The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces

We study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\det D u =f$$\end{document}detDu=f, where f is integrable and bounded away from zero. In particular, we take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in L^p$$\end{document}f∈Lp, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document}p>1, or in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\log L$$\end{document}LlogL. We prove that for a Baire-generic f in either space there are no solutions with the expected regularity.


Introduction
For n > 1 let us take a bounded domain ⊂ R n which is both smooth and uniformly convex. We consider the prescribed Jacobian equation, that is, In fact, for a Baire-generic f ∈ X p , (1.2) has no solution in the space n≤q, p<q W 1,q ( , R n ).
We briefly describe the proof of the main theorem. Assume for simplicity that = B is a ball. The starting point of our proof is the observation that if f ∈ L p (B) is a radially symmetric function then the unique spherically symmetric solution of (1.2) is, in general, in W 1, p \W 1,q for any q > p, see Proposition 3.1. Hence we first show that any L p function admits perturbations which are radially symmetric in a small neighbourhood of ∂ B and such that the W 1,q -norm of the symmetric solutions in that neighbourhood is unbounded. The crucial step of the proof is to show that, for these perturbations, the symmetric solutions have comparable q-Dirichlet energy to that of the energy-minimiser: hence we deduce that the W 1,q -norm of q-energy-minimal solutions [see equation (4.2)] is unbounded at every f ∈ X p . The weak continuity of the Jacobian, together with an application of the Baire Category Theorem, then concludes the proof.
It will be clear from the proof that the boundary condition u = id on ∂ in (1.2) can be relaxed to the requirement that u(x) ∈ ∂ for H n−1 -a.e. x ∈ ∂ . Under this generalised boundary condition, our arguments also deal easily with the case where the compatibility condition in (1.3) is replaced by the condition 1 | |´ f (x) dx = k for some fixed k ∈ N. In general, the number k gives an essential upper bound on the multiplicity function of solutions.
Our main theorem raises the following question, which we do not address here: Question 2 is natural from the viewpoint of the regularity theory for optimal transport maps. In Optimal Transportation, the underdetermined equation (1.1) is usually turned into a determined elliptic equation by imposing the constraint u = Dφ for some convex scalar function φ: thus φ satisfies the Monge-Ampère equation det D 2 φ = f , see [8,18,49]. The relation between the prescribed Jacobian equation and the Monge-Ampère equation is analogous to that of their linear counterparts, the divergence equation and the Poisson equation, see Table 1. Moreover, the estimates for the Monge-Ampère equation scale as in these linear problems, which is precisely the scaling in Question 2.
However, the failure of a certain estimate for the determined problem does not imply that the same happens for the underdetermined problem, as shown by Bourgain-Brezis [7]. Surprisingly, even in the linear setting, solutions of the determined problem do not have optimal regularity for the underdetermined problem! This is also the case for the non-linear problem (1.1), at least if we only require inf f ≥ 0. Indeed, at [52, page 293], Ye gives an example in = B ⊂ R 2 such that (1.2) has a smooth solution but the corresponding solution of the Monge-Ampère equation is only C 1,1 regular.
In the case n 2 /(n + 1) ≤ q < n, Müller has posed an outstanding problem about the range of the distributional Jacobian u → Det Du, see [44, page 659]. In [35] it was shown that for any 1 < q < n and any f ∈ L q/n ( ) there is a pointwise solution u ∈ W 1,q ( , R n ) of Ju = f and one may prescribe any boundary datum g ∈ W 1−1/q,q (∂ ). The free choice of the boundary datum precises the well-known fact that the pointwise Jacobian loses its geometric content when q < n. Nevertheless, for q ≥ n, as in this paper, the distributional Jacobian and the pointwise Jacobian agree. We do not discuss distributional Jacobians further, instead referring the reader to [1,4,9,42].
To conclude the introduction we briefly discuss the motivation for the work initiated on this paper, which is twofold. Firstly, the Jacobian plays an important role in Continuum Mechanics, specially in nonlinear elasticity, see e.g. [5,16,20,28]. This is not surprising since, for a Sobolev map, positivity of the Jacobian implies a weak form of local invertibility [21]. Secondly, in R n , there is an intimate connection between the Jacobian equation, commutator estimates and factorization theorems in Hardy and L p spaces [29,30,36]; an outstanding open problem in the area is to determine whether the Jacobian is onto the Hardy or L p space [14,31,38]. Positive partial results can be found in [14,30,36,37] and negative ones in [38]. We refer the reader to our forthcoming work [23,24] for further details and progress towards the possible non-surjectivity of the Jacobian.

Notation and background results
In this section we gather a few preliminary results for the convenience of the reader.
The set will always denote a bounded smooth domain in R n . Moreover, in Sect. 4, we will further assume that is uniformly convex, in the sense that the second fundamental form of ∂ is uniformly positive.
We will often write Ju ≡ det Du. As is customary, the symbols a ≈ b and a b are taken to mean that there is some constant C > 0 independent of a and b such that C −1 a ≤ b ≤ Ca and a ≤ Cb, respectively. We denote by S n−1 the unit sphere in R n and, for r < R, we write Here | · | denotes the Euclidean norm of a vector in R n ; for a matrix A ∈ R n×n , we also write |A| for its operator norm. We represent by L n the n-dimensional Lebesgue measure, by H n−1 the (n − 1)-dimensional Hausdorff measure and we write ω n ≡ L n (B 1 (0)).

Mappings of finite distortion
In this section we gather a few results from the theory of mappings of finite distortion, which is detailed in [26,32], see also [2] for the planar case.
If u has finite distortion, we can set Ku(x) = |Du| n Ju(x) if Ju(x) = 0 and Ku(x) = 1 otherwise; this function is the (optimal) distortion of u.
We state some basic properties of the topological degree, referring the reader to [21] for further details.

Lemma 1
For u ∈ C 0 ( , R n ) and x ∈ R n \u(∂ ), the topological degree of u at x with respect to , denoted by deg(x, u, ), has the following properties: For our purposes, the following classical result [51] is particularly relevant:

a map of finite distortion. Then u has a continuous representative with the Lusin (N) property.
Recall that a map has the Lusin (N) property if it maps L n -null sets to L n -null sets. Theorem 2.2 gives no improvement in the supercritical case, since the continuous representative of a map u ∈ W 1, p , p > n, has the Lusin (N) property.
Theorem 2.2 shows that solutions of (1.2) satisfy the change of variables formula: Proof The non-degeneracy assumption (1.3) implies that u is a map of finite, and even integrable, distortion. Hence, by Theorem 2.2, the coarea formula applies, see for instance [21,Theorem 5.23] . Hence the conclusion follows from (2.1).
Although this will not be needed for the proof of the Main Theorem, it is worthwhile mentioning that solutions of (1.2) in a sufficiently good Sobolev space are homeomorphisms. The sharp statement follows for instance from the main result in [27]: Recall that a map is discrete if the preimage of every point is locally finite. A counterexample due to Ball [5] shows that the conclusion of Theorem 2.4 does not hold if the distortion is in L q ( ) for q < n − 1. We then deduce: Proof It follows from our assumptions that the distortion of u is in L n−1 ( ). Consider a sufficiently small neighbourhood δ of and extend u to be the identity in δ \ . Theorem 2.4, applied in δ , shows that u| is open and discrete and it follows that u ∈ Hom( , u( )), see e.g. [26,Theorem 3.27]. Moreover, u( ) = . Indeed, it suffices to prove that u( ) ⊆ , since where the unions are disjoint since u is a homeomorphism, and ⊆ u( ) by Lemma 1. Suppose that there is x ∈ such that u(x) ∈ R n \ . Take y ∈ such that u(y) ∈ and consider a continuous path in joining x and y. The image of such a path under u must cross ∂ somewhere, which contradicts (2.2).
In fact, a simple argument using the change of variables formula shows that the inverse map u −1 is in W 1,n ( , ). Note that the situation in the planar case is particularly pleasant, since then n − 1 = 1. Other results in the direction of Corollary 2.5 can be found in [5,48].

Improved integrability of the Jacobian
In this section we recall standard facts concerning the improved integrability of the Jacobian. This phenomenon was first noticed by Müller in [43] and then generalised by Coifman-Lions-Meyer-Semmes in [14]. The latter paper shows that the Jacobian determinant of maps inẆ 1,n (R n , R n ) lies in the Hardy space H 1 (R n ), a fact which was extended in [25] to a large class of polynomials and differential operators. These results are essentially local and, as we are interested in the behaviour of Jacobians on domains, we will need the following global version from [28]: Theorem 2.6 Let u ∈ W 1,n ( , R n ) be such that Ju ≥ 0 a.e. in . Then, for any p > n, There is a more general version of Theorem 2.6 when Ju changes sign, but we will not need it here. The space L log L( ) in the theorem is a particular case of an Orlicz-Zygmund space and it can be defined as the space of those measurable functions f : → R such that The above expression defines a complete norm, although it is not immediate that it satisfies the triangle inequality, see [33, §8]. The standard Luxemburg norm in L log L( ) is different but equivalent to the one in (2.3); we chose the latter since it is easier to use in calculations. We refer the reader to [47] for the relation between the space L log L and local Hardy spaces.

Radial data and radial stretchings
In this section we give a description of the regularity of radial stretchings solving (1.2).
and we identify any such function with a function f : [0, +∞) → R in the obvious way. Under natural assumptions, and when the data is radially symmetric, equation (1.2) admits a unique spherically symmetric solution, that is, a solution of the form u(x) = ρ(|x|) x |x| ; we refer to these maps as radial stretchings. More precisely, c.f. [6, Lemma 4.1], we have: and only if ρ is absolutely continuous on (0, 1) and a similar statement holds for p = ∞. In this case, for a.e. x in B R (0), and writing r = |x|, we have the formulae We also record here the standard notation ∂ r u(x) ≡ (Du(x)) · x r ; in particular, if u is a radial stretching as in Lemma 2, then 3) The following proposition follows easily from Lemma 2, see also [52,Theorem 6]. Further, u ∈ W 1, p (B R (0)) and Moreover, this inequality is sharp: in general u / ∈ W 1,q for any q > p.
Proof The existence and uniqueness of a radial stretching u solving (1.2) follows immediately from (3.1). We turn to the regularity properties and assume that 1 ≤ p < ∞, as the case p = ∞ is similar. Using (3.4) and Jensen's inequality, we deduce that and therefore, integrating in r , Since f satisfies (1.3), we deduce using (3.4) that (inf f ) 1 n ≤ ρ(r )/r . Thus, from (3.1), we haveρ p(n−1) n and the desired estimate follows from Lemma 2. To see that this inequality is sharp, it suffices to note using (3.1) that for δ > 0 there is a constant C δ such that C δ f (r ) ≤ρ(r ) for r ∈ (δ, R).

Non-solvability of the prescribed Jacobian equation
This section is dedicated to the proof of the main result. We begin by taking = B 1 (0). Throughout this section we fix c ∈ (0, 1). For p ∈ [1, +∞) and η ∈ [0, 1), let : (4.1) and note that X p ≡ Z 0 p . We make a few immediate remarks about the sets Z η p : (iii) if c = 1 then the only elements of X p are functions which are 1 a.e. in B 1 . Thus we assume throughout this section, without loss of generality, that c < 1.
We write B Z η p ( f , ε) for a ball of radius ε around f , under this distance. Given f ∈ X p and max{ p, n} ≤ q, we consider the q-energy of the datum f , defined by Thus (1.2) admits a W 1,q solution if and only if E q ( f ) < ∞. We say that w is a q-energyminimal solution for f if w solves (1.2) and moreover We begin with the following lemma: (ii) f j is radially symmetric in A(R j , 1); (iii) if u j is the radial stretching such that Ju j = f j in A(R j , 1) and u j = id on S n−1 , then Proof For γ, R ∈ ( 3 4 , 1), consider the functions The choice of M ensures that ffl Fix α ∈ (− q p , −1) and choose γ = γ (R) in such a way that in particular, we have γ (R) → 1 as R → 1. Take any sequence R j 1 and consider the associated numbers γ j = γ j (R j ). We will prove that, for j large enough, f j ≡ f γ j ,R j satisfies the properties above.
Concerning the lower bounds of the sequence, for r ∈ (R, 1), straightforward algebraic manipulations show that thus f γ,R (r ) > c for γ sufficiently close to 1 . For r ∈ (0, R), clearly we have Hence the lower bounds are satisfied. Moreover, (ii) clearly holds.
For (iii), denote by u γ,R (x) = ρ γ,R (r ) x r the unique radial stretching solving Ju γ,R = f γ,R in A(R, 1) and such that u γ,R = id on S n−1 . By Lemma 2 we find that and so, by (3.3), since r n−1 dr ≈ dr for r ∈ ( 1 2 , 1), and recalling the definition of M in (4.3) as well as our choice of γ in (4.5), Thus, since α < −1, we see that (iii) also holds.
Hence it remains to prove (i), and we split the proof into two cases.
Thus, since´A (R,1) | f (x)| p dx → 0 as R → 1, in order to prove that f γ,R → f in L p (B 1 ) it suffices to show that, as γ, R 1, Then, using the estimates on f γ,R from the previous case and (4.4), we obtain The right-hand side converges to zero as R → 1, since α/q + 1 > 0.
Our goal is to show that E q ( f γ,R ) → +∞ as γ, R → 1. The idea is that energy-minimal solutions are controlled by the radial solution. .3) and (4.5). There is a constant C = C(n, q) > 0 such thatˆA where u = u γ,R is the radial stretching such that Ju γ,R = f γ,R in A(R, 1) and u = id on S n−1 .
Proof Throughout the proof θ will denote an element of S n−1 and we write r θ for the corresponding element in a sphere of radius r .
Since R > 1 2 , from Hölder's inequality, On the other hand, since ∂ r u(r θ) = Mθ for all r ∈ (R, 1), c.f. (4.3) and (4.6), we can estimate Thus, the proposition will be proved once we show that For λ ∈ (0, 1) to be chosen later, let By the ACL property of Sobolev functions, H n−1 ( ) = H n−1 (S n−1 ) and so the set S n−1 \ is an H n−1 -null set. Fubini's theorem and the fact that v satisfies the Lusin (N) property, c.f. Theorem 2.2, then implies (A(R, 1)).

Remark 1
It is clear from the proof that the boundary condition v = id on ∂ B 1 (0) can be weakened to the requirement that v(θ) ∈ S n−1 for H n−1 -a.e. θ ∈ S n−1 . Note that this condition is independent of the representative of the equivalence class of v ∈ W 1,q (B 1 (0), R n ). The argument above carries through simply by replacing the set with ∩ {v(θ) ∈ S n−1 }.
Combining Lemma 3 with Proposition 4.1, we immediately obtain: Corollary 4.2 Let 1 ≤ p < q < ∞ and n ≤ q. For any ε, η > 0 and any f ∈ X p , there is a sequence f j ∈ B Z η p ( f , ε) such that dist p ( f j , f ) → 0 and E q ( f j ) → ∞. We are ready to prove our main result, an equivalent formulation of which we restate here for convenience of the reader.
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