A NNALES DE L’I. H. P., SECTION C




 B ERNARD DACOROGNA
 J ÜRGEN M OSER
 On a partial differential equation involving
 the jacobian determinant
 Annales de l’I. H. P., section C, tome 7, no 1 (1990), p. 1-26
 <http://www.numdam.org/item?id=AIHPC_1990__7_1_1_0>




    © Gauthier-Villars, 1990, tous droits réservés.
    L’accès aux archives de la revue « Annales de l’I. H. P., section C »
    (http://www.elsevier.com/locate/anihpc) implique l’accord avec les condi-
    tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-
    lisation commerciale ou impression systématique est constitutive d’une
    infraction pénale. Toute copie ou impression de ce fichier doit conte-
    nir la présente mention de copyright.




                   Article numérisé dans le cadre du programme
                 Numérisation de documents anciens mathématiques
                               http://www.numdam.org/
Ann. Inst. Henri Poincaré,

Vol. 7, n° 1, 1990, p. 1-26.                                                      Analyse non linéaire




       On a partial differential equation involving the
                   Jacobian determinant


                                    Bernard DACOROGNA
                               Ecole Poly technique, Lausanne, Suisse



                                         Jürgen MOSER
                             Eidg. Techn. Hochschule, Zurich, Suisse




    ABSTRACT. - Let Q c f~n a bounded open set andf>O in Q satisfying

                      We study existence and regularity of diffeomorphisms
f(x)    dx = meas  o.
u : SZ -~ Q such that



.
   Key words : Volume preserving diffeomorphism, nonlinear PDE, elliptic equations, Jaco-
bian determinant.


    RESUME. - Soit Q c ~n un ouvert borne et soit f > 0 dans SZ satisfaisant

                         On etudie 1’existence et la regularite de diffeomorphis-

mes u :        Q tels que




                Annales de /’Institut Henri Poincaré - Analyse non linéaire - 0294-1449
                            Vol. 7/90/01/01/26/$ 4.60/ © Gauthier-Villars
2                              B. DACOROGNA AND J. MOSER



                                 I. INTRODUCTION

    Consider a bounded connected open set Q of fR" and two n-forms a, P


with f, g > 0. We shall prove that, under some regularity assumptions on
Q, f and g, there exists a diffeomorphism cp : S2 Q, keeping the boundary
                                                           --~


pointwise fixed and such that


where ~,   _


               ~iI a.
    In analytical form, the above result is equivalent to

    THEOREM 1.       -



                         Let            be an integer, 0 a 1, S2 have a Ck + 3, «
boundary                   denoting the  usual Holder spaces). Letf, g E Ck° « (SZ) with
f, g > 0      in     SZ.    Then    there    exists a diffeomorphism cp             with
                + 1, «
 cp, cp -1 E Ck        (S~;    and  satisfying



where 03BB=     g   f.
   Remarks. - (i) This scalar equation is clearly underdetermined and
uniqueness does not hold as the following trivial example shows. Let Q
be the unit disk of I~2, f = g =1,                           with               N an
integer,  and   cp (x) = (r cos (9 + a (r2)), r sin (8 + a (r2))) where (r, 8) denote
the polar coordinates. It is clear that such a cp is a solution of ( 1.1 ).
   (ii) For notational convenience we shall denote for k >_ 0 an integer,
           and Q a bounded open set of                   the set of diffeomorphisms
(homeomorphisms if k = o) cp : S2 -~ SZ with (p, cp -1 E C’~° °‘ (SZ; f~n) by
         (Q) [if a 0 we j ust set Diffk (Q)].
                    =



   This theorem is a stronger version of a known result. For two volume
forms on a smooth compact manifold without boundary it was established
in [M] (under stronger smoothness assumptions) and by Banyaga [B] for
manifolds with boundaries (in the C~ case). For the special case of the
ball in dimension 2 or 3, see [T], [D] respectively. Our purpose here is to
present a simple proof of this theorem, using standard properties of the
Laplacian for domains in !R". This allows us to avoid the use of differential
forms and the notion of manifolds. At the same time we obtain precise
regularity results. As one would expect the solution cp is one differentia-
bility class smoother than’ f and g if one works with Holder norms. In
our proof, given in Section 2, this point requires special attention. For


                                         Annales de l’Institut Henri Poincaré - Analyse non linéaire
                           THE JACOBIAN DETERMINANT                               3

manifolds without boundaries this gain in smoothness was established by
Zehnder [Z], under the additional assumptions that f, g are sufficiently
close in C°~ « norm and g is in C4. It goes without saying that our proof
can be carried over without difficulty to manifolds Q with boundaries.
                 As an application of Theorem 1 we can construct a volume preserving
diffeomorphism with given boundary data. In other words we claim
that if Q is as in Theorem 1 and ~° E Diffk+ 1 ~ « (S2), then there exists
03C8~Dif k+1,03B1(03A9)        such that




Indeed if we use Theorem 1 with g= 1,                       then ~, =1 (since
             is a diffeomorphism) and we can therefore find cp satisfying
( 1.1 ). We obtain a solution of the above problem by
   This type of problem plays a role in the construction of volume preserv-
ing mappings with prescribed periodic orbits and ergodic mappings, as
worked out by Alpern [A], Anosov-Katok [AK]. In [D] it was shown how
to apply the result to minimization problem in the calculus of variations
with further applications to nonlinear elasticity.
   Finally it is interesting that the corresponding problem for non compact
manifolds leads to additional topological obstructions as shown by Greene-
Shiohama [GS].
   In Section 3 we present an alternate proof under different regularity
assumptions. Here we avoid the use of elliptic partial differential equations
and work only with the implicit function theorem. For this reason we use
the Ck norm,            an integer, instead of the Holder norms Ck~ °‘. This

approach    allows   even  the treatment of continuous functions f and g in
which case cp turns out to be a homeomorphism and not a diffeomorphism
since this approach fails to give the expected derivative gain which we
have for Holder norms in Theorem 1. For this reason the differential
equation has to be interpreted in a weaker form, namely



for all open sets E c Q. It is clear that if cp is C 1, then ( 1. 2) is equivalent
to (1.1) with ~, =1.
   This second approach is also used to study a question related to
Theorem 1, where we ask for regularity only in the interior, i. e.,
cp E Diffk (Q) n Diff0 (SZ), satisfying ( 1.1 ) (i. e. cp is a diffeomorphism of Q
and it extends as a homeomorphism of Q, keeping the boundary pointwise
fixed). For thispurpose one needs much weaker regularity requirements
for        we shall allow, for example, Lipschitz boundaries or isolated


Vol. 7, n° 1-1990.
4                         B. DACOROGNA AND J. MOSER



boundary points as in a punctured disk. These cases cannot be treated by
the potential theoretical methods used in Section 2.
   The proof of Section 3 is related to an argument given in [M]. However
as A. Katok pointed out to one of us, this argument is incorrect (on

p. 291 it was not provided that v = w on the boundary, as was required).
Our purpose here is to rectify this argument and at the same time to take
care of domains with boundary. As before this approach applies to general
manifolds and not only to Q which are embedded in (~n.
   We conclude the introduction with an open problem which we were not
able to resolve: le Q be a connected open set with smooth boundary, say

C~ smooth. Let f E C (SZ), f > 0 and          fdx=measo. Does there exist a
diffeomorphism u : 0 -+ Q with u (x)     =
                                             x on aSZ   solving det V u = f in Q?


                    II. A DEFORMATION APPROACH


    We now state the main result.

  THEOREM I’. - Let                  integer and 0 a 1. Let Q c                 be a
bounded open set with Ck +3, °‘ boundary ~03A9. Let f E Ck° °‘ (SZ), f> 0 in SZ and




Then there exists u E Diff + 1, " (SZ) satisfying




  Remark. - Note that Theorem 1 stated in the introduction follows at
once from Theorem     1’. Indeed we have the theorem by setting cp = v -1 ~ u
where u and v satisfy




    We now describe roughly the idea of the proof

                                      Annales de l’Institut Henri Poincaré - Analyse non linéaire
                             THE JACOBIAN DETERMINANT                            5

   STEP 1 (Theorem 2). - We write u (x) = x + v (x). The linearized problem
is then




We solve the above problem by setting v = grad a + c, where c takes into
account the boundary condition and is divergence free. Using standard
existence theory and Schauder estimates for elliptic equations we obtain
the result. Although the solution of this problem is clearly not unique,
our construction provides a well defined solution.

  STEP 2 (Lemma 3). - We then find a Ck~ °‘ solution by a deformation
argument, i. e. by solving the ordinary differential equations




where v is  as in Step 1. Standard properties of ordinary differential

equations give that u (x) _ ~ 1 (x) is a solution of (2 . 2), but it is only in
    « .




   The two last steps are used to obtain the Ck + 1, « regularity.

  STEP 3 (Lemma 4). - Using Step 1 and a smallness assumption on the
C° ~ ~ norm, 0 ~i a  1, of f -1, we obtain a Ck + 1, « solution by linearizing
the equation around the identity.

   STEP 4. - We remove the smallness assumption in Step 3 on            norm

of f - l, by composing    two  deformations.   The   first one (using Step 3)
which allows to pass from f E Ck~ « to g E Ck + 1 ° « with f - g small in the
      norm. The second one by applying Step 2 to g.
   We start with Step 1 and give a theorem concerning the existence and
regularity of solutions of the linearized problem.
   THEOREM 2. - Let Q c             be a bounded open set with Ck + 3, « boundary,
k >_ 0 being an integer and 0       a  1. Let g E Ck~ « (SZ) with




Then there exists v E Ck + 1;.°‘ (SZ;    satisfying


Furthermore there exists K = K (oc, k, SZ) > 0 such that



Vol. 7, n° 1-1990.
6                            B. DACOROGNA AND J. MOSER


    Remarks. - (i) We have denoted by ]) .       the Ck~ °‘ norm.
    (ii) In fact the proof of the theorem will show that if




then our construction will provide a bounded linear operator L : X - Y
which associates to every geX, a unique v = L g E Y satisfying (2 . 4).
  Before proving Theorem 2 we introduce some notations

  NOTATIONS. - Let             ~" ~n -1 »2) with w          =
                                                                      1 i ; n. For nota-
tional convenience we define wr~ for         by letting                             We then
define


by



  Remarks. - (i) If w~~ are the components of an (n - 2) form a over ~8n
then (curl* w)~ are the components of da.
  (ii) For every WE C2 ([Rn; ~n ~n- l02) we have

which corresponds to the relation dda        =
                                                 o.

  Proof of Theorem 2. - We decompose the proof into two steps. In
both steps we shall use standard results of elliptic operators with Neumann
boundary condition and we refer, for example, to Ladyzhenskaya-
Uraltseva [LU] (Section 3 of Chapter 3) for details.


     STEP 1.   -


                   Let               be the unique solution with                              of

the Neumann problem




where v is the outward unit normal. Furthermore there exists K > 0 with


We then let c E Ck + 1 ~ «     [Rn) be defined by


                                       Annales de l’lnstitut Henri Poincaré - Analyse non linéaire
                                  THE JACOBIAN DETERMINANT                        7

Observe that


where (.;. ) denotes the scalar product in f~n.
  Suppose (cf. Step 2) that we can find b E Ck + 2, °‘       ~n ~n -1 »2) and K > 0
such that



If we then set       ~
                          .




we have indeed solved the    problem (2 . 4) and (2.5), by combining (2) and
(5). (Note that to write v as in (6) is not unusual in magnetism or in
elasticity, cf. for example Abraham-Becker [AB] or Love [L].)
  STEP 2. - We now consider for                         tR") the problem

with the additional assumption that c is tangential to           i. e.,



Equation (4)         is                      and we can assume that
                              again underdetermined
                                         has
                              on oQ so that
                                 gradbij      the direction of the normal.
More precisely, under the condition (3) we can assume this vector in the
form


Indeed if the above holds we have on ~Q, with the convention that




since (3) holds and v== 1.
  We therefore have reduced the problem to finding                            satis-
fying

where                    stands here for
  Note that (7) is not a differential equation for bij but merely the
prescription of the normal derivative of bij on ~03A9. Since the gradient of
the distance function            is - v whenever              one obtains a

solution of our problem in the form



Vol. 7, n° 1-1990.
8                          B. DACOROGNA AND J. MOSER



with ç (0) 0, ç’ (0) = 1 and § E C~ with - 0 outside a small neighbourhood
          =



of 0. However this solution lies only in Ck + 1 ° «.
   In order to find a smoother solution in Ck + 2, « we solve the following

Neumann problem
                      with

                       i

Using the standard results (cf [LU]) we find                                     and K > 0
with


We let x E      be such that x’ (0)      =
                                             0, x (0) = 1 and x --_ 0 outside a small
neighbourhood of 0. We finally let

where                                             Observe that since JQ is Ck + 3, «
and dj~ E C~ + 2, «, then b~~ E Ck + 2, «. From (8) it also follows that


obtaining thereforeimmediately (5). We then only need to show (7) [and
hence (4)] and this follows from the fact that if x E oQ (denoting by Sk~
the Kronecker symbol)




   Note finally that if we fix the above function x we have constructed a
definite solution v of (2 . 4), thus defining a linear operator L : X -~ Y with
the properties given in Remark (ii) above..
   The next step in proving Theorem 1, is to establish it with a weaker
regularity than stated.
  LEMMA 3. - Let                     integer and 0 a 1. Let Q c (~" be a
bounded open set with Ck + 3 ° " boundary ~03A9. Let f E Ck,03B1 (SZ) with f> 0 in SZ

                                      Annales de l’Institut Henri Poincaré - Analyse non linéaire
                               THE JACOBIAN DETERMINANT                       9

and




Then there exists u E             (SZ) satisfying


  Remark. - The following proof is based on a deformation argument
described in [M], which automatically ensures that the solution is a diffeo-
morphism.
  Proof of Lemma 3. - We decompose the proof into two steps.
   STEP 1.     -


                     Let for t~[0,1], zeQ




where v E Ck + 1, « (Q; !R") (but vt E Ck~ « (SZ; (l~n)) satisfies




(Such a v exists by Theorem 2.)
  We then define ~t (x) : [0, 1] x SZ --~ I~" as the solution of




First note that    E Ck~ « (SZ;  for every t and that is uniquely defined
on [0, 1]. Observe also that for every ~e[0,1]we have




[This follows from the observation that if x E oQ, then x is a solution of
(3), since v=O on     the uniqueness implies then that           for every

  We now show that u (x) --_     (x) is a solution of (2 . 2). The boundary
condition has already been verified so we need only to check that
det V    (x) f (x). To prove this we let
                   =




If we show (cf. Step 2) that




we shall have the result from the fact that          h ( 1, x) = h (o, x) .
Vol. 7, n° 1-1990.
10                        B. DACOROGNA AND J. MOSER



     STEP 2. - We therefore only need to show (5). Let A be an n x n
matrix, then it is a well known fact (cf Coddington-Levinson [CL], p. 28)
that if B)/ satisfies ~r’ (t) A (t) B)/ (t) then
                         =




where tr (A) stands for the trace of A. We therefore get that




We now differentiate (4) to get




Using (3) and (6) we obtain




Using the definition of vt [cf. (1)] we deduce that

Combining the two identities we have


The definition of v [cf. (2)] gives immediately (5) and thus the lemma..
  The third step in proving Theorem 1 is to establish the result with a
smallness assumption on the           norm of f -1.

   LEMMA 4. - Let Q, k, a and f E Ck~ " (S2) be as in Theorem 1 [in particular
(2 .1) is satisfied]. Let 0 [i _ a 1. Then there exists
such that    I f -1 ~ I °, ~ _- E, then there exists E
                                                    u                  (SZ) such that


  Remark. - (i) A similar result can be found in Zehnder [Z].
  (ii) We shall use below some elementary properties of Holder continuous
functions and we refer for a proof of such facts to Hormander [H]. In
particular we shall use the fact that if f,        °‘, then there exists a
constant C > 0 such that




                                       Annales de l’Institut Henri Poincaré - Analyse non linéaire
                           THE JACOBIAN DETERMINANT                             11

where ( I . I ~ o denotes the C° norm.
   Proof of Lemma 4. - We start by defining two constants K1, K2 as
follows.
   (i) Let



As seen in Theorem 2 we can then define a bounded linear operator
L : X - Y which associates to every bE X an a E Y such that




Furthermore there exists K~> 0 such that




   (ii) Let for §, any n x n matrix,

where I stands for the identity matrix and tr (~) for the trace of ~. Note
that Q is a sum of monomials of degree t, 2 _ t __ n. We therefore can find
KZ > 0, such that if w 1, w2 E Ck° °‘ with I ~w1 I ( o, ~w2 0~ 1, then

   In order to solve (2 . 2) we set v (x) --_ u (x) - x and we rewrite (2 . 2) as




If we set


then (5) is satisfied for any v~Y with


Note first that the equation is well defined (i. e., N : Y --~ X), since if v   =
                                                                                    0

on a~2 then                       Indeed from (3) it follows that




Vol. 7, n° 1-1990.
12                        B. DACOROGNA AND J. MOSER




since v=O on aSZ and
                         f=meas 03A9,
hand side of the above identity is 0.
                                        it follows immediately that the right


  We now solve (7) by the contraction principle. We first let for r > 0


We shall show that by choosing              and r small enough then
           is a contraction mapping (with respect to the Ck + 1 ~ °‘ norm).
The contraction principle will then immediately lead to a solution

     Indeed if we let




and if v, w E Br we then have



                                ..... ,_




The first inequality, which is also valid for k   =
                                                      0 and a    =
                                                                     ~i, follows from




To obtain (11) we observe that




and hence combining (10) for k=O and                   with the above inequality
we have     immediately (11).
   It now remains to show that u (x) = x + v (x) is a diffeomorphism, this is
an   easy consequence of the fact that                  and u (x) = x on 0Q
(see for example Corollary 2, p. 79 in [MO])..
   We may now conclude the proof of Theorem 1’.
   Proof of Theorem 1’ (Step 4). - By density of C°° functions in
with the       norm (0           1) we can find gECoo (Q), g > 0 in Q such
that             .




                                           de 1’In,stitut Henri Poincaré - Analyse non linéaire
                           THE JACOBIAN DETERMINANT                           13




where E is as in Lemma 4.
   We then define b E Diffk+ 1 ° °‘ (SZ) to be a solution [which exists by (1),
(2) and Lemma 4] of




We further let                     to be a solution of




Such a solution exists by Lemma 3 since                               and




  Finally observe        that the function                   has all the claimed
properties..


                         III. ANOTHER APPROACH

  We now present a second approach for solving (1.1) which is more
elementary, in the sense that it does not require the existence theory and
Schauder estimates for elliptic partial differential equations. It will use, as
a main tool, the implicit function theorem.
  We can now state the first theorem of this section.

   THEOREM 5.      Let k >__ 0 be an integer, S2 be a bounded connected open
                     -




set oflRn with Ck n C1 boundary         Letf,              g > 0 in SZ with



Then there exists cp E Diffk (SZ) with cp (x)   =
                                                    x on aS2 and such that




for every open set E c S~..
   Moreover if supp ( f - g) c Q, then supp ( cp id } c Q where id stands for
                                                       -




the identity map.

Vol. 7, n° 1-1990.
14                         B. DACOROGNA AND J. MOSER


  Remarks. - (i) Recall first that if cp E Diff ° (Q), then it is understood
that cp is a homeomorphism from Q onto Q.
  (ii) As already observed in the introduction if k >_ 1, (3 . 2) is equivalent
to ( 1.1 ) by a change of variables




Since E is arbitrary, it follows that g (cp) det ~03C6=f in Q.
  (iii) We can also rewrite (3 . 2) in the following equivalent form



for every 03BE~C~ (Q) with compact support. Approximating the characteris-
tic function xE by ç we obtain (3 .2). Therefore the above identity can be
viewed as the "weak form" of the equation g (cp (x)) det ~ cp (x) =f(x).
   (iv) Aside from the weaker boundary smoothness requirement, the
above theorem is for              weaker than Theorem 1, since it does not
provide any     smoothness   gain for cp. The main points of this theorem are
firstly  that k = 0  is admitted  and secondly that cp can be chosen so as to
be the identity near aSZ if f= g near         These results cannot be obtained
by   the  method    presented  in  Section 2. For the equivalence problem of
measures under homeomorphisms one has, of course, the stronger theorem
of Oxtoby and Ulam (see [A] for references).
   We now turn to the second result where we search for
cp Diffk (Q) ~ Diff0 (SZ) (k >_ 1) with cp (x) x on ~03A9, i. e., cp is a diffeo-
                                                   =
   E

morphism of Q which extends as a homeomorphism of Q keeping the
boundary pointwise fixed. This requires much weaker regularity
 assumptions on           For the following we shall require that Q c !R" has,
 with respect to the volume form T dx, the property (Hk) defined below.
                                       =




  DEFINITION. - Let Q c f~n be a bounded connected open set and k >_ 1
be an integer. Q is said to satisfy (Hk) if it can be covered by finitely
many open domains Q~ such that for every j (see the figure below),
  (i) there exists a Ck diffeomorphism

 where                   and               where Qn = (o,1 )" is the unit cube of
 f~n and                          if          and q=0 if                Moreover
 det ~03C8j~Ck   and there exists A >_-1 with




     (ii) The map 03C8j is proper, i. e. if K c P" is compact so is                   (K) and
       extends to a continuous map of Pn with                 (x) E D~ if x E q.
                                       Annales de I ’Institut Henri Poincaré - Analyse non linéaire
                                 THE JACOBIAN DETERMINANT
                                                                                  15




       Remark. - The property (Hk) is of local character. In the appendix
                                                                              we

    shall show that the following domains    have this property:
           if aSZ is locally given as a graph of a Lipschitz function, e. g. any
~
       (i)
                                             A. 2);
    open convex polyhedron (see Propositon
                                                                  A.
      (ii) domains with isolated boundary points (see Proposition 3),
                                                                        or a

    combination of both.
      Before stating our second result, we give a proposition which explains

    why the above definition is required.




       PROPOSITION 6. - With the above notations if              Qn U q - Q" U q is
     continuous with cp (x) x for x E q, then
                             =




     extends continuously to t~~ with cp (x)for x E ~~.
                                               =
                                                   x

                                                               and     We let
         Proof. - We drop for simplicity the index j in
             be a sequence of points such that p‘’ ~ p* E ~. We have to show
     that


     so that   cp (p*)=p*.
         Since B)/ is a proper map we conclude that

     i. e.,               we must   have ri - 0 where r‘’ (ri,
                                                        =
                                                                 ...,rn) otherwise for
                                                                   with ri > o. There-
     a subsequence, still denoted r‘’, we would have rv     - r*
     fore K= { rB r* ~ would be compact in P and thus W-1 (K) _ ~ p~, ~ -1 (r*) ~
     would be compact in 0 - 0), which contradicts the fact that p" ~ p* E ~.

     Vol. 7, n° 1-1990.
 16                               B. DACOROGNA AND J. MOSER


      Since cp is continuous on q with cp (x) = x, it follows from the fact that
 r i -> 0, that

 and hence by the continuity of ~r -1 that


 (3) implies then immediately (1). N
    We now have the following
      THEOREM 7. - Let k~0 and             Q satisfy (Hk’) with k’=max{                1,k}.
                        ~>0,/+ -,/~+~-        bounded and satisfying




 Then there exists cp E Diffk (Q) n Diffo (SZ) with cp (x)                =
                                                                              x on aS2 such that




 for every open set E c Q.
    Moreover if supp {f-      g} c Q, then supp {(p - id } c Q.
    Remark. - With the help of Theorem 7 it is easy to construct a volume
 preserving mapping of a convex polyhedron Q in (~n which permutes
 finitely many of its points, say Pl,          PN, in a prescribed manner and
                                                  ...,


 keeps     the    boundary pointwise fixed (see Alpern [A]). To do so one first
 constructs any diffeomorphism 03C8 of Q permuting the Pj in the desired
 manner. This diffeomorphism takes then the volume form i            dx 1... dxn
                                                                                      =

          .

                                                                      N
into                             We then apply the above theorem to Q- U Pj and
       j=i
 find a diffeomorphism cp with (p* r = (det V (p) T = fT where f = det V ~r -1,
 which keeps the boundary fixed. The desired volume preserving mapping
 is then    B)/.
    We next turn to the proof of Theorems 5 and 7. It can be reduced to
 the case of a cube, using a covering of Q by sets Q~ as described in the
 above definition. Using Lemma 1 of [M] one can construct a sequence of
 functions f~ > 0 such                and f~ + 1- f~ has support in Q~ or in

 mapping
                                  and satisfying

                                                     with
                                                                  03A9fj+1 dx.    Thus using the

                                                                                           if
 and q= 0 if                      we can
                                                         f~ + 1 into
                                1
                                  corresponding to the volume forms f * dx,
              defined in Pn. Therefore it will suffice to construct

                                             Annales de l’In.rtitut Henri Poincaré - Analyse non linéaire
                                 THE JACOBIAN DETERMINANT                             17

with                        c   Pn such that




for every open set E in Qn. Applying this argument to 5~~, j =1, 2, . ~ . , N
we obtain Theorems 5 and 7; Proposition 6 ensuring that                   ~~  cp~ ~
keeps the boundary ~~ pointwise fixed.
  Without further elaboration of this patching argument we formulate,
for the case of the cube, the precise conditions and statements needed for
the proof of Theorems 5 and 7. We have to distinguish between two cases
depending on whether SZ~ meets the boundary aSZ or not. This corresponds
to the cases                      or q = Q~ respectively. We also denote by

                                      if q~~ and q=0                We also
drop for simplicity the index n  in Pn.

  PROPOSITION 8. - Let                                          with either q= 0 or
                     =0~.
                   Assume that
    (i) Of,                            g,           bounded in P.



  (111~ supp t.~ g ~ c ys U q.                                                _




Then there exists cp E Diffk         (P)
                                 which extends as a homeomorphism of Qn
with cp (x) x for x E q with supp ( cp id ~ c Qn U q and
              =                                 -




for every open set E c Qn.
   If, in addition, f,                         then the above conclusions hold and
(p Diffk (Qn).
  E

Remark. - The additional statement is appropriate for the proof of
Theorem 5, where cp is Ck up to the boundary, while the first statement
fits the proof of Theorem 7.



           IV. AN ELEMENTARY PROOF OF PROPOSITION 8

  In this section we proceed with the proof of Proposition 8, thus com-
pleeting the proof of Theorems 5 and 7. The argument can be considered
as an analogue of the separation of variables for the differential equation
det V u = f where at each stage an ordinary differential equation has to be
solved. Moreover this differential equation admits an integral and can be
solved via the implicit function theorem. For this reason even the case of

Vol. 7, n° 1-1990.
18                        B. DACOROGNA AND J. MOSER



continuousf and g can be handled. In this sense the argument is quite
elementary though a bit tricky.
  To give an idea of the proof of Proposition 8, we first try to find cp as
a mapping preserving the line segments xi = a, i. e. we take cp of the form




with a function v monotone increasing in xl. The relation (3.3), when
                          n


applied to E = [o, a] X fl            leads for b j - a j -~ 0 to the equivalent
                         j=2
requirement


for every x’ = (X2, ... , xn) E Qn -1. Since g > 0 this equation defines v (x)
uniquely with v monotone in                 for x 1= 0 and         for x’ near
~Qn-1 (since f-g=O       for  x’ near           However   in order  to achieve
      for xl = 1 we need the condition




                                              av
for every x’ E         We note that v     ~         E   Ck (P), but no such assertion
                                              axl
                                                                                av
holds for the other derivatives. Observe also that

bounded, since f’, , f ‘ 1, g, g-l are bounded.
                                                                     ~x1(~x1)   ( ~        are




     We have therefore proved.

   PROPOSITION 9. - Iff and g satisfies the hypotheses of Proposition 8 as
well as (4 .1 ), then there exists cp satisfying all the assertions of Proposition 8.
Moreover cp preserves all line segments parallel to the xl-axis in Qn.
   In order to conclude the proof of Proposition 8, it therefore suffices to
transform g by an appropriate diffeomorphism 03C8 such that the condition
(4 ..1 ) is realized. This will be achieved in the following.
     PROPOSITION 10. - If 0    f, g E Ck (Qn) with supp ~, f ’- g ~ c Qn U q and


then there exists 03C8 E Diffk (Q") with supp { ’" - id} c Q", and gl E Ck (P)
such that




                                      Annales de l’Institut Henri Poincaré - Analyse non linéaire
                              THE JACOBIAN DETERMINANT                       19

for every open set E c Q" and

                                           ’   ~




   Remarks. -    (i)               near aQn, it follows from (4 . 3) that

                    aQ".
                     near

  (ii) In this proposition no restriction about the boundary behaviour
of f and g near q are required (such restrictions are only needed in
Proposition 9).
  (iii) The combination of Propositions 9 and 10 give immediately Propo-
sition 8.
Proof of Proposition 10. - We shall construct ~ as

and define g n = g and for s = 2, 3, ..., n


if k >_ 1, or equivalently




for every open set E c Qn. (The latter definition holds also for k = 0.)
  We shall construct cpn,    cps + 1 inductively in such a way that
                               ...   ,




where                xs) and x’ _ (xs + 1, ... ,
                       ...,                       i. e. such that integrals over
s-dimensional cubes            for j > s match. For s = n this corresponds
exactly to (4. 2) and for s =1 this is our desired assertion (4 . 4).
  We proceed by induction and assume that cpn,           ...,       are already
constructed so that (1) and (2) hold and that they agree with the identity
near the boundary. We therefore have




   To complete the induction we construct cps as follows


with


where the functions u and § will be defined below. Observe that cps
preserves all line segments Xj = Cj (j ~ s) parallel to the xs-axis.

Vol. 7, n° 1-1990.
20                          B. DACOROGNA AND J. MOSER


  We now define ç as a cut-off                                             with compact
support and satisfying




where E > 0 is chosen so that




   We next construct u. To derive the condition u has to satisfy, we first
set u (0, x’) o. We then would like (2) to be satisfied for (s -1 ) in place
              =



of s, i. e.




where gs _ 1 satisfies ( 1’). Integrating (7) over 0         xs a and denoting by
Qa = ~ xs E QS : 0 xs a }    we have




Using (3), (4) and the fact that u (0, x’) 0, we have
                                            =




                            with
Combining (8) and ( 1’) we have




(From now on we drop, for convenience, the dependence on x’, since x’
appears only as a parameter.)

  We can rewrite (9) in the following form. Letting bE Ia *
                                                            [-a 1+~,1

     ( xs e Qs: 0 xs + § (xs-1) b ) and defining
R:b={rEQs:Oxsa+Ç(r-1)b}
Rsab =
                        a
                                                                             l +E I +E  -a 1+~],


                                     Annales de l’Institut Henri Poincaré - Analyse non linéaire
                          THE JACOBIAN DETERMINANT                           21

 we have then that   (9) is equivalent to the equation


 [Note that since gs > 0 it follows from (10) that u(0) = 0 is the unique
 solution of (10) for
   We now claim that (10) has a unique solution
                                                       u~Ixs with u and
             Moreover, we have
 n




If we can achieve this, we shall have with the help of (3) and (4) constructed
cps with the appropriate conditions. Indeed the only thing which remains
to be checked is that             for every x near aQ". For          near
this is ensured by the cut-off function ç. For xj near 0 or                 this
follows from the assumption that f = g for x~ near 0 or 1               so that
by the uniqueness of u we have u = 0 if x~ is near 0 or 1. Thus cps has all
the desired properties and the induction is completed. The proposition
therefore is proved.
   It remains to show that we can find u solving (10) with the claimed
properties. To see that (10) has at most one solution           it is sufficient
to observe that G is monotone in b, indeed




The existence follows from the fact that for a E (o,1 ), the function
b - G (a, b) - F (a) has opposite signs at the end points of Ia. At the left
end point we have




where we have used (5) and (6). Similarly one shows that




Indeed, using the induction hypothesis (2) we have


Vol. 7, n° 1-1990.
22                                                              B. DACOROGNA AND J. MOSER


and therefore (15) is equivalent to




which is straightforward [as in (14)], therefore (15) is established.
  Collecting (13), (14) and (15) we have indeed shown the existence of a
unique u satisfying (10). Moreover u (o) = 0 and u(l)=O by construction
and from (2) respectively. Thus the
takes the interval [0, 1]into itself.
     We now show (12) and that                                                   E Ck.   Indee ~u exists          as is seen      by
                                      xs                                                           axs
differentiation of the relation (10), i. e.




where 2014                                             is given by (13) and
       ~~




                                                 au
Thus u and                                             E   Ck. To conclude the proof it remains therefore to show
                                                 axs
( 12). By ( 16) it suffices to prove that

For this purpose we need (5). From (13) and (17) we obtain




and if ç E [0, 1]we obviously have

        03B6Fa+(1-03B6)Ga~min{Fa,Ga}~min{min f                                                   , min gs} >Emaxgs.
The combination of the two                                                 inequalities proves (18) if ~ E [0,1 ] . If
~ E ( 1,1 + E] we obtain

where we have used (6) in the last inequality. Therefore (18) is established
and this concludes the proof of Proposition 1 0..
  We conclude with some comments. The diffeomorphism 03C8 of Proposi-
tion 10 is given in the form t~ cpn       ° ... ° cp2. The diffeomorphism cp
                                                                       =     °




                                                                           Annales de l’Institut Henri Poincaré - Analyse non linéaire
                         THE JACOBIAN DETERMINANT                          23

of Proposition 9, which we rename (pi, is constructed by the same process
as cps for s >_ 2. The diffeomorphism of Proposition 8 is then




In the above exposition we prefered to separate 03C61 from B)/
near the boundary and the boundary behaviour of cp is controlled by (pi

alone, which was obtained by integration along the lines parallel to the

  Incidentally, if we replace § by § * 1, the above construction of agrees
with that of [M], but then the condition                near aQn cannot be

ensured.
  Finally we point out that the Theorems 5 and 7 are, of course, valid
for arbitrary compact Ck’-manifolds, k’ = max (1, k) with boundaries satis-
fying the condition (Hk,).


                                 APPENDIX

  Here we want to verify the remarks about Condition (Hk) stated in
Section 3. We first start with
  PROPOSITION A .1. - Let Q be a domain with Ck boundary, k >_ 1. Then
Q can be covered by open domains S2~ such that there exists a diffeomorphism
  E Diffk    Qn) with det V E Ck
   Proof. - It suffices to consider the neighbourhood point, which we
may take as the origin in     Without loss of generality we may assume
that the boundary aS2 near y = 0 is locally given as the graph y 1 = b (y’),
y’ _ (y2,   yn) of a Ck function b with b (o) = o. Let E, b > o, we let SZ~ be
            ...,




Denoting the Lipschitz constant of b by L, we have I b (y’) ~ _ L E and we
choose E so small that 0 4 E L ~, hence



We then define the   mapping 03C8-1j: Qn -+ 03A9j by
              is a cut-off function satisfying § (t) =1 for t __ o, ~ (t) 0=




for     and~’ ~ _ 2. It is clear that           and maps Qn onto SZ~.
Moreover


which is Ck (Qn) and is positive since |b ç’     2 EL 03B4/2.
Vol. 7, n° 1-1990.
24                           B. DACOROGNA AND J. MOSER


  PROPOSITION A. 2. - Let Q be a domain with Lipschitz boundary, then
Q satisfies (Hk) for every k >_ 1.

  Proof. - We adopt the same notations as the above proposition. Again
we can assume    that the boundary is locally near the boundary point 0
represented by             b (o) = 0 and SZ~ is given by (A.I), but b is now
a Lipschitz function defined for                For convenience we extend b
as a Lipschitz function of ~n -1 ~ ~, with the same Lipschitz constant L.
  To define 03C8-1j we have to modify (A. 2) since b is not Ck.
Therefore we replace b by a mollified function                        (where
                                                               ac
R)   =
         {y e        0}) with c (0,0 y’)
                y1 > 0                ’ = b  ’ and |~c ~y1|~
                                          b (y’)
                                          =
                                                                The construction
                                                          - L. (The

of such a c is standard and is done below.) We then define




By construction this is a C~-diffeomorphism from                                 onto
with the property that
                             «                         ,   «




Thus Q satisfies (Hk) for every k.
  It therefore remains to construct c. Let                                      with compact




so that C E      ( I~ ~ ), c (o, v‘) = b (v’) and




Hence for >      0 we have
                                 I a~ _ L. .
                                 ay ~
  PROPOSITION A. 3. - A domain obtained by removing a finite number of
isolated points from a domain satisfying (Hk) satisfies also (Hk).

  Proof. - Since the question is a local one it suffices to consider a
punctured disk in Q, i. e.,

                         °




                                          Annales de l’Institut Henri Poincaré - Analyse non linéaire
                                THE JACOBIAN DETERMINANT                            25

  Instead of verifying (Hk) directly it is more convenient to use a "blow
up" map with the property that Q~ is mapped smoothly into a domain
with smooth boundary and by a mapping whose Jacobian is bounded
away from 0 and oo . This is obviously sufficient for the Property (Hk) to
hold near y = 0. Let p > 0, the mapping in question is defined by


with




This mapping has a Jacobian 1, since the volume form




(with     ] and dro the area on the (n - 1 )-dimensional sphere Sn -1 ) is
mapped into
                 n
                            with      ( since the mapping is radial
with I z                  and hence
  Remark. - Clearly, the class of domains Q satisfying (Hk) is much
larger than indicated by these cases. We can also admit for example lower
dimensional boundaries as well as cusps.


                                 ACKNOWLEDGEMENTS

  We would like to thank M. Struwe and E. Zehnder for interesting
discussions.


                                    REFERENCES

[AB]    M. ABRAHAM and R. BECKER, Classical Theory of Electricity and Magnetism, Glas-
           gow, 1950.
[A]     S. ALPERN, New Proofs that Weak Mixing is Generic, Inventiones Math., Vol. 32,
           1976, pp. 263-279.
[AK]    D. V. ANOSOV and A. B. KATOK, New Examples in Smooth Ergodic Theory. Ergodic
          Diffeomorphisms, Trudy Moskov Mat. Obsc. Tom, Vol. 23, 1970, Trans. Moscow
          Math. Soc., Vol. 23, 1970, pp. 1-35.
[B]     A. BANYAGA, Formes volume sur les variétés à bord, Enseignement Math., Vol. 20,
           1974, pp. 127-131.
[CL]    E. A. CODDINGTON and N. LEVINSON, Theory of Ordinary Differential Equations,
          MacGraw-Hill, New York, 1955.
[D]     B. DACOROGNA, A Relaxation Theorem and its Application to the Equilibrium of
          Gases, Arch. Ration. Mech. Anal., Vol. 77, 1981, pp. 359-385.

Vol. 7, n° 1-1990.
26                            B. DACOROGNA AND J. MOSER


[GS]   R. E. GREENE and K. SHIOHAMA, Diffeomorphisms and Volume Preserving Embed-
          dings of Non Compact Manifolds, Trans. Am. Math. Soc., Vol. 255, 1979, pp. 403-
          414.
[H]    L. HÖRMANDER, The Boundary Problems of Physical Geodesy, Arch. Ration. Mech.
          Anal., Vol. 62, 1976, pp. 1-52.
[LU]   O. A. LADYZHENSKAYA and N. N. URALTSEVA, Linear and Quasilinear Elliptic
          Equations, Academic Press, New York, 1968.
[L]    A. E. H. LOVE, Treatise on the Mathematical Theory of Elasticity, New York, 1944.
[MO]   G. H. MEISTERS and C. OLECH, Locally One to One Mappings and a Classical
          Theorem on Schlicht Functions, Duke Math. J., Vol. 30, 1963, pp. 63-80.
[M]    J. MOSER, On the Volume Elements on a Manifold, Trans. Am. Math. Soc., Vol. 120,
         1965, pp. 286-294.
[T]    L. TARTAR, private communication, 1979.
[Z]    E. ZEHNDER, Note on Smoothing Symplectic and Volume Preserving Diffeomor-
          phisms, Springer, Lect. Notes Math., Vol. 597, 1976, pp. 828-855.

                                              ( Manuscript received December 6, 1988.)




                                         Annales de l’Institut Henri Poincaré - Analyse non linéaire
