Div-curl fields of finite distortion

C. R. Acad. Sci. Paris, t. 327, SCrie I, p. 729-734, Equations aux d&ivCes partielleslfartial Differential

Div-curl Tadeusz

fields

IWANIEC

of finite

a, Carlo

a Department of Mathematics, E-mail: tiwaniec@mailbox.

le 3 septembre

Abstract.

1998,

Regularity

curl E=O

Equations

distortion

SBORDONE

Syracuse syr.edu

accept&

b

University,

b Dipartimento di Matematica e .4pplicazioni, E-mail: [email protected] (Requ

1998

Syracuse,

Universiti

NY

di Napoli,

le 14 septembre

13244, Via

USA Cintia,

80126

Napoli,

Italy

1998)

results for div-curl fields (B, E) ELB,,(Q, W” x W), 9 > 1, div B =O, of finite distortion K : 0 c W” --f [l, co), i.e. satisfying a.e. in R IB(x)I”

+ IE(z)12

5 (K(x)

+ K-‘(x))(B(+E(z)),

aregiven, in analogywith the theory of quasiconformalmappingsin the plane.Almost optimal integrability theoremsfor the gradientof weak solutionsto somelinear and nonlinearpde’sfollow. 0 AcadCmiedesScienceslElsevier,Paris Ch.amps div-rot RCsum&

de distortion

jinie

On donne des r&&tats de r&ularite’pour des champs div-rot (B: E) E I$,, (Cl, R” xR”), 4 > 1, div B = 0, curl E=O de distortion $nie K : S C R” -+ [l! w), c’est-&dire satisfaisant presque partout dans Cl

IB(z)l* + IE(z)I’ 5 (K(z) + K-‘(~)HBbLE(~)), par analogie avec la thkorie des applications quasi-conformes dans le plan. On dome ensuite des thkorkmes presque optimaux d’int&rabilite* concernant le gradient de solutions faibles d’&quations aux de’rivkes partielles 1inLaires et non likaires.

0 AcadCmiedes Sciences/Elsevier,Paris

Version

frangaise

abrt$$e

En suivant la voie des applications quasi-conformes dans le plan complexe nous Ctendonsce concept 2 une dimension arbitraire, nous obtenons entre autres une version indkpendante de la dimension, et presque optimale, de thbor&mes d’indgrabilitC. Note prbsentbe par Jacques-Louis 0764~4442/98/03270729

0 AcadCmie

LIONS. des Sciences/Elsevier,

Paris

729

T. Iwaniec, TH~OF&ME

C. Sbordone

1. - Pour 1 5 K 5 o;), soient q et p lespuissancesconjugue’esde Hijlder d&inies par : 2-p

2 7K-5

2

=q<2*:P=2+7K-7.

Alors, toute solution distributions u E Wrr$(fI)

de l’e’quation :

div A(z)Vu. = 0

dans

0,

Oh appartient 6 W:;:(0).

Une consequence de ce theoreme est le rbultat d’existence et d’unicid de Dirichlet pour une equation non homogbne :

qui concerne le probleme

div A(:c)Vu = CL, avec une mesure ,u au second membre. TH~OREME

3. - Supposonsque A = A(x)

ve’rifie (1) avec

2 1 I K < 1 + 7(n _ 2). Alors, pour toute mesure ,LLde variation totale finie, il existe une solution unique de l’tfquation

div A(z)Vn avec u E W, l12i

= p

(Q), c’est-h-dire

Nous obtenons aussi des resultats de regularite pour des equations non uniformement pour lesquelles la constante K est remplacee par une fonction de norme BMO petite. TI&OR~ME

4. - I1 existe E = c(n) > 0 teE que si A(a)

elliptiques

v&ijie (1) (R = R”) avec K = K(z),

IIKIIBMo(R~J 2 c(n), afors l’kquation

div A(rc)Vu = div f admet une solution unique Vu E L’(W”,

R”) h condition que Kf IP4l2

E L2(R”q5, R”). De plus,

L Wfll2.

This Note is concerned with regularity theory of a class of linear and nonlinear elliptic PDEs. An interplay between such equations and quasiconformal mappings is well known in dimension 730

Div-curl

fields of finite

distortion

n = 2, due to the work of many researchers beginning with R. Finn and J. Serrin [9], L. Bers and L. Nirenberg [4], B. Bojarski [6], etc. In the present paper we continue this theme from a more general perspective, that led us to the concept of quasihamtonic jields in W’. Of particular relevance to our study are the so-called div-curl fields in 0 c W”: &(R,

Iw” x R”) = {T = (B, E) E LyO,(R, R” x ET) ; div B = 0 and curl E = O}.

Let us first illustrate how such fields relate to the theory of linear elliptic PDEs of the form: div A(x)Vu where A : R -+ Wx” all c E W”

= 0,

(1)

is a measurable function with values in symmetric

matrices such that for

(2)

Uniform ellipticity means that 1 5 K(z) < K for some constant K. Now the pair T = (B! E), with E = Vu and B = A(x)Vu, is a div-curl field. Although it is not apparent at this point it is true that condition (2) is equivalent to the so-called distortion inequalify for T IFI where, by analogy to quasiconformal J(z,F)

I [K(z) + K.-l(+J(x;

F),

(3)

mappings, we introduce the notation IFI

= IB12 + IEl2 and

= (B, E).

It is important to realize that for some classical nonlinear PDEs, such as Leray-Lions equation: div d(z; 0~) = 0, under suitable coercivity and growth conditions, the substitution as above leads to the inequality (3). Now, guided by these examples, an arbitrary div-curl field F E dP(R, Iw” x W) is said to have$finite distortionficnction K(z) if (3) holds almost everywhere in R. 3 is said to be K-quasiharmonic if 1 < K(z) 5 K. Needless to say, the natural integrability exponent here is p = 2, though the essence of our work is to examine quasiharmonic fields with exponents p different from 2. A fruitful idea when studying such fields is to introduce the so-called f components -T’- = $ (B - E) and 3+ = $ (B + E) which parallel precisely to the Cauchy-Riemann components of the differential of a function in the complex plane. Continuing this analogy we arrive at the Beltrami type system:

F’-(x) = M(x)F-‘+(x) with M(X)

a symmetric

(4)

matrix of the form: M(x)

= X(z)[I

- ‘2e(x) ~3e(x)]:

(3

here, e = e(x) is a measurable unit vector field on R and K(z) - I

X(x) = Ill

I Ko+l

Conversely, the gradient field E = 3 + - 3-

<1

a.e.

(6)

= Vu solves the equation

div d(x:)Vu

= 0,

(7)

731

T. Iwaniec,

C. Sbordone

where we gain a rather special structure of the coefficient matrix

e(x). 1e(z)C3

I-M(z) d(z) = I + M(z)

Note that this matrix verifies.the same ellipticity det d(s) =

(8)

condition as A(z) does at formula (2); moreover, 1 -X(z) -~ 1 +X(z)

n-2

1 1

(9)

.

The point to be made here is that even when the original equation was nonlinear the transition via the Beltrami system leads us back to a linear elliptic equation. Thus many of our results apply to non-linear equations as well. We argue by analogy with the complex methods in dimension 7~= 2. For this purpose we introduce a singular integral operator S : LP(R”, Wn) -+ LP(R”, IV), called Hilbert trunsjiimt in R”. It acts as identity on divergence free vector fields and minus identity on the curl free fields. Thus S(F-) = F’+ and S(F+) = FT-, whenever a div-curl field F belongs to Lr(W”, R” x IV) with 1 < p < co. This approach demands that one must identify the p-norms of the Hilbert transform. Based on recent developments [l l] and [3] we give a dimension free estimate:

11~11~ = IISIIP5 1 + 7b

(10)

- 2)>

where 1 < q 5 2 5 p < oc are Holder conjugate exponents. As a side benefit, we obtain dimension free and nearly optimal improvement of Meyers’ theorem (see [14]) THEOREM 1 (Integrability

dejned

theorem). - For 1 5 K < co, let q and p be Hiilder conjugate exponents

by: 2 2- -=q<2.:p=2+-.

2

7K-5

Then every distributional

7K-7

solution u E W:;:(0)

(11)

of the equation

divA(z)Ou

:= 0 in R,

where

K-l ItI 5 M4l,

E) I KIE12>

(12)

actually belongs to W:;:(R). Following the lead of J. Serrin’s work [17] one might expect that the range of exponents q and p for this theorem is in fact determined by: 2K K+l


2K

(13)

This is known in dimension2idue to a resu2hKof K. Astala [l] (see also [S], [13] and [2] for a In higher dimensions the integrability problem borderline case when q = and p = -). for the full range of expone%sTaIs given by (%). Xmains one of the outstanding questions in PDEs. We explore our estimates to obtain some partial results. 732

Div-curl

fields of finite

distortion

THEOREM2. - Let 0 be a cube in R” and let A satisfy condition (12) in s1. Then for each vector jield f E L’(R, Wn) the Dirichlet problem

div A(z)Vtt has unique solution u E WiFr(R),

= div f

(14)

whenever

2 2-p 7K-51r<2+-.

2 7K-7

(1%

We have also a uniform bound (16)

Obviously, the cube fl can be replaced by other domains with sufficiently regular boundary. One interesting inference from this theorem concerns the equation div A(x)VU

= /J

(17)

with an arbitrary signed measure ,u in 0. THEOREM3. - Suppose A : R -+ FFx”

satisfies condition (12) with

2 1 5 K < 1+ 7(n _ 2).

(18)

Then for every measure p of jinite total variation there exists unique solution u E W, equation (17); That is to say,

lGw(q

of

(19)

This theorem can be viewed as a far reaching complement We conjecture that Theorem 3 remains valid when l
n-2

of some earlier results [5], [ 161.

(20)

and that this bound for K is sharp. We are also dealing with the div-curl fields whose distortion function K(z) is unbounded, but has small BMO-norm. Since a constant function has BMO-norm zero, the class of fields we consider contains K-quasiconformal fields. Among other things, we show that every div-curl field in the Orlicz class L2 log-’ L actually belongs to L2 log L, on compact subsets. As a note of warning, no reasonable conclusions can be drawn if the BMO-norm of K := K(z) fails to be small. The major innovation, as compared with the existing theory, is that the uniform ellipticity bounds on the equation div A(~)021 = div f

(21)

are no longer valid and, therefore, the higher integrability properties of the gradient can only be observed in the category of some Orlicz spaces close to L2(Q IF). 733

T. Iwaniec,

C. Sbordone

Our next result seems to be the first of its kind: the uniform above being relaxed.

ellipticity

bounds from below and

THEOREM 4. - There exists a number E= t(n) > 0 such that if A satisfies (2) with IIKIIB~~(~~) < C, then the equation (21) in R” admits a unique solution (up to a constant) with Vu E L2(W”, P), provided K f E L2(R”, W). Furthermore, we have

IF’+

I 4ll~fll2.

G-9

As a matter of fact the energy integral is also bounded. That is to say

s

R”

(A(z)Vu,

Vu)dll: 5

s R”

W1(4f,

f)dx 5

IKf 12.

(23)

Our study culminates in the following regularity theorem for very weak solutions of the homogeneous equation div A(z)Vu = 0 in R c W”, (24) with finite energy, Jo(A(z)Vu,Vu)dz THEOREM

< co.

5. - For each compact G

c

R we have

IVu12 log(e $- IVul) < x. s G

We expect that for every a 2 1 there is an E = ~(72,o) > 0 such that ]VU] E L2 log” L(G), whenever I IK I Inkto 5 E(n, cz). For similar results we refer to a forthcoming paper [lo]. q

References [I] Astala K., Area distortion of quasiconformal mappings, Acta Math. 173 (1994) 37-60. [2] Astala K., Iwaniec T., Saksman E., Beltrami operators, (1998) (to appear) [3] Bafnrelos R., Lindeman A., A martingale study of the Beurling-Ahlfors transform in R”, J. Funct. Anal. 145 (1997) 224-265. [4] Bers L., Nirenberg L., On linear and nonlinear elliptic boundary value problems in the plane, Conv. Int. le Eq. Lineari a detivate parziali, Trieste 1954, Cremonese (Ed.), Roma. 1995, pp. 141-167. [5] Boccardo L., Gallouet T., Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal, 87 (1989) 149-169. [6] Bojarski B., Generalized solution of a system of first order differential equations of elliptic type with discontinuous coefficients, Math. Sb. 43 (1957) 451-503. [7] Carozza M., Moscariello G., Passarelli A., Linear elliptic equations with BMO coefficients, (1998). [S] Eremenko A., Hamilton D., On the area distortion by quasiconformal mappings, Proc. Amer. Math. Sot. 123 (1995) 2793-2797. [9] Finn R., Serrin J., On the Holder continuity of quasiconformal and elliptic mappings, Trans. Amer. Math. Sot. 89 (1958) l-15. [lo] Iwaniec T., Koskela P., Martin G., Mappings of BMO-distortion and Beltrami type operators, (1998) (to appear). (111 Iwaniec T., Martin G., Riesz Transforms and related singular integrals, J. Reine Angew Math. 473 (1996) 25-57. [12] Iwaniec T., Sbordone C., Quasihannonic fields, (1998) (to appear). [13] Leonetti F., Nesi V., Quasiconformal solutions to certain first order systems and the proof of a conjecture of G.W. Milton, J. Math. Pures Appl. 76 (1997) I-16. [ 141 Meyers N., An LP-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scu. Norm. Sup. Pisa 17 (1963) 189-206. [15] Migliaccio L., Moscariello G., Mappings with unbounded dilatation, (1997) (to appear). [16] Murat F., Equations non lineaires avec second membre L’ ou mesure, preprint,l994. [ 171 Serrin J., Pathological solutions of elliptic differential equations, Ann. Scu. Norm. Sup. Pisa 18 (1964) 385-387.

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