CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQUATION IN TWO DIMENSIONS

Transcription

1 CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQATION IN TWO DIMENSIONS MARTA LEWICKA AND MOHAMMAD REZA PAKZAD Abstract. This paper concerns the questions of flexibility and rigidity of solutions to the Monge- Ampère equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on degenerate i.e. flexible weak solutions that can be constructed through methods of convex integration à la Nash & Kuiper and establish the related h-principle for the Monge-Ampère equation in two dimensions. Contents 1. Introduction The weak determinant Hessian. 1.. Convex integration for the Monge-Ampère equation in two dimensions Rigidity versus flexibility Notation Acknowledgments. 5. The C 1 approximations - preliminary results The C 1 approximations - a proof of Theorem The C 1,α approximations - a proof of Theorem 1.1, preliminary results and some heuristics towards the proof of Theorem The C 1,α approximations - a step and a stage in a proof of Theorem The C 1,α approximations - a proof of Theorem Rigidity results for α > /3 - a proof of Theorem References 9 1. Introduction. In this paper we study the C 1,α solutions to the Monge-Ampère equation in two dimensions: 1.1 Det v := 1 curl curl v v = f in Ω R. Our results concern the dichotomy of rigidity vs. flexibility, in the spirit of the analogous results and techniques appearing in the contexts of: the low co-dimension isometric immersion problem [37, 3, 3, 4, 1], and the Onsager s conjecture for Euler equations [4, 1, 13, 9, 15]. In the first, main part of the paper we show that below the regularity threshold α < 1/7, the very weak C 1,α Ω solutions to 1.1 as defined below, are dense in the set of all continuous functions see Theorems 1.1 and 1.. These flexibility statements are a consequence of the convex integration h-principle, that is a method proposed in [17] for solving certain partial differential relations and that turns out to be applicable to our setting of the Monge-Ampère equation as well. 1

2 MARTA LEWICKA AND MOHAMMAD REZA PAKZAD Here, we directly adapt the iteration method of Nash and Kuiper [37, 3], in order to construct the oscillatory solutions to In the second part of the paper we prove that the same class of very weak solutions fails the above flexibility in the regularity regime α > /3. Our results are parallel with those concerning isometric immersions [3, 1, 38], Euler equations [9, 15], Perona-Malik equation [5, 6], the active scalar equation [], and should also be compared with results on the regularity of Sobolev solutions to the Monge-Ampère equation [38, 41, 31, 4] whose study is important in the context of nonlinear elasticity and with the rigidity results for the Monge-Ampère functions [, 3] The weak determinant Hessian. Let Ω R be an open set. Given a function v W 1, loc Ω, we define its very weak Hessian denoted by H in [1, 16] as: Det v = 1 curl curl v v, understood in the sense of distributions. A straightforward approximation argument shows that if v W, loc then L 1 loc Ω Det v = det v a.e. in Ω, where v stands for the Hessian matrix field of v. We also remark that this notion of the very weak Hessian is distinct from the distributional Hessian Det v = Det v denoted by Hu in [1, 16], that is defined through the distributional determinant Det: Det ψ = div ψ ψ 1 = ψ 1 ψ 1 1 ψ ψ 1 for ψ = ψ 1, ψ W 1,4/3 Ω, R. Contrary to the distributional Hessian, the very weak Hessian is not continuous with respect to the weak topology. Indeed, an example of a sequence v n W 1, Ω is constructed in [1], where Det v = 1 while v n converges weakly to. One consequence of the proof of our Theorem 1.1 below is that Det is actually weakly discontinuous everywhere in W 1, Ω see Corollary 6.. Here is our first main result: Theorem 1.1. Let f L 7/6 Ω on an open, bounded, simply connected Ω R. Fix an exponent: α < 1 7. Then the set of C 1,α Ω solutions to 1.1 is dense in the space C Ω. More precisely, for every v C Ω there exists a sequence v n C 1,α Ω, converging uniformly to v and satisfying: 1. Det v n = f in Ω. When f L p Ω and p 1, 7 6, the same result is true for any α < 1 1 p. In order to better understand Theorem 1.1, we point out a connection between the solutions to 1.1 and the isometric immersions of Riemannian metrics, motivated by a study of nonlinear elastic plates. Since on a simply connected domain Ω, the kernel of the differential operator curl curl consists of the fields of the form sym w, a solution to 1.1 with the vanishing right hand side f can be characterized by the criterion: 1.3 w : Ω R 1 v v + sym w = in Ω. 1 We remark that the recent work of De Lellis, Inaunen and Szekelyhidi [11] showed that the flexibility exponent 1 can be improved to 1 in the case of the isometric immersion problem in dimensions. We expect similar 7 5 improvement to be possible also in the present case of equation 1.1; this will be investigated in our future work.

3 CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQATION 3 The equation in 1.3 can be seen as an equivalent condition for the following 1-parameter family of deformations, given through the out-of-plane displacement v and the in-plane displacement w albeit with different orders of magnitude ε and ε : φ ε = id + εve 3 + ε w : Ω R 3 to form a nd order infinitesimal isometry bending, i.e. to induce the change of metric on the plate Ω whose nd order terms in ε disappear: φ ε T φ ε Id = oε. In this context, we take the cue about Theorem 1.1 from the celebrated work of Nash and Kuiper [37, 3], where they show the density of co-dimension one C 1 isometric immersions of Riemannian manifolds in the set of short mappings. Since we are are now dealing with the nd order infinitesimal isometries rather than the exact isometries, the classical metric pull-back equation: y g e = h, for a mapping y from Ω, h into R 3 equipped with the standard Euclidean metric g e, is replaced by the compatibility equation of the tensor T v, w = 1 v v + sym w with a matrix field A that satisfies: curl curl A = f: 1.4 T v, w = A Note that there are many potential choices for A, for example one may take A x = λxid with λ = f in Ω. Again, equation 1.4 states precisely that the metric φ ε T φ ε agrees with the given metric h = Id + ε A on Ω, up to terms of order ε. The Gauss curvature κ of the metric h satisfies: κh = κid + ε A = ε curl curl A + oε, while κ φ ε T φ ε = ε curl curl 1 v v + sym w + oε, so the problem 1.1 can also be interpreted as seeking for all appropriately regular out-of-plane displacements v that can be matched, by a higher order in-plane displacement perturbation w, to achieve the prescribed Gauss curvature f of Ω, at its highest order term. In this paper, similarly as in the isometric immersion case, we show that solutions to 1.4 are ample. We design a scheme inspired by the work of Nash and Kuiper, which pushes a short infinitesimal isometry, i.e. a couple v, w such that T v, w < A, towards an exact solution to 1.4 in successive small steps. Note that both y g e = y T y and the term v v in T v, w have a quadratic structure, which is crucial in the analysis of [37, 3] and also of this paper. Here, not only the presence of the linear term sym w in T u, w does not destroy the adaptation of the Nash-Kuiper scheme, but it actually allows for this construction to work. 1.. Convex integration for the Monge-Ampère equation in two dimensions. As we will see in section 4, Theorem 1.1 follows easily from the statement of our next main result: Theorem 1.. Let Ω R be an open and bounded domain. Let v C 1 Ω, w C 1 Ω, R and A C,β Ω, R sym, for some β, 1, be such that: 1.5 c > A 1 v v + sym w > c Id in Ω.

4 4 MARTA LEWICKA AND MOHAMMAD REZA PAKZAD Then, for every exponent α in the range: < α < min { 1 7, β }, there exist sequences v n C 1,α Ω and w n C 1,α Ω, R which converge uniformly to v and w, respectively, and which satisfy: 1.6 A = 1 v n v n + sym w n in Ω. The above result is the Monge-Ampère analogue of [1, Theorem 1], where the authors improved on the Nash-Kuiper method to obtain higher regularity within the flexibility regime. In our paper, we adapt the same methods to our problem. Convex integration was originally developed by Gromov [17] to deal with finding weak solutions of a differential inclusion Lux K in Ω, by investigating certain classes of sub-solutions, e.g. functions u that satisfy Lux conv K where the original constraint set K is replaced by its appropriate convex hull conv K. nder specific circumstances, it leads to establishing the density of very weak solutions, satisfying Lu L Ω, in the set of sub-solutions, and in case the constraint set is a continuum the regularity might be improved to Lu C Ω. Recently, these methods were applied in the context of fluid dynamics and yielded many interesting results for the Euler equations. In [1], De Lellis and Székelyhidi proved existence of weak solutions with bounded velocity and pressure, their non-uniqueness and the existence of energydecreasing solutions. In [13], using iteration methods à la Nash-Kuiper, the same authors proved existence of continuous periodic solutions of the 3-dimensional incompressible Euler equations, which dissipate the total kinetic energy. These results are to be contrasted with [9, 15], where it was shown that C,α solutions of the Euler equations are energy conservative if α > 1/3. There have been several improvements of [1, 13] since, towards a possible proof of the Onsager s conjecture which puts the Hölder regularity threshold for the energy conservation of the weak solutions to the Euler equations at C,1/3 [18, 19, 5, 6, 7, 8]. The stationary incompressible Euler equation has been studied in [8] where the existence of bounded anomalous solutions have been proved. The authors indicate that in dimensions, the relaxation set corresponding to the appropriate subsolutions is smaller than in the case of the evolutionary equations. In this context, we noticed a connection between our reformulation of the Monge-Ampère equation and the steady state Euler equation, which lead to our modest Corollary 4.1. In this paper we use a direct iteration method to construct exact solutions of 1.1. The recasting of the statement and the proof in the language of convex integration might shed more light on the structure of the Monge-Ampère equation, but it would not improve the results and therefore we do not address this task. We note, however, that constructing Lipschitz continuous piecewise affine approximating solutions to 1.6 for A is quite straightforward and could be used to prove a convex integration density result via the Baire category method as was done in [1] for the Euler equations see also Figure.1 and the corresponding explanation Rigidity versus flexibility. The flexibility results obtained in view of the h-principle are usually coupled with the rigidity results for more regular solutions. Rigidity of isometric immersions of elliptic metrics for C 1,α isometries [3, 1] with α > /3, or the energy conservation of weak solutions of the Euler equations for C,α solutions with α > 1/3, are results of this type. For the Monge-Ampère equations, we recall two recent statements regarding solutions with Sobolev regularity: following the well known unpublished work by Šverák [41], we proved in [31] that if

5 CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQATION 5 v W, Ω is a solution to 1.1 with f L 1 Ω and f c > in Ω, then in fact v must be C 1 and globally convex or concave. On the other hand, if f = then [38] likewise v C 1 Ω and v must be developable see also [, 3, 4]. A clear statement of rigidity is still lacking for the general f, as is the case for isometric immersions, where rigidity results are usually formulated only for elliptic [1] or Euclidean metrics [38, 34, 4]. In this paper, we prove the rigidity properties of solutions to 1.1 in the Hölder regularity context when f. Namely, we prove: Theorem 1.3. Let Ω R be an open, bounded domain and let: 3 < α < 1. If v C 1,α Ω is a solution to Det v = in Ω, then v must be developable. More precisely, for all x Ω either v is affine in a neighbourhood of x, or there exists a segment l x joining Ω on its both ends, such that v is constant on l x. We also announce the following parallel rigidity result for f c >, that will be the subject of the forthcoming paper [3]: Theorem 1.4. Let Ω R be an open, bounded domain and let: 3 < α < 1. If v C 1,α Ω is a solution to Det v = f in Ω, where f is a positive Dini continuous function, then v is convex. In fact, it is also an Alexandrov solution to det v = f in Ω. In proving Theorem 1.3, we use a commutator estimate for deriving a degree formula in Proposition 7.1. Similar commutator estimates are used in [9] for the Euler equations and in [1] for the isometric immersion problem; this is not surprising, since the presence of a quadratic term plays a major role in all three cases, allowing for the efficiency of the convex integration and iteration methods. Let us also mention that it is still an open problem which value of α is the critical value for the rigidity-flexibility dichotomy, and it is conjectured to be 1/3, 1/ or / Notation. By R sym we denote the space of symmetric matrices, and by R sym,> we denote the cone of symmetric, positive definite matrices. The space of Hölder continuous functions C k,α Ω consists of restrictions of functions f C k,α R to Ω R. Then, the C k Ω norm of such restriction is denoted by f k, while its Hölder norm C k,α Ω is f k,α. By C > we denote a universal constant which is independent of all parameters, unless indicated otherwise Acknowledgments. The authors would like to thank Camillo De Lellis for discussions about this problem. This project is based upon work supported by, among others, the National Science Foundation. M.L. was partially supported by the NSF grants DMS and DMS M.R.P. was partially supported by the NSF grant DMS A part of this work was completed while the authors visited the Forschungsinstitut für Mathematik at ETH Zurich, Switzerland. The institute s hospitality is gratefully acknowledged.. The C 1 approximations - preliminary results. In this and the next section we prove a weaker version of the result in Theorem 1.. Namely:

6 6 MARTA LEWICKA AND MOHAMMAD REZA PAKZAD Theorem.1. Let Ω R be an open and bounded domain. Let v C Ω, w C Ω, R and A C Ω, R sym be such that:.1 c > A 1 v v + sym w > c Id in Ω. Then there exist sequences v n C 1 Ω and w n C 1 Ω, R which converge uniformly to v and w respectively, and which satisfy:. A = 1 v n v n + sym w n in Ω. We start with a series of preliminary lemmas whose details we provide for the sake of completeness. The first lemma is an observation in convex integration, pertaining to solving an appropriate differential inclusion to be used for constructing the 1-dimensional oscillatory perturbations in v n and w n. As always, C > is a universal constant, independent of all parameters, in particular independent of the function a below. Lemma.. Let a C Ω be a nonnegative function on an open and bounded set Ω R. There exists a smooth 1-periodic field Γ = Γ 1, Γ C Ω R, R such that the following holds for all x, t Ω R: Γx, t + 1 = Γx, t,.3 1 tγ 1 x, t + t Γ x, t = ax, together with the uniform bounds:.4 Γ 1 x, t + t Γ 1 x, t Cax, Γ x, t + t Γ x, t Cax, x Γ 1 x, t C ax, x Γ x, t C ax ax. Proof. Firstly, note that there exists a smooth 1-periodic function γ C R, R, such that for all t R there holds: γt + 1 = γt, 1 γt dt =,, { γt P := s 1, s R 1 } ; s 1 + s = 1, s 1. Existence of γ is a consequence of the fundamental lemma of convex integration, since the intended average, lies in the convex hull of the parabola P see Figure.1. Indeed, one can take: γt = cosπt, cos4πt P. It is now enough to ensure that t Γ 1 = axγ 1 x and t Γ = ax γ x to obtain.3. Namely: Γ 1 x, t = ax π sinπt, Γ x, t = ax sin 4πt. 4π We see directly that the bounds in.4 hold. To compare with the problem of isometric immersions, note that in that context, a 1-dimensional convex integration lemma is similarly proved in [4, Figure, p. 11], where instead of a parabola, the constraint set consists of a full circle. We will also need a special case of [1, Lemma 3] about decomposition of positive definite symmetric matrices into rank-one matrices.

7 CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQATION 7 Figure.1. The parabola P in the 1d convex integration problem of Lemma.. Lemma.3. There exists a sufficiently small constant r > such that the following holds. For every positive definite symmetric matrix G Rsym,>, there are three unit vectors {ξ k R 3 } 3 and three linear functions {Φ k : R sym R} 3, such that: for any G R sym we have 3.5 G R sym G = Φ k Gξ k ξ k, and that each Φ k is strictly positive on the ball BG, rg R sym with radius rg = Proof. 1. First, assume that G = Id. Set: ζ 1 ζ 1 = 1 1 ζ 1 = 1 1 +, +, ζ = 1 1 +, +, ζ 3 = 1 1, 1. In order to check that the following matrices form a basis of the 3-dimensional space R [ [ we validate that: det 1 1 ], ζ ζ = ], ζ 3 ζ 3 = 1 r G 1/. sym: [ Consequently, there exist linear mappings {Ψ k : R sym R} 3 yielding the unique decomposition: 3.6 G R sym G = Ψ k Gζ k ζ k. Now, since Id = 3 4 ζ 1 ζ ζ ζ + 1 ζ 3 ζ 3, the continuity of each function Ψ k implies its positivity in a neighborhood of Id of some appropriate radius r.. For an arbitrary G R sym,> we set: 1 k = ξ k = G 1/ ζ k G1/ ζ k and Φ k G = G 1/ ζ k Ψ k G 1/ Then, in view of.6 we obtain.5: G R sym G = G 1/ 3 Ψ k G 1/ GG 1/ ζ k ζ k G 1/ = GG 1/. 3 Φ k Gξ k ξ k, ],

8 8 MARTA LEWICKA AND MOHAMMAD REZA PAKZAD Finally, if G G < rg then G 1/ GG 1/ Id G 1/ G G < r, and so indeed Φ k G >, since Ψ k G 1/ GG 1/ >. The above result can be localized in the following manner, similar to [4, Lemma 3.3]: Lemma.4. There exists sequences of unit vectors {η k R } and nonnegative smooth functions {φ k Cc R sym,>}, such that:.7 G R sym,> G = φ k G η k η k and that: i For all G R sym,>, at most N terms of the sum in.7 are nonzero. The constant N is independent of G. ii For every compact K R sym,>, there exists a finite set of indices JK N such that φ k G = for all k JK and G K. Proof. 1. Let r be as in Lemma.3 and additionally ensure that:.8 r < 1 8. Recall that for each G R sym,> we have denoted rg = r and that BG, rg R G 1/ sym,>. We first construct a locally finite covering of R sym,> with properties corresponding to i and ii. Since the set R sym,> is a cone, we have:.9 R sym,> = k C, where C = {G Rsym,>; 1/ G 1}. k Z The collection {BG, rg} G C covers the sector C by balls that have uniformly bounded radii: G rg r r. Hence, by the Besicovitch covering theorem, it has a countable subcovering G = σ σ=1 Gσ, consisting of σ N countable families {G σ}σ σ=1 of pairwise disjoint balls. Note that for all c > one has: rcg = crg and so: BcG, rcg = cbg, rg. Consequently, the collections Gk σ = {k B; B G σ } each consist of countably many pairwise disjoint balls, and G k = σ σ=1 Gσ k is a covering of the dilated sector k C, for every k Z. Define:.1 σ = 1... σ Geven σ = Gk σ and Godd σ = Gk σ. k k+1 Clearly, in view of.9, the σ families in.1 form a covering of R sym,>, namely: G = σ σ=1 G σ even We now prove that each of the families in G consists of pairwise disjoint balls. We argue by contradiction. Assume that: σ σ=1 G σ odd. G BG 1, rg 1 BG, rg for some BG 1, rg 1 G σ k 1, BG, rg G σ k. Without loss of generality we may take k 1 = and k = k 1, so that: 1 G 1 1 and k 1 G k.

9 CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQATION 9 This yields a contradiction with.8, in view of: k 1 1 G G 1 G G 1 G G + G G 1 1 r rg + rg 1 = r G + G 1 r k + 1, G 1/ G 1/ Note that G can be assumed locally finite, by paracompactness. We write: G = {B i = BG i, rg i } i=1 and let {θ i Cc B i } i=1 be a partition of unity subordinated to G. For each i N, let {ξ k,gi } 3 and {Φ k,g i } 3 be the unit vectors and the linear functions as in Lemma.3. Then: G R sym,> G = θ i GG = 3 θ i GΦ k,gi Gξ k,gi ξ k,gi, i N i N and we see that.7 holds by taking: η i,k = ξ k,gi and φ i,k = θ i Φ k,gi. Since supp φ i,k B i and since each G belongs to at most σ balls B i, we see that i holds with N = 6σ. On the other hand, condition ii follows by local finiteness of G. 3. The C 1 approximations - a proof of Theorem.1. The first result in the approximating sequence construction is what corresponds to a step in Nash and Kuiper s terminology. Proposition 3.1. Let Ω R be an open and bounded set. Given are: functions v C Ω and w C Ω, R, a nonnegative function a C Ω, and a unit vector η R. Then, for every λ > 1 there exist approximations ṽ λ C Ω and w λ C Ω, R satisfying the following bounds: 1 ṽ 1 λ ṽ λ + sym w λ v v + sym w + a η η 3.1 C λ a a + C v + λ a, 3. ṽ λ v C λ a and w λ w C λ a a + v, 3.3 x Ω ṽ λ x vx Cax + C λ a, w λ x wx Cax a + v + C a a + v + a v. λ Proof. sing the 1-periodic functions Γ i from Lemma., we define ṽ λ and w λ as λ-periodic perturbations of v, w in the direction η: 3.4 ṽ λ x = vx + 1 λ Γ 1x, λx η w λ x = wx 1 λ Γ 1x, λx η vx + 1 λ Γ x, λx ηη.

10 1 MARTA LEWICKA AND MOHAMMAD REZA PAKZAD The error estimates in 3. follow immediately from.4. The pointwise error estimates 3.3 follow from.4 in view of: ṽ λ x = vx + 1 λ xγ 1 x, λx η + t Γ 1 x, λx ηη, w λ x = wx 1 λ vx xγ 1 x, λx η t Γ 1 x, λx ηη vx 1 λ Γ 1x, λx η vx Finally, we compute: 1 ṽ λx ṽ λ x 1 vx vx + 1 λ η xγ x, λx η + t Γ x, λx ηη η. = 1 λ sym vx x Γ 1 x, λx η + t Γ 1 x, λx ηsym vx η + 1 tγ 1 x, λx η η η and: + 1 λ tγ 1 x, λx ηsym η x Γ 1 x, λx η + 1 λ xγ 1 x, λx η x Γ 1 x, λx η, sym w λ x sym wx = 1 λ sym vx x Γ 1 x, λx η t Γ 1 x, λx ηsym vx η 1 λ Γ 1x, λx η vx + 1 λ sym η x Γ x, λx η + t Γ x, λx ηη η. We see that the terms in boxes cancel out, while the terms in double boxes add up to ax η η in virtue of.3. Consequently: 1 1 ṽ λx ṽ λ x + sym w λ x vx vx + sym wx + ax η η = 1 t Γ 1 x, λx ηsym η x Γ 1 x, λx η Γ 1 x, λx η vx + sym η x Γ x, λx η λ + 1 λ xγ 1 x, λx η x Γ 1 x, λx η, which implies 3.1 in view of the bounds in.4. We now complete the stage in the approximating sequence construction. Proposition 3.. Let Ω R be an open and bounded domain. Let v C Ω, w C Ω, R and A C Ω, R sym be such that the deficit function D defined below is positive definite in Ω: 3.5 c > D = A 1 v v + sym w > cid in Ω.

11 CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQATION 11 Fix ε >. Then there exist ṽ C Ω and w C Ω, R such that the new deficit D is still positive definite, and bounded by ε together with the error in the approximations ṽ, w, namely: 3.6 c > D = A 1 ṽ ṽ + sym w > cid in Ω, 3.7 D < ε and ṽ v + w w < ε. Moreover, we have the following uniform gradient error bounds: 3.8 ṽ v CN 1/ D 1/ and w w CN v + D 1/ D 1/, where the constant N N is as in Lemma.4. Proof. 1. Note that the image D Ω is a compact subset of R sym,>. By Lemma.4 and rearranging the indices, if needed, so that JD Ω = {1... N} in ii, we get: 3.9 x Ω N Dx = b k x η k η k where b k = φ k D C Ω. Let now a k = 1 δ 1/ b k, with δ > so small that: N 3.1 D a k η k η k = δd and δ D < ε. We set v 1 = v, w 1 = w. For k = 1... N we inductively define v k+1 C Ω and w k+1 C Ω, R, by means of Proposition 3.1 applied to v k, w k, a k, η k and with λ k > 1 sufficiently large as indicated below. We then finally set ṽ = v N+1 and w = w N+1.. To prove the estimates , we start by observing that since by Lemma.4 i at most N terms in the expansion 3.9 are nonzero, there holds: 3.11 N a k x N b k x N 1/ N 1/ N b k x 1/ 1/ 1/ = N Trace Dx Dx 1/ CN 1/ D 1/. Further, by 3.1 and 3.1: 1 D = D ṽ ṽ + sym w 1 v v + sym w N 1 = D v 1 k+1 v k+1 + sym w k+1 v k v k + sym w k = D = δd + N a k η k η k N 1 v k+1 v k+1 + sym w k+1 1 v k v k + sym w k + a k η k η k N 1 O ak a k + a k + a k v k. λ k

12 1 MARTA LEWICKA AND MOHAMMAD REZA PAKZAD Choosing at each step λ k sufficiently large with respect to the given a k and the already generated v k, we may ensure the smallness of the error term in the right hand side above and hence the positive definiteness of D in 3.6, because of the uniform positive definiteness of: δd > cδid in Ω. Likewise, the first inequality in 3.7 follows already when the error is smaller than ɛ/. The same reasoning proves the error bounds on ṽ v and w w in 3.7, in view of 3.: ṽx vx = wx wx = N v k+1 x v k x = N w k+1 x w k x = N 1 O a k, λ k N 1 O ak + a k v k. λ k 3. To obtain the first error bound in 3.8, use 3.3 and 3.11: N N N 1 ṽx vx v k+1 x v k x C a k x + O a k λ k CN 1/ D 1/, where again, by adjusting λ k at each step, we ensure the controllability of the error term with respect to the nonnegative quantity N 1/ D 1/. Likewise: k 1 k = 1... N v k x vx + v i+1 x v i x v + CN 1/ D 1/, and obviously by 3.11: i=1 i=1 k 1 a k x a i x CN 1/ D 1/, which yield by 3.11: N a k x a k + v k C v + N 1/ D 1/ N a k x CN v + D 1/ 1/ D. Consequently and by 3.3, there follows the last gradient error bound in 3.8: N wx wx w k+1 x w k x C N a k x a k + v k + N 1 O ak a k + a k v k + a k v k λ k CN v + D 1/ 1/ D. This concludes the proof of the stage approximation construction. We now finally give: Proof of Theorem.1.

13 CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQATION Fix ε >. It suffices to construct v C 1 Ω and w C 1 Ω, R such that: 3.1 A = 1 v v + sym w in Ω and: 3.13 v v + w w < ε. The exact solution v, w of 3.1 will be obtained as the C 1 limit of sequences of succesive approximations {v k C Ω, w k C Ω, R } k=, where v and w are given in the statement of the Theorem and satisfy.1, while v k+1 and w k+1 are defined inductively by means of Proposition 3. applied to v k, w k and ε k >, under the following requirement: 3.14 ε k < ε and ε 1/ k < 1. In agreement with our notation convention, we introduce the k-th deficit D k, which is positive definite by 3.6: k D k := A 1 v k v k + sym w k C Ω, R sym,>. By 3.7 it follows that: k 1 k 1 k 1 v k v + w k w v i+1 v i + w i+1 w i < ɛ i < i= i= i=1 ɛ i. Thus, {v k } k= and {w k} k= converge uniformly in Ω, respectively, to v and w which satisfy 3.13 in view of We now show that this convergence is in C 1. Indeed, by 3.7: D k < ɛ k, so by 3.8: 3.15 v k+m v k m 1 i=k v i+1 v i CN 1/ m 1 i=k i=1 D i 1/ CN 1/ m 1 In particular, in view of 3.14 the sequence { v k } k= is bounded, so we further have: 3.16 w k+m w k m 1 i=k m 1 CN w i+1 w i i=k vi + D i 1/ Di 1/ CN m 1 i=k i=k ε 1/ i, ε 1/ i. where the constant C is independent of k and m. Through the above assertions 3.15 and 3.16, in view of the second condition in 3.14, we conclude that {v k } and {w k} k= are Cauchy sequences that converge in C 1 Ω to v C 1 Ω and w C 1 Ω, R, respectively. Finally: A 1 v v + sym w = lim k D k lim k ε k = implies 3.1 and completes the proof of Theorem.1.

14 14 MARTA LEWICKA AND MOHAMMAD REZA PAKZAD Remark 3.3. In addition to the uniform convergence postulated in Theorem.1, one also has: n v n v + CN 1/. sing notation as in the proof above and recalling 3.15 and 3.14, this bound follows by: v v = lim v k v lim k k CN 1/ k 1 i= ɛ 1/ i CN 1/. 4. The C 1,α approximations - a proof of Theorem 1.1, preliminary results and some heuristics towards the proof of Theorem 1.. Theorem 1.1 follows easily from Theorem 1., that will be proved in the next section. Proof of Theorem 1.1. Since C 1 Ω is dense in C Ω, we may without loss of generality assume that v C 1 Ω. Set w = and A = λ + cid C,β Ω, Rsym where c is a constant and λ is constructed as follows. Extend the function f to f L p Ω ɛ defined on an open smooth set Ω ɛ Ω and solve: λ = f in Ω ɛ, λ = on Ω ɛ. Since λ W,p Ω ɛ, then Morrey s Theorem implies that λ C,β Ω for every β, 1 when p, and for β = p when p 1,. Also, for c large enough, condition 1.5 on the positive definiteness of the defect is satisfied. On the other hand: curl curla = λ + c = f, so the result follows directly from Theorem 1., since /p 1 7 is equivalent to p 7 6. Our next simple Corollary concerns the steady-state Euler equations with the exchanged roles of the given pressure q and the unknown forcing term g. Corollary 4.1. Let Ω R be an open and bounded domain. Let q C,β Ω for some β, 1 and fix ɛ >. Then for every exponent α in the range: < α < min{ 1 7, β }, there exist sequences {u n C,α Ω, R } n=1 and {g n C,α Ω} n=1 solving in Ω the following system: 4.1 divu n u n q = g n, div u n =, and such that u n = v n and g n = curl w n, where each v n C 1,α Ω and w n C 1 Ω, R, while the sequence {v n } n=1 is dense in C Ω and w n < ɛ for every n 1. Proof. As before, since C 1 Ω is dense in C Ω, it is enough to take v C 1 Ω and approximate it by a sequence {v n C 1,α Ω} n=1 with the properties as in the statement of Corollary. Let w = and let c > be a sufficiently large constant, so that q + cid v v is strictly positive definite in Ω. By Theorem 1., there exists sequences v n C 1,α Ω and w n C 1,α Ω, R which converge uniformly to v and w and which satisfy: q + cid = v n v n + sym w n in Ω. Taking the cofactor of both sides in the above matrix identity, we get: q + cid = v n v n + cof sym w n. Taking the row-wise divergence, we obtain 4.1 with u n = v n and g n = curl w n, since div cof w n =, while div cof w n T = curl wn.

15 CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQATION 15 Towards a proof of Theorem 1. we will derive a sequence of approximation results, and then combine them with Theorem.1 in section 6. For completeness, we first prove a simple, useful: Lemma 4.. Let Ω R be an open and bounded domain. Given are functions: f C N Ω, R n and ψ C R n, R m. Then: k =... N ψ f k M f k, where the constant M > depends on the dimensions n, m, the differentiability order N, the domain Ω, the norm ψ N on the compact set f Ω and the norm f, but it does not depend on the higher norms of f. Proof. The statement is obvious for k =. Fix k {1... N} and let m = m 1,, m k be any k-tuple of nonnegative integers such that k i=1 im i = k. Denoting m = k i=1 m i and using the interpolation inequality [1]: i = 1... k f i M f 1 i/k f i/k k, valid with a constant M > depending on n, N and Ω, we get: k i=1 i f m i M m k i=1 f m i im i /k f im i/k k = M m f m 1 f k. with m := m m j. Calculating the partial derivatives in k ψ f by the Faà di Bruno formula, gives hence the desired estimate: k ψ f M k i f m i M f k. m Above, the summation extends over all multiindices m = m 1,, m k with the properties listed at the beginning of the proof. We recall the following estimates which have been proved in [1]: Lemma 4.3. Let ϕ C c B, 1, R be a standard mollifier supported on the ball B, 1 R n, that is a nonnegative, smooth and radially symmetric function such that R n ϕ = 1. Denote: l, 1 ϕ l x = 1 l n ϕx l. Then, for every f, g C R n there holds: k, j i=1 f ϕ l k+j C l k f j k f ϕ l f k C l k f α, 1] f ϕ l f Cl α f,α α, 1] f ϕ l 1 C l 1 α f,α 4.6 k α, 1] fg ϕ l f ϕ l g ϕ l k C l k α f,α g,α, with the uniform constants C > depending only on the smoothness exponents k, j, α.

16 16 MARTA LEWICKA AND MOHAMMAD REZA PAKZAD Proof. The estimate 4. follows directly from the definition of convolution. To prove 4.3, note that for every x R n : k f ϕ l fx = ϕ l y k fx y k fx dy R n = k ϕ l y fx y fx dy = 1 1 R n l k R n l n k ϕ y l fx y + r x y dy = 1 1 l k R n l n k ϕ y l r xydy C l k sup r x y C x R n ; y <l l k f, where we integrated by parts, discarded the contribution with the symmetric term fx y which integrates to, and estimated the Taylor s formula remainder term: The proof of 4.4 follows similarly by: r x y = fx y fx fx y = f O y. k α fx y fx f ϕ l fx = ϕ l y y R n y α dy Cl α f,α R n ϕ l ydy Cl α f,α, while for 4.5 we write: f ϕ l x = fx y 1 R n l n+1 ϕ l y l dy 1 fx y fx y α 1 = l R n y α l l n ϕ l y l dy Cl α 1 1 f,α R n l n ϕ l y l dy C l 1 α f,α. Finally, for the crucial commutator estimate 4.6 we refer to [1, Lemma 1]. A heuristic overview of the next two sections. Let us attempt to follow the construction in sections and 3, but with the goal of controlling the higher Hölder norms of the iterations, and hence also quantifying the growth of the C norms of v, w. Let A C Ω, R sym be the target matrix field and let v 1 C Ω, w 1 C Ω, R be given at an input of a stage. As in Proposition 3., we decompose the defect D = A 1 v 1 v 1 + sym w 1 into a linear combination N a k η k η k of rank-one symmetric matrices with smooth coefficients given by Lemma.4. We define: v k+1 x = v k x + 1 λ Γ 1x, λx η k, w k+1 x = w k x 1 λ Γ 1x, λx η k v k x + 1 λ Γ x, λx η k η k.

17 CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQATION 17 This yields, by applying Lemma 4. to ψx = x and f = a k : m :... 3 m v k+1 m v k C a k i λ j 1, m :... m w k+1 m w k C i+j=m; i,j m i+j=m; i,j m + C a k i λ j 1 i+j+s=m; i,j,s m a k i λ j 1 s+1 v k, On the other hand, applying Lemma 4. to ψ = φ k defined in Lemma.4 and to f = D, we get: k : 1... N a k C v w A. Now, in order to control the C 1,α norm of v N+1 through interpolation, we need to control the norm v N+1, which in turn depends on a k. The above estimate shows that at the end of each stage, the C norm of a k is determined by the C 3 norms of the given v 1 and w 1 of the previous stage. Further, the C norm of w N+1 is only controlled by the C 3 norm of v and also of all the a k s. One might hope to control a k 3 if the deficit D is small enough, but the dependence of w N+1 on v 3 cannot be easily bypassed. Recalling that we need infinitely many stages in the construction, this implies that a direct estimate cannot be obtained in this manner, unless we deal with analytic data similar as in [4]. We thus need to modify the previous simplistic approach. The appropriate modification is achieved by introducing a mollification before each stage. Indeed, we note that the loss of derivatives in the above estimates is accompanied by a similar gain in the powers of λ, in a manner that the total order of derivatives, plus the order of powers needed to control v N+1 and w N+1 is constant. If we replace v 1 and w 1 by their mollifications on the scale l λ 1, each derivative loss can be estimated by one power of λ, and v and w will control v N+1 and w N+1. One problem still remains to be taken care of: does the deficit D decrease at the end of each stage? As the calculation below will show, a mollification of order λ 1 does not suffice to this end, and we need to mollify at a larger scale of l > λ 1. This is indeed how we want proceed. In practice, we let the mollification scale to be l = δ/m and we treat v like a, controlling its j-th norm by δl j. We then sacrifice one l in order to gain one δ; instead of v ϕ l j C v 1 l j, we use v ϕ l j C v ll j, choosing l such that l v < δ and obtaining the desired bound 5.. Finally, note that the loss of N powers of λl > 1 in the control of the C norms at the end of each stage, is the main reason why the described scheme does not deliver better than C 1,1/7 estimates, even for the optimal N = 3 from the decomposition in Lemma The C 1,α approximations - a step and a stage in a proof of Theorem 1.. In this section, we develop the approximation technique that will be used for a proof of Theorem 1. in the next section. The first result is a variant of Proposition 3.1 in which we accomplish the step of the Nash-Kuiper construction with extra estimates on the higher derivatives. Proposition 5.1. Let Ω R be an open, bounded set. Given are functions: v C 3 Ω, w C Ω, R, a nonnegative function a C 3 Ω and a unit vector η R. Let δ, l, 1 be two parameter constants such that: 5.1 a m δ l m m =... 3, and v m δ m = 1,. lm

18 18 MARTA LEWICKA AND MOHAMMAD REZA PAKZAD Then for every λ > 1/l there exist approximating functions ṽ λ C 3 Ω and w λ C Ω, R satisfying the following bounds, with a universal constant C > independent of all parameters: 5. 1 ṽ 1 λ ṽ λ + sym w λ v v + sym w + a η η C δ λl, 5.3 ṽ λ v m Cδλ m 1 m =... 3, 5.4 w λ w m Cδλ m v Proof. We define ṽ λ, w λ as in the proof of Proposition 3.1: ṽ λ x = vx + 1 λ Γ 1x, λx η, m =.... w λ x = wx 1 λ Γ 1x, λx η vx + 1 λ Γ x, λx ηη. Firstly, 5. follows immediately from 3.1 in view of 5.1, because λl > 1: 1 λ a a + 1 v + λ a δ δ λ l + 1 δ λ l 3δ λl. To check 5.3, we compute directly as in Lemma.: m ṽ λ v C λ m Γ 1 x, λx η C λ i+j=m; i,j m a j λ j C λ by 5.1 and noting again λl > 1. Similarly: m w λ w C m Γ x, λx η + m Γ 1 x, λx η v λ C a i λ j + a i λ j v s λ C λ C λ i+j=m, i,j m m i=1 m i=1 δ l i λm i + i+s m, i,s m i+j+s=m, i,j,s m δ l i λm i+s δ l s + m i= δ l i λm i v Cδλ m v, δ l i λm i Cδλ m 1 i+j=m, i,j m δ l i λj v where we applied Lemma 4. to ψx = x and f = a in view of 5.1 yielding a 1, so that: a i C a i Cδ/l i. This achieves 5.4 and completes the proof of Proposition. We now accomplish the stage in the Hölder regular approximation construction. Proposition 5.. Let Ω R be an open, bounded domain. Let v C Ω, w C Ω, R and A C,β Ω, R sym for some β, 1, be such that the deficit D is appropriately small: 5.5 D = A 1 v v + sym w, < D < δ 1. Then, for every two parameter constants M, σ satisfying: 5.6 M > max{ v, w, 1} and σ > 1,

19 CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQATION 19 there exists ṽ C Ω and w C Ω, R such that the following error bounds hold for ṽ, w and the new deficit D = A 1 ṽ ṽ + sym w : 5.7 D A,β C D β/ + 1 σ D, 5.8 ṽ v 1 C D 1/ and w w 1 C1 + v D 1/, 5.9 ṽ CMσ 3 and w C1 + v Mσ 3. M β The constant C > is universal and independent of all parameters. Proof. Analogously to [1, Proposition 4], the proof is split into three parts. 1. Mollification. Let ϕ C c B, 1 be the standard mollifier in d, as in Lemma 4.3. Since v, w and A can be extended on the whole R, with all their relevant norms increased at most C times C depends here on the curvature of the boundary Ω, we may define: v = v ϕ l, w := w ϕ l, A := A ϕ l with l = D 1/ M < 1. Applying Lemma 4.3 and noting 5.6, we immediately get the following uniform error bounds for v, w, A and for the induced deficit D = A 1 v v + sym w : v v 1 + w w 1 Cl v + w C D 1/, A A Cl β A,β, 5.1 D m D ϕ l m + v ϕ l v ϕ l v v ϕ l m C l m D + C l m v C l m D m = In the proof of the last inequality above, we used 4.6 with the Hölder exponent α = 1. We note that so far we have simply exchanged the lower regularity fields v, w, A with their smooth approximations, at the expense of the error that, as we shall see below, is compatible with the that postulated in The following estimate, however, reflects the advantage of averaging through mollification that results in the control of C 3 norm of v by the C norm: 5.11 m = 1, v m v m+1 C l m 1 v C D 1/ lm 1, where again we used Lemma 4.3 and 5.6. Note that the scaling bound 5.11 is consistent with the second requirement in 5.1 of Proposition 5.1. We also record the following simple bound: 5.1 w C w CM.. Modification and positive definiteness. Contrary to the stage construction in the proof of Proposition 3., we do not know whether the original defect D and hence the induced defect D is positive definite, so that Lemma.4 could be used. In any case, we need to keep the number of terms in the decomposition 3.9 into rank-one matrices as small as possible. We now further modify w in order to use the optimal decomposition in.5. Let r be as in Lemma.3 and define: w = w D + D id, D = A 1 r v v + sym w.

20 MARTA LEWICKA AND MOHAMMAD REZA PAKZAD Clearly, by 5.1 we get: 5.13 w w C D + D C D. Note now that: D x = D + D Id + Dx = D + D r Id + r r D + D D By Lemma.3 we may apply.5 to the scaled defect G = Id D x = where 3 { a k = D + D r Φ k G 1/ } 3 D + D r Φ k Gxξ k ξ k = x Ω. r D + D D and arrive at: 3 a k xξ k ξ k x Ω, are positive smooth functions on Ω. We claim that: 5.15 k = m =... 3 a k m C l m D 1/. Indeed, for m = this inequality follows directly by D C D. For m = we use Lemma 4. to each ψ = Φ 1/ k and f = G, where noting that G C and recalling 5.1 yields: a k m D + D 1/ C G m r 5.16 C D + D 1/ C + C D + D 1/ + r D + D D m 1 1 D + D 1/ l m D C D 1/ + 1 l m D 1/ and hence achieves Note that the scaling bound 5.15 is consistent with the first requirement in 5.1 of Proposition Iterating the one-dimensional oscillations. We set v 1 = v, w 1 = w and inductively define v k+1 C 3 Ω and w k+1 C Ω, R for k = 1,, 3 by means of Proposition 5.1 applied to v k, w k, the function a k and the unit vector ξ k appearing in 5.14, with the parameters: l k = l σ k 1 < 1, λ k = 1 > 1, l k+1 l k and with the remaining three parameters: { 5.17 δ 3 δ δ 1 = max l m } { v m + max l m } a k m m=1, m=...3,...3 as indicated below. We then finally set: ṽ = v 4 and w = w 4. We start by checking that the assumptions of Proposition 5.1 are satisfied. Namely, we claim that δ k, l k, 1 together with: 5.18 a k m δ k l m k m =... 3 and v k m δ k l m k m = 1,, at each iteration step k = 1,, 3, if only the constant δ in 5.5 is appropriately small. Indeed, δ 1 C D 1/ in view of 5.11 and 5.15, so δ 1 < 1 if only δ 1. Further, by the definition 5.17 it follows that: a k m = 1 l l m a m k m δ 1 l, so the first assertion in m δ k l m k

21 CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQATION holds. For the second assertion, we see directly that it holds when k = 1, as: v 1 m = 1 l l m v m m δ 1 l. On the other hand, using induction on k and exploiting 5.3, we get: m v k+1 m v k m + v k+1 v k m δ k 1 δ k lk+1 m + C lk+1 m = C δ k lk+1 m l m k δ k+1 l m k+1 + Cδ k λ m k m = 1, k = 1,. The proof of 5.18 is now complete for the choice δ k+1 = Cδ k, where C > 1 is, as always, an appropriately large universal constant. Consequently: δ, δ 3 C D 1/ < 1 if only δ We now directly verify the concluding estimates of Proposition 5.. We have, in view of the definition of D and 5.14: D = A A + D + 1 v 1 1 v 1 + sym w 1 v 4 v 4 + sym w = A A v 1 k+1 v k+1 + sym w k+1 v k v k + sym w k + a k ξ k ξ k, and thus by 5.1, 5. and the definition of l, there follows 5.7: D 3 δ 3 k A A + C C l β A,β + δ 1 3 λ k l k λ k l k β/ D C M β A,β + 3 δ β/ 3 D C σ M β A,β + 1 σ D. We now check 5.8, using 5.1, 5.13 and 5.4: 3 3 ṽ v 1 v v 1 + v k+1 v k 1 C D 1/ + C δ k C D 1/ 5.19 w w 1 w w 1 + w w 1 + C D 1/ + D + 3 w k+1 w k 1 3 δ k 1 + v k C D 1/ 1 + v + v v 1 + C D 1/ 1 + v + D 1/ C D 1/ 1 + v k+1 v k 1 C D 1/ 1 + v. Finally, the first bound in 5.9 follows by 5.11 and 5.3: 3 ṽ v + v k+1 v k C 3 l D 1/ + C δ k λ k C l D 1/ + Cδ 3 3 σ k l C l D 1/ 1 + σ 3 CMσ 3, 3 v k

22 MARTA LEWICKA AND MOHAMMAD REZA PAKZAD while the second bound is obtained by: 3 3 w w + w w + w k+1 w k C M + D + δ k λ k 1 + v k 3 C M + δ 3 CMσ σ 3 l 1 + v k 3 CM 1 + σ 3 + σ 3 v k 3 v k CMσ v. in view of 5.1, 5.13 and reasoning as in The C 1,α approximations - a proof of Theorem 1.. We are now in a position to state the final intermediary approximation result, parallel to [1, Theorem 1]. Theorem 6.1. Assume that Ω R is an open, bounded domain. Given are functions v C Ω, w C Ω, R and A C,β Ω, R sym for some β, 1, such that the deficit D below is appropriately small: 6.1 D = A 1 v v + sym w, < D < δ 1. Fix the exponent: 6. { 1 < α < min 7, β }. Then, there exist v C 1,α Ω and w C 1,α Ω, R such that: v v + sym w = A, 6.4 v v 1 C D 1/ and w w 1 C1 + ṽ D 1/, where C > is a constant depending on α but independent of all other parameters. Proof. The exact solution to 6.3 will be obtained as the C 1,α limit of sequences of successive approximations {v k C Ω, w k C Ω, R }. 1. Induction on stages. We set v = v and w = w. Given v k and w k, define v k+1 and w k+1 by applying Proposition 5. with parameters σ and M k that will be appropriately chosen below and that satisfy: 6.5 M k > max{ v k, w k, 1} and σ > 1. Following our notational convention, we define the k-th deficit D k = A 1 v k v k +sym w k. In view of Proposition 5., we get: 6.6 D k+1 C A,β M β k D k β/ + 1 σ D k, 6.7 v k+1 v k 1 C D k 1/ and w k+1 w k 1 C1 + v n D k 1/,

23 CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQATION v k+1 CM k σ 3 and w k+1 C1 + v k M k σ 3, provided that 5.5 holds for each D k. We shall now validate this requirement, with the parameters: 6.9 M k = C1 + v σ 3 k M. In fact, we will inductively prove that one can have: 6.1 D k 1 σ sk D with any < s < min { 6β } 1,. β Fix s as indicated in 6.1. Clearly, 6.1 and 6.5 hold for k =. By 6.6 and the induction assumption we obtain the bound: 6.11 σ sk+1 D k+1 C A,β D β/ 1 σ s 6β 1 β/s 1 σ β k D M β C kβ 1 + v β + Cσ s 1. We see that in view of the condition on s in 6.1, both σ s 1 6β 1 β/s and σ β are smaller than 1. Further, it is possible to choose σ > 1 so that the second term in 6.11 be smaller than 1/ and that the quotient term in parentheses above is also smaller than 1. Then, choose M so that 6.5 holds for k = together with: C A,β D β/ 1 σ s M β This results in the first term in 6.11 being smaller than 1/ if only C 1. Consequently, we get σ sk+1 D k+1 / D 1 as needed in 6.1. Observe now that by 6.7 and by the established 6.1: 6.1 i= < 1. k 1 k 1 k v k v + v i+1 v i 1 v + C D i 1/ v + C i= i= 1 σ si/ D 1/ = v + C D 1/ 1 σ s/ v + C D 1/, if only, say, σ s > 4 which can be easily achieved through the choice of σ. Now, by 6.8 and 6.1: v k+1 M k+1 1 C C 1 + v and w k+1 M k+1 1 C1 + v k 1 C 1 + v C C1 + v + D 1/ 1 + v Hence, taking the constant C 1 large enough, we see that both quantities above can be made smaller than 1, proving therefore the required C 1,α control of the approximating sequences v n and w n. Let now α be an exponent as in 6.. Choose s satisfying 6.1 and: 6.13 α6 + s s <. It is an easy calculation that s satisfying 6.1 and 6.13 exists if and only if the exponent α is in the range 6.. Indeed, 6.13 is equivalent to α < s 6+s, while 6.1 is equivalent to: < s { s < min 7, β }..

24 4 MARTA LEWICKA AND MOHAMMAD REZA PAKZAD We will prove that sequences {v k, w k } k= are Cauchy in C1,α Ω. Firstly, by 6.7, 6.1,6.1: 6.14 v k+1 v k 1 C D k 1/ C D 1/ σsk/, w k+1 w k 1 C1 + v k D k 1/ C 1 + v σ sk/ + D 1/ 1/ D, so we see right away that they are Cauchy in C 1 Ω. On the other hand, by 6.8, 6.1, 6.1: v k+1 v k + w k+1 w k C1+ v k M k σ 3 C 1+ v + D 1/ C1+ v σ 3 k M, so the sequences have the tendency to diverge in C Ω. Interpolating now the C 1,α norm by [1]: we obtain: 6.15 f,α f α 1 f 1 α, v k+1 v k,α + w k+1 w k,α C α C σ 3 kα M α C 1 α = C M α C α h σ α6+s s 1 σ sk1 α/ where by C we denoted an upper bound of all quantities involving C, v, D. It is clear that choosing σ sufficiently large so that C σ 3 s/ < 1, the resulting bound 6.15 implies that { v k, w k } k= are Cauchy in C,α Ω, provided that 6.13 holds. We see that the choice of exponent range in 6. so that the above construction technique works, is optimal. 3. Concluding, we see that {v k, w k } k= converge to some v C1,α Ω and w C 1,α Ω, R. Since the defects in the approximating sequence obeys: lim k D k = by 6.1, we immediately get 6.3. Additionally, by 6.14: v v 1 w w 1 k= k= completing the proof of 6.4. v k+1 v k 1 C k= k= 1 σ sk/ D 1/ = k, C D 1/ 1 σ s/ C D 1/ 1 w k+1 w k 1 C σ sk/ 1 + v D 1/ C1 + v D 1/. We are now ready to give: Proof of Theorem 1.. Fix a sufficiently small ε >. We will construct v C 1,α Ω and w C 1,α Ω, R such that: 6.16 A = 1 v v + sym w in Ω and: 6.17 v v + w w < ε. In order to apply Theorem 6.1, we need to decrease the deficit A 1 v v + sym w so that it obeys 6.1. This will be done in three steps.

25 CONVEX INTEGRATION FOR THE MONGE-AMPÈRE EQATION 5 First, let ṽ C Ω, w C Ω, R and à C Ω, R sym be such that: 6.18 ṽ v 1 + w w 1 + à A < ε c > A 1 ṽ ṽ + sym w > c Id in Ω. Second, by Theorem.1 and Remark 3.3, there exists v C 1 Ω and w C 1 Ω, R such that: 6.19 à = 1 v v + sym w in Ω, v ṽ + w w < ε and v ṽ C. Third, let ṽ C Ω and w C Ω, R be such that: 6. v ṽ 1 + w w 1 < ε. By 6.19, 6. and 6.18, we get: 6.1 A 1 ṽ ṽ + sym w A à + 1 ṽ ṽ + sym w 1 v v + sym w A à + v + ṽ v ṽ + w w ε + v + ε + C ε + ε < δ, as required in Theorem 6.1, if only ε is small enough. We now apply Theorem 6.1 to ṽ, w and the original field A, and get v C 1,α Ω and w C 1,α Ω, R satisfying 6.16 and such that: v v + w w C 1 + ṽ A 1 ṽ ṽ + sym w + 3ε C 1 + ε + v ε + 3ε, by 6.4, 6.1, 6., 6.19 and Clearly 6.17 follows, if ε is small enough. The following Corollary is of independent interest: Corollary 6.. Let Ω, f, p, α be as in the statement of Theorem 1.1. Let q. Then, for all v W 1,q Ω, there exists a sequence v n C 1,α Ω weakly converging to v in W 1,q Ω, and such that: Det v n = f in Ω. Proof. Let v n C 1 Ω converge to v in W 1,q Ω. For every v n, consider the approximating sequence {v n,k C 1,α Ω} as in Theorem 1.1, converging uniformly to v n. Define now {v n } to be an appropriate diagonal sequence, so that it converges to v in L q Ω. We will check that {v n } is bounded in W 1,q. The boundedness of v n L q is clear from the convergence statement. On the other hand, the proof of Theorem 1. gives, by 6.4, 6.18, 6.19, 6. and 6.1: v n x v n x + ɛ + C + Cδ 1/ v n x + C x Ω. Consequently, v n L q v n L q + C C, which concludes the proof.

26 6 MARTA LEWICKA AND MOHAMMAD REZA PAKZAD 7. Rigidity results for α > /3 - a proof of Theorem 1.3. The crucial element in the proof of the rigidity Theorems 1.3 and 1.4 is the following result, that is the small slope analogue of [1, Proposition 6]: Proposition 7.1. Let Ω R be an open, bounded, simply connected domain. Assume that for some α /3, 1, the function v C 1,α Ω is a solution to: Det v = f in Ω, where f L p Ω and p > 1. Then the following degree formula holds true, for every open subset compactly contained in Ω and every g L R with supp g R \ v : 7.1 g vf = gy deg v,, y dy. R Above, degψ,, y denotes the Brouwer degree of a continuous function ψ : Ū R at a point y R \ ψ. Proof. 1. Fix and g as in the statement of the Proposition. We first recall [35] that deg v,, is well defined on the open set R \ v. In fact, this function is constant on each connected component { i } i= of R \ v and it equals on the only unbounded component = R \ vū. Thus, without loss of generality, we may assume that g is compactly supported and that: supp g k. By compactness, there must be: supp g N k for some N, and consequently the integral in the right hand side of 7.1 is well defined. Let now {g i Cc N k} i=1 be a sequence pointwise converging to g and such that g i g L for all i. It is sufficient to prove the formula 7.1 for each g i and pass to the limit by dominated convergence theorem. To simplify the notation, we drop the index, and so in what follows we are assuming that g Cc R \ v vū. As in the proof of Theorem 1.1, let A W,p Ω C,β Ω be such that curlcurla = f. Here, we take β = min{ p, α}, 1. Consequently, in view of the simple connectedness of Ω, there exists w C 1,β Ω, R such that: A = 1 v v + sym w. For a standard d mollifier ϕ Cc B, 1 as in Lemma 4.3, define: l, 1 v l = v ϕ l, w l = w ϕ l, A l = A ϕ l, and apply the area formula change of variable formula [14, ] to the smooth functions g and v l : 7. g v l det v l = gy deg v l,, y dy. R We see that v l converge uniformly to v, so the degrees converge pointwise [35] and thus: lim gy deg v l,, y dy = gy deg v,, y dy. l R R To conclude the proof in view of 7., it suffices now to show that: 7.3 lim g v l det v l = g vf. l