~a:[~ Dr {-D~(wkrl)DJlwi + D,(w~)Darlw k} dx

+ ~a~ D~rl D ~tlwiwk dx.

By virtue of (2), the first integral is bounded from below by 2~1V(wrl)12dx. The boundedness of a~(x) and the properties of t / imply the estimate

K K ~lV(wrl)lNdx< R ~ I V ( w ~ ) l l w l d x + ~ ~ Iwladx,

T2R Jt't T2R

where K is a constant independent of R.

Using the inequality a. b N ea 2 +l-b2, the second integral may be estimated. e

Since t l - 1 on BR, and w = u - co, we finally arrive at

IVu?dx<__K; S lu-co?dx. (13) BR T2R

Poincar~'s inequality now implies that

IVu l2dx<K ~ [Vul2dx. (14) BR T2R

As R-+ oo, the integral on the right hand side of (14) approaches zero, because of the finiteness of ~ I Vul 2 dx.


Therefore we conclude that I lvul 2 dx=O, and Lemma 3 is proved. R n

Proof of Theorem 1. Let C O be the constant appearing in Lemma 1. It will be shown that the conclusion of the theorem holds for any c5 which satisfies

First of all, we note that

1 ( n - l ) " - t _ _ 2 ,

0 < 6 < ~ 8nCo (15)

~][7uIZdx<=CR, 2 for all R>0 , (16) B R

where the constant C depends on the various parameters, and on u, but not on R. This can be seen from (13) in the proof of Lemma 3, taking into account that lu-cof is uniformly bounded by a quantity independent of R.

Secondly, let L and L o denote the elliptic operators D~(a~[(x)De) and D~(b~Dp), respectively. With this notation, the system (3) may be rewritten as Lu=O.

Consider an arbitrary ball BR, and let veH12(BR, IR u) be the unique solution of the problem

L o v = 0 on BR, (17)

u -- v e / J ~ (B R, ]RN).

Moreover, set w =u--v~I2112(BR, IRN). By virtue of Lemma 1,

5[Vvl2dx<=Co(PRf 5]Vv[2dx for all pe(O,R). (18) B 0 BR

Furthermore, L o w = L o u - L o v = L o u = (L o - L) u. Testing this equation with w, we get

bO~D~wkD~,widx= ~. ~,e ~,~ k i (bik --aik (x))Deu D~,w dx, BR BR

(19)

and using (6), (9), and the Schwarz inequality, we obtain the estimate

IVwl2dx~c 52 ~ IVul2dx+Ko ~ IVul2dx, BR BR BR 0

(20)

the constant K o being independent of R. Because of [Vv[2~2]Vul2+Rl[Twl 2, IVul2~2[vvl2+2[Vwl 2, and ~<1, (18)

and (20) imply that

Bp BR o

(21) for all pe(0,R).

128 M. Meier

If ~lVul2dx< o% the assertion of the theorem follows from Lemma 3. Now

we show that the case S lVul2dx= oo cannot occur if (15) holds. Otherwise there

would be an R~ > R o such that

IVulZdx e: = 2 6 z + 2Ko(1 + 2 Co) B~~

[. [Vul~clx B~ 1

1 ( n - - 1 ~ n-1 < - (22)

n \SnCo/

Setting ~b(p)= ~lgul2dx, we infer from (21) and (22) that Bt,

(23)

holds for all p, R with R I < p < R . Application of Lemma 2 (with ~0 = n, e = n - 1) yields the estimate

j ' lVu[2dx<C [IVul2dx (24) B o B•

for all p, R with R ~ < p < R, where C is independent of p and R. For fixed p > R~,

we may let R ~ o o in (24), and from (16) it follows that [.[VulZdx=O. Since Bo

p >R~ is arbitrary, we get a contradiction to our assumption that ~lpul2dx = oo. This concludes the proof of the theorem. ~"

Remark. Although our method also works if (15) is replaced by slightly weaker inequalities, we do not obtain the best possible value for the admissible deviation 6 in Theorem 1. The optimal constant can be determined by the technique of Sperner [11], provided that the coefficients b~ satisfy an additional splitting condition (see Theorem 4 below).

Further Results and Example

1. A Liouville theorem also holds for systems (3) which come sufficiently close to a diagonal system for large values of the argument x. The statement of this result is the same as in Theorem 1, except that bi~ are not constant but of the form ~- ~P bik --5ik 91 i (x). Here, 9.I~(x)~Lo~ (IR n) are assumed to satisfy the inequality .~[~fl(X) ~a ~/~ ~.~0 l~12 for all ~elR n, all ie{1, . . . ,N}, and a.e. xelR n.

For the proof, only minor modifications of the above reasoning are necessary. Instead of Lemma 1, we now use the De Giorgi-Nash theorem which implies that the estimate (8) holds with n replaced by some exponent c~ o > n - 2 . The same alterations have to be made in (18), (21), and (23). Moreover, if ~ is sufficiently small and R 1 is chosen large enough, e in (22) will be so small that


Lemma 2 is applicable with some exponent c~e(n-2, %). This yields the estimate

5 l V u l 2 d x < C ~ ]Vu[2dx for all p,R with R I < p < R . Bp

The rest of the proof is the same as before. 2. It is well known that an entire solution u of Laplace's equation satisfying

the condition u=o(Ixl)(x-+oo) has to be a constant. By virtue of (8) and (13), a corresponding theorem holds for entire solutions of elliptic systems with constant coefficients. Now let us consider an elliptic system (3), the coefficients alk (x) of which converge to constants b~~ as x ~ 0o. In this situation there may exist nonconstant entire solutions u with the property u =o(Ixl)(x~oo), as the example below will show. Therefore, the following theorem is best possible.

Theorem 2. Let ~e aik (x) be bounded measurable coefficients on IR ~, the associated bilinear form A of which is coercive.

Moreover, assume that for x~o~ the ~ aik (x) converge to constants bi~ satisfying the inequality (4).

I f u~H~,loo(IR ~, IR n) is an entire solution of the system

with the property

D~(a~[~(x)De u k) = 0 (i = 1,..., N) (3)

u=O(lxl 1 ~) (x--,oo) for some 7~(0,1), (25)

then u is a constant vector.

For the demonstration, we can again take over the major part of the proof of Theorem 1; only a few details have to be changed. First of all, (16) must be replaced by the inequality

5 Igul zdx<KR~-2~ (for all R>I), (16') BR

which follows from (13) and (25). ~ n I - n - ? \ " ~ Now pick an arbitrary 3 with 0 < 3 < ~8nCo) , where

constant from Lemma 1. Then there exists a number R o > 1 such that

C O is the

I[(% (x)--bik)ll ~3~o for all xelR"-BRo .

We carry out the same argument as before, assuming that

[VU[ 2dx B~o <z ( n-~ ? r

e = 2 3 2 + 2 K ~ 1 7 6 SIVul2dx n \ 8 n C o !

BR 1

(22')

130 M. Meier

for some R 1 > R o. By virtue of (23) and (22'), Lemma 2 yields the estimate

~ ] Vule d x < C I VulZdx (24') Bp

for all p, R with R~ <p <R. The conclusion follows by combining (16') and (24').

3. Example. This example shows that, even in the case of one equation (N = 1), it is impossible to replace (25) in Theorem 2 by the condition u =o(]x])(x-* oe).

Let ~ > 0 be a number to be determined later, and set

__ X l

u(x) (loglxl)2§ , p~(x)=D~u(x),

n �89

Moreover, define

f Y/ X 1 X~ a " a ( x ) = ( ~ + n - 1 Ixl 2

Since

gl~_ "~ 2log Ixl pp(x) n - l J [(log [X[) 2 @0"] 2 ]p(x)] ]p(x)l '

1 f x21 log I xl IP(x)12 =~iog lx[)2 + ~] 2 .1 - 4 txi2 (log lxJ)2+~r

we can choose o > 0 in such a way that

(26)

(27)

x 2 (log Ixl) 2 )

+ ixi2 [(loglxj)2+o] ~ ; '

1 1 Ip(x) l > - (28)

=2 (log Ix l )2 + ~ '

I I ( S ( x ) - ~)11 < �89 (29) for all x~lR".

By virtue of (29), aC'P(x) are bounded functions on P,." which satisfy a~(x) ~ (~ =2 > • (12 for all (elR" and all x elR". In particular, the associated bilinear form A is coercive. Because of (27) and (28), the coefficients a~'(x) converge to c5 ~' as x - * ~ . Finally, a straightforward calculation shows that the function u defined in (26) is a nonconstant entire solution of the equation

D~(a~'(x) Dr =0

with the property u=o(lx[)(x-~oo).

Liouville Theorems for Nonhomogeneous Systems

Let us consider nonhomogeneous elliptic systems of the form

D~(a~(x)Dpu k) =fi(x , u, Vu) (i = 1 ... . . N). (30)


Here, ~ % (x) are bounded measurable coefficients on IR", the associated bilinear form A of which satisfies (2), and f = ( f z , .. . ,fN) is a function on IR" x 1R N x 1R "N which grows at most quadratically in the derivatives Vu, i.e.:

If(x,u,p)l<=alpl 2 for all (x ,u ,p)e lR"xlRNxlR "N (31)

with some constant a > 0. Under these assumptions, an easy modification of the proof of Lemma 3

yields

Lemma 4. Let ucH~,lo~Loo(lR",IR N) be a bounded entire solution of the system (30), and set M = [lullg~-,w~). Moreover, suppose that

2 a M < 2 . (32)

Then there exists a constant C, independent of R, such that

~ [ V u l 2 d x ~ C R "-2 foral l R~(O, oo). BR

If, in addition, ~ [ Vu]2 dx < o% then u is a constant vector.

Let b~ be either constant coefficients satisfying the Legendre-Hadamard condition (4) or coefficients of the form ~ ~ b~k =6ik N i (X), where 9.l~'~eL~o(lR" ) and ~-[~fl(X)~fl~.~.O]~l 2 for all (elR", all ie{1 . . . . . N} and a.e. xslR".

Under these assumptions, we prove that a bounded entire solution u of (30) has to be a constant vector provided that

(i) the ~ ag k (x) are sufficiently close to b~ for large values of the argument x, (ii) the expression a Ilul[g~o ~) is small enough.

Theorem 3. Suppose that ~ ~ alk, bik and f are as above, and that (2) and (31) are satisfied. / 1 \

Then there exists aconstant 6 * = 6 * ( n , N , @ b ~ } > O with the following property: \ 1 ~ 0 1

I f ueH~,xor N) is a bounded entire solution of (30) satisfying the condition

2 a M < 2 , (32) where

M : = Ilu I 1 ~ o w,),

and if there are nonnegative numbers R o and 61 such that

[L(a~k--b~k)(X)[[ <~512 o for a.e. x~IR"-BRo, (33)

2 a M �9 a l + ~ - 0 <6 , (341


Proof. We treat only the case of constant coefficients b~. Let B R c lR" be an arbitrary ball, and define v, w as in the proof of Theorem 1. Then the estimate

132 M. Meier

(18) holds, and the maximum principle for elliptic systems with constant coefficients yields

II VlIL~(BR,~,) _-- < C1 M (35)

with some constant C I = C 1 (n,N, l b ~ [ ) > l (for a proof, we refer to E2], Theorem 7.1). \ /VO /

Moreover, L o w = (L o - L) u + f (x , u, Vu). Testing this equation with w, and using (9), (33), (31), and the Schwarz

inequality, we obtain the estimate

~ lVu l2dx+Ko ~ ]Vul2dx, (36) 2 a M ~ \

B ~

where we have set m~ = IlWllL| Since w = u - v , we conclude from (35) that M * < ( C ~ + I ) M . By virtue of

Lemma 4, the same reasoning as in the proof of Theorem 1 implies that u is a constant vector if the condition

62 2(C1+1)aM 1 (n-1 ~n-1 2o < ~ n \8nCo!

is satisfied. In particular, the conclusion of Theorem 3 holds for

g),_ 1 ( n - l t" 4n(Ca + 1) \ 8 7 o ! "

We consider now the special case that the coefficients b~[ are of the form b ~ - ~ a~P with constants A "~ satisfying the condition A=~=A ~. Then it is ik - - ~ i k ~

possible to give an explicit upper bound for the admissible deviation 6a in (33) in terms of the various parameters.

Theorem 4. Let ~ aik (x) be bounded measurable coefficients on ]R ~, n > 3, let A ~t3 be constants with the property A ~ =A ~, and assume that f = ( f ~ . . . . ,fN) satisfies the condition (31).

Suppose also that the following inequalities hold:

Ao 1~]2~ A=' ~= ~, ~#o I(I 2

I[(aT~ (x) - ~kA =~) [I ~ ,5~ ~o

for all ~eIR '~u and a.e. xelR ~, (37)

for all (elR", (38)

for a.e. xelR"-BRo, (39)

where 2, 2o, #o and (~1 a r e positive constants, and Ro>O. I f ueH~,lo o c~L~o(lR", IR u) is a bounded entire solution of the system (30), and if

M = IlU]lL=C~,,~)satisfies the conditions

2 a M < 2 , (32)

o- oo /n-1 ~1 < ( 2 - 2 a M ) n - 2 ' (40)



Remark. A cor respond ing regular i ty t heo rem for weak solut ions of (30) has been p roved by Sperner (cf. El l ] ) . The same example as in [11] also shows that cond i t ion (40) is " a lmos t necessary" for ob ta in ing a Liouvi l le result. Moreover , it turns out tha t this b o u n d for 61 is op t ima l in the h o m o g e n e o u s case ( f - 0 ) .

W e shall only sketch the p r o o f of T h e o r e m 4, omi t t ing the details . If R o = 0, the conclus ion follows from the Ca-est imate in [11] and the a l r eady men t ioned a rgumen t due to Iver t (cf. [7]). In the general case, we can p roceed as in the p r o o f of [11]. No te tha t the smallness a s sumpt ion abou t I[(a~k(X)--(}ikA~#)ll is only used to es t imate cer ta in b o u n d a r y integrals over 0Bp involving the terms a~ei~r-lx)--61kA~. Here, N=(N~/~)~,/~= 1 ...... is the symmetr ic , posi t ive definite i k \ •

matr ix sat isfying N 2 = ( A ~B) 1. By vir tue of (39), we ob ta in these es t imates for sufficiently large values of p. Moreover , the remain ing ca lcula t ions in [11] can be taken over wi thou t any add i t iona l res t r ic t ion on p, since in our case A =~ are cons tan t coefficients, and therefore no local cons idera t ions are necessary.

F r o m this we infer tha t there exist posi t ive numbers R1, c~, and C, such tha t

n - 2 + 2ct

~ lVu[2 dx<= C 5 [Vu[2 dx B o B R

for all p, R with RI<p<R. (41)

Final ly , cond i t ion (37) implies (2), and hence L e m m a 4 is appl icable . The asser t ion of T h e o r e m 4 follows now at once.

References

1. Campanato, S.: Equazioni ellittiche del II ~ ordine e spazi L (2'~). Ann. Mat. Pura e Appl. 69, 321- 381 (1965)

2. Canfora, A.: Teorema del massimo modulo e teorema di esistenza per il problema di Dirichlet relativo ai sistemi fortemente ellittici. Richerche di Mat. 15, 249-294 (1966)

3. Frehse, J.: Essential selfadjointness of singular elliptic operators. Boll. Soc. Bras. Mat. 8.2, 87-107 (1977)

4. Giaquinta, M., Modica, G.: Regularity results for some classes of higher order non linear elliptic systems. Journal Reine Angew. Math. 311/312, 145-169 (1979)

5. Giusti, E., Miranda, M.: Sulla Regolarit~ delle Soluzioni Deboli di una Classe di Sistemi Ellittici Quasi-lineari. Archive Rat. Mech. Anal. 31, 173-184 (1968)

6. Ivert, P.-A.: Regularit~itsuntersuchungen yon LiSsungen elliptischer Systeme yon quasilinearen Differentialgleichungen zweiter Ordnung. Link6ping Studies in Science and Technology. Disser- tations No. 31 (1978)

7. Ivert, P.-A.: On quasilinear elliptic systems of diagonal form. Math. Z. 170, 283-286 (1980) 8. Hildebrandt, S., Widman, K.-O.: S~itze vom Liouvilleschen Typ fiir quasilineare elliptische

Gleichungen und Systeme. Nachrichten der Akad. Wiss. G6ttingen, Nr. 4, 41-59 (1979) 9. Kadlec, J., Ne~as, J.: Sulla Regolarit/~ delle Soluzioni di Equazioni Ellittiche negli Spazi H k' 4

Annali Scuola Norm. Sup. Pisa 21, 527-545 (1967) 10. Meier, M.: Liouville theorems for nonlinear elliptic equations and systems. Manuscripta Math.

29, 207-228 (1979) 11. Sperner, E.: Uber eine scharfe Schranke in der RegularitMstheorie elliptischer Systeme. Math. Z.

156, 255-263 (1977)

Received February 1, 1980

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Liouville theorems for nondiagonal elliptic systems in arbitrary dimensions