Jacobian of weak limits of Sobolev homeomorphisms

1 Introduction

The main goal of this note is to establish when the sign of the Jacobian is preserved under W1,p-weak convergence. Such a question pops out naturally in the variational approach to Geometric Function Theory (GFT) [2, 14, 22] and Nonlinear Elasticity (NE) [1, 4, 6, 19, 24, 25]. Both theories GFT and NE deal with minimizing sequences of Sobolev homeomorphisms. In the context of NE, one typically deals with two-dimensional or three-dimensional models and require that the deformation gradients belong to M+n×n, where Mm×n={real ⁢m×n⁢ matrices}, and M+n×n={A∈Mn×n:det⁡A>0}. The infimum of the energy is not attained, in general, at a homeomorphism; interpenetration of matter may occur. Even in a special case of Dirichlet energy injectivity is often lost when passing to the weak limit of the minimizing sequence, [3, 13, 15, 16]. Further examinations are needed to know the properties of such singular minimizers.

Throughout this text Ω will be a domain in ℝn. The class of Sobolev mappings f:Ω→ℝn with nonnegative Jacobian determinant, J⁢(x,f)=det⁡D⁢f⁢(x)⩾0 almost everywhere, is closed under the weak convergence in W1,p⁢(Ω,ℝn) provided p⩾n (see [14, Theorem 8.4.2]). However, if p<n, passing to the weak W1,p-limit of a sequence with nonnegative Jacobians one may lose the sign of the Jacobian. Indeed, there exists a sequence of Sobolev mappings fk:Ω→ℝn with J⁢(x,fk)>0 almost everywhere such that the sequence converges weakly in W1,p⁢(Ω,ℝn), p<n, to the mapping f⁢(x)=(-x1,x2,…,xn), see [14, p. 181]. Moreover, following the construction in [18] such mappings fk can be made continuous. However, it is not obvious at all as to whether one can make a similar example with fk being homeomorphisms. This is the subject of our result here. Here [n2] denotes the integer part, i.e. [22]=1, [32]=1 and so on.

It is worth noting that in Theorem 1 the Jacobian J⁢(x,f) can have very different behavior than the Jacobians in the sequence without knowing that J⁢(x,fk)>0 on a set of positive measure. Indeed, there exists a sequence of Sobolev homeomorphisms fk with J⁢(x,fk)=0 a.e., converging weakly in W1,p⁢(Ω,ℝn), 1⩽p<n, to the mapping f⁢(x)=x. Let us briefly sketch this using the construction from [10]: we cover Ω by cubes of diameter less than 1k and on each cube we follow the construction from [10] to obtain a homeomorphism with zero Jacobian a.e. It is possible to make the W1,p-norm of the sequence uniformly bounded and hence find a weakly convergent subsequence. Furthermore, it follows from the construction that the sequence fk converges uniformly to the identity. This also shows that there is a sequence with J⁢(x,fk)=0 a.e. converging weakly in W1,p⁢(Ω,ℝn), 1⩽p<n, to f⁢(x)=(-x1,x2,…,xn).

Recently it was shown in [12] and [5] that a Jacobian of a Sobolev homeomorphism can change sign in dimension n≥4 for 1≤p<[n2].

2 Preliminaries

3 Proof of Theorem 1 for p>n-1, n⩾3, and p⩾1, n=2

Each homeomorphism fj is either sense-preserving or sense-reversing. Under our assumptions there exists a point xj such that fj is differentiable at xj, see Section 2.2, and J⁢(xj,fj)>0. By (2) we know that the degree of fj is one and hence each fj is sense-preserving.

As fj⇀f in Lp, p>1, we know that ∫Ω|D⁢f|p is uniformly bounded and hence we can find a Radon measure μ and a subsequence (which we will denote again as fj) such that

Moreover, for p=1 we can use De La Vale Pousin characterization of weak convergence in L1 and we can find an continuous convex function Φ:[0,∞)→[0,∞) such that

It follows that we can find a Radon measure μ and a subsequence (which we will denote again as fj) such that

It is well known that for almost every x∘∈Ω we have

Let δ>0. For the contrary we suppose that there is x∘∈Ω such that (4) and (6) hold at x∘ and

Without loss of generality we may and do assume that

Using (4) we can find 0<r1 small enough such that for all 0<r<r1 we have

Since the sequence of mappings fj converges to f weakly in W1,p⁢(Ω,ℝn), we have that the sequence of mappings fj converges to f strongly in Lloc1⁢(Ω,ℝn). Now, we may pick up an index j∘ large enough such that for all j≥j∘,

The last two inequalities imply that for all 0<r<r1 we have

Our next goal is to prove the following:

1. if p>n-1, then there exists a constant C (depending only on p and n) such that for all 0<r<r1 and j≥j∘,

   δn-1-p⁢rn⩽C⁢∫B⁢(x∘,r)|D⁢fj|p,

2. if n=2 and p=1, there exist a constant C and such that for all 0<r<r1 and j≥j∘ there is a set A⊂B⁢(x∘,r) such that

   |A|<C⁢δ⁢|B⁢(x∘,r)|  and  r2⩽C⁢∫A|D⁢fj|.

These would lead to a desired contradiction. Indeed, choose 0<r<r1 such that μ⁢(∂⁡B⁢(x∘,r))=0 and in case (i) we obtain after passing to a limit in j that

After passing δ→0+ we obtain a contradiction with (6). In case (ii) we can use Jensen’s inequality and (5) to obtain

Similarly as above we obtain in the limit that

and now passing to a limit δ→0+ we obtain a contradiction using (5).

The above proof was based on the Sobolev embedding theorem on spheres and therefore does not work for p<n-1. To overcome these difficulties we follow Hencl and Malý [11] and use the theory of linking numbers and its topological invariance. For the convenience of the reader we recall the needed properties of linking numbers here.

4 Linking number

We use the notation 𝔹d for the unit ball in ℝd and 𝕊d-1 for the unit sphere. By 𝔹¯d⁢(c,r) we denote the closed ball with center c and radius r>0.

Let n,t,q be positive integers with t+q=n-1. Let us consider the mapping Φ⁢(ξ,η):𝔹¯t+1×𝔹¯q+1→ℝn defined coordinatewise as Φ⁢(ξ,η)=x, where

Denote by 𝔸 the anuloid

Of course, given x∈𝔸¯ we can find a unique ξ∈𝕊t and η∈𝔹¯q+1 such that Φ⁢(ξ,η)=x. We will denote these as ξ⁢(x) and η⁢(x).

A link is a pair (φ,ψ) of parametrized surfaces φ:𝕊t→ℝn, ψ:𝕊q→ℝn. The linking number of the link (φ,ψ) is defined as the topological degree

where the mapping L=Lφ,ψ:𝔸¯→ℝn is defined as

or equivalently

where ψ¯ is an arbitrary continuous extension of ψ to 𝔹¯q+1 (of course, the degree does not depend on the way how we extend ψ, it depends only on the values on the boundary ∂⁡𝔸=Φ⁢(𝕊t×𝕊q)). Geometrically speaking, for t=q=1, the linking number is the number of loops of a curve φ around a curve ψ counting orientation into account as +1 or -1. For the introductions to the linking number in ℝ3 and its application to the theory of knots see [23].

The canonical link is the pair (μ,ν), where

For example in dimension n=3 we get that

It is well known that the linking number is a topological invariant. The simple proof of the following proposition can be found in [11].

Analogously, we can pick a∈𝔹¯q+1⁢(0,110) and b∈𝔹¯t+1⁢(𝐞1,110)∩𝔹¯t+1 and consider the pair

Similarly to the previous proposition we have:

5 Proof of Theorem 1 for p>[n2], n⩾3, and p≥1, n=3

Our argument is similar to the proof given in Section 3 and therefore some details are only sketched. By μ we again denote the w* limit of (some subsequence) ∫|D⁢fj|p for p>[n2] and of ∫Φ⁢(|D⁢fj|) for p=1 and n=3.

By C1 and C2 we denote a fixed constants whose exact value will be determined later. We fix δ>0 and we choose a point x0 such that (4) and (6) hold and without loss of generality we assume that the derivative of f at x∘ is given by (7).

We fix r1>0 such that for all 0<r<r1 we have

and again for all j≥j∘ we obtain

We fix t,q⩽[n2] such that t+q=n-1 (e.g. t=q=n-12 for n odd and t=n-22, q=n2 for n even). Our goal is to prove the following:

1. if p>[n2] and n⩾3, then there exists a constant C (depending only on p and n) such that for all 0<r<r1 and j≥j∘,

   δmin⁡{t,q}-p⁢rn⩽C⁢∫B⁢(x∘,4⁢r)|D⁢fj|p,

2. if p=1 and n=3, we have A⊂B⁢(x∘,4⁢r) such that

   |A|<C2⁢δ⁢|B⁢(x∘,4⁢r)| and r3⩽C⁢∫A|D⁢fj|.

Analogously to reasoning in Section 3 we obtain a contradiction using min⁡{t,q}-p<0 for p>[n2] and (5) for p=1 and n=3.