A simple sufficient condition for the quasiconvexity of elastic stored-energy functions in spaces which allow for cavitation

In this note we formulate a sufficient condition for the quasiconvexity at $x \mapsto \lambda x$ of certain functionals $I(u)$ which model the stored-energy of elastic materials subject to a deformation $u$. The materials we consider may cavitate, and so we impose the well-known technical condition (INV), due to M\"{u}ller and Spector, on admissible deformations. Deformations obey the condition $u(x)= \lambda x$ whenever $x$ belongs to the boundary of the domain initially occupied by the material. In terms of the parameters of the models, our analysis provides an explicit upper bound on those $\lambda>0$ such that $I(u) \geq I(u_{\lambda})$ for all admissible $u$, where $u_{\lambda}$ is the linear map $x \mapsto \lambda x$ applied across the entire domain. This is the quasiconvexity condition referred to above.


Introduction
Since the seminal work of Ball [3], the phenomenon of cavitation in nonlinear elasticity has been studied by many authors, with significant advances [9,10,13] having been made in the case that an appropriately defined surface energy be part of the cost of deforming a material. In this note we consider the original case of a purely bulk energy where as usual u : Ω ⊂ R n → R n represents a deformation of an elastic material occupying the domain Ω in a reference configuration, and where n = 2 or n = 3. Our goal is to give a straightforward, explicit characterization of those affine boundary conditions of the form where λ is a positive parameter, which obey the quasiconvexity inequality 1 I(u) ≥ I(u λ ). (1.2) In the case of radial mappings [3] it is this inequality which must be violated in order that a global minimizer of I might cavitate (i.e. where a hole is created in the deformed material), a crucial ingredient of which is the application of a large enough stretch on ∂Ω (i.e. taking λ sufficiently large). When deformations are not restricted to any particular type we are still interested in whether the quasiconvexity inequality holds for a given λ since it rules out the possibility that a global energy minimizer cavitates. Thus the largest λ for which (1.2) holds is sometimes referred to as a critical load. Our chief inspiration for this work is [12], where bounds for the critical load are given in terms of constants appearing in certain isoperimetric inequalities. We use a different technique to find an explicit upper bound on the critical load in the two and three dimensional settings. 1 Strictly speaking, this is a W 1,q -quasiconvexity inequality; the term quasiconvexity usually refers to the case in which I(u) ≥ I(u λ ) holds for all Lipschitz u agreeing with u λ on ∂Ω. See, e.g., [4] for the distinction. 1 Our method also yields conditions on ∇u for the inequality (1.2) to be close to an equality in the sense that if δ(u) := I(u) − I(u λ ) is small and positive then, in the two dimensional case Ω min{|∇u − λ1| 2 , |∇u − λ1| q } dx ≤ c δ(u), (1.3) where 1 < q < 2 is an exponent governing the growth of the stored-energy function W appearing in (1.1). See Theorem 2.11 for the latter. The corresponding condition in three dimensions is where 2 < q < 3: see Theorem 3.5 for details. In both cases the Friesecke, James and Müller rigidity estimate [8,Theorem 3.1] (see also [5,Theorem 1.1]) is used in conjunction with the boundary condition to recover information apparently lost in deriving sufficient conditions for (1.2). We also note that these conditions are invariant under the elasticity scaling in which a function v(x), say, is replaced 2 by v ǫ (x) = 1 ǫ v(ǫx), where ǫ > 0. This is important in view of the example in [16,Section 1]. The latter says, among other things, that, in the absence of surface energy, a deformation which cavitates at just one point in the material can have the same energy as another deformation with infinitely many cavities.
The setting we work in is motivated by [13] in the sense that we impose condition (INV), a topological condition which is explained later. Cavitation problems must be posed in function spaces containing discontinuous functions. In particular, Sobolev spaces of the form W 1,q (Ω, R n ) with q ≥ n are not appropriate, since their members are necessarily continuous. In the case q > n this follows from the Sobolev embedding theorem, while if q = n then well-known results [18,17], applying to maps u with det ∇u > 0 a.e., imply that u has a continuous representative. Thus we work in W 1,q (Ω, R n ), where n−1 < q < n, and in so doing we are able to take advantage of existing results, including but not only those of [13].
The stored-energy functions we consider in the two dimensional case have the form where 1 < q < 2 and where h : R → [0, +∞] satisfies (H1) h is convex and C 1 on (0, +∞); (H2) lim t→0+ h(t) = +∞ and lim inf t→∞ In three dimensions the appropriate class of W is detailed in Section 3. In both cases we define a set of admissible deformations It is made clear in [3] and [15] that when λ is sufficiently large there are maps u 0 belonging to A λ of the form with r(0) > 0, such that I(u 0 ) < I(u λ ).
(1.5) The growth of h(t) for large values of t is pivotal in ensuring that such an inequality can hold. Thus the integrand W is not (W 1,q -)quasiconvex at λ1. The loss of quasiconvexity is typically associated with so-called cavitating maps like u 0 , whose distributional Jacobian Det ∇u 0 is proportional to a Dirac mass, a remark first made by Ball in [3].
For later use, we recall that the distributional Jacobian of a mapping in W 1,p (Ω, R n ), with p > n 2 /(n + 1), is defined by where ϕ belongs to C ∞ 0 (Ω). When u is C 2 the distributional Jacobian coincides with the Jacobian det ∇u. The same is true if, more generally, u ∈ W 1,p (Ω) with p ≥ n 2 /(n + 1) and Det ∇u is a function (see [14]).
The paper is arranged as follows: after a short explanation of notation, we consider the two and three dimensional cases separately in Sections 2 and 3 respectively. Subsection 2.1 contains the bulk of the estimates needed for (1.3); the relevant estimates in the three dimensional case draw on these results and are presented succinctly in Section 3. Along the way, we give a slight improvement of [19,Lemma 2.15], and, as a byproduct of our work in three dimensions we are led to a conjecture concerning the quasiconvexity of a certain function which, to the best of our knowledge, has not yet been considered in the literature.
1.1. Notation. We denote the n × n real matrices by R n×n and the identity matrix by 1. Throughout, Ω ⊂ R n is a fixed, bounded domain with Lipschitz boundary, B(a, R) represents the open ball in R n centred at a with radius R > 0 and S(a, R) := ∂B(a, R). Other standard notation includes L n for the Lebesgue measure in R n .
The inner product of two matrices A, B ∈ R n×n is A · B := tr (A T B). This obviously holds for vectors too. Accordingly, we make no distinction between the norm of a matrix and that of a vector: both are defined by |ν| := (ν ·ν) 1 2 . For any n×n matrix we write adj A := (cof A) T , while tr A and det A denote, as usual, the trace and determinant of A, respectively. Other notation will be introduced when it is needed.

The two dimensional case
The relevance of the distributional Jacobian to the loss of quasiconvexity can be seen using the following argument, the first part of which is due originally to Ball [2]. Firstly, the convexity of A → |A| q and of h implies that which, when u ∈ A λ , can be integrated over Ω; the result is then I(u) ≥ I(u λ ) follows. This can be ensured, for example, by imposing further conditions on u guaranteeing that for any bounded continuous function f , whereū represents the trace of u, here assumed to possess a continuous representative in order that the degree is well-defined. The idea behind this originates inŠverák's work [17], and was later refined by Müller, Qi and Yan [11].Šverák showed, among other things, that functions obeying (2.3) have a continuous representative, and in particular cannot cavitate. 3 In fact, the discrepancy between Ω det ∇u dx and Ω det ∇u λ dx can be measured using Det ∇u and interpreted in terms of cavitation provided some additional conditions are imposed on u. Initially following the approach in [12], we appeal to a result in [13]. Before that, we recall the definition of Müller and Spector's condition (INV), stated in terms of a general dimension n and domain Ω.
The map u : Ω → R n satisfies condition (INV) provided that for every a ∈ Ω there exists an L 1 -null set N a such that, for all R ∈ (0, dist (a, ∂Ω)) \ N a , u| S(a,R) is continuous, The topological image of B(a, R) under the mapping u, im T (u, B(a, R)), is defined below.
in Ω and that u * , the precise representative 4 of u, satisfies condition (INV). Then Det ∇u ≥ 0 and hence Det ∇u is a Radon measure. Furthermore, where m is singular with respect to Lebesgue measure and for L 1 -a.e. R ∈ (0, dist (a, ∂Ω)), Reverting to the two dimensional case Ω ⊂ R 2 , the assumption that u ∈ W 1,q (Ω) for q > 1 implies (by Sobolev embedding) that u| S(a,R) is continuous for L 1 -a.e. R ∈ (0, dist (a, ∂Ω)). Hence, for such R, the topological image im T (u, B(a, R)) = {y ∈ R 2 \ u(S(a, R)) : deg(u, S(a, R), y) = 0} is well-defined. Following [12] Finally, by applying (2.6) to inequality (2.1), we obtain It is clear that when h ′ (λ 2 ) ≤ 0 or m(Ω) = 0 we have I(u) ≥ I(u λ ). Summarising the above, we have the following: Assume that u e satisfies the hypotheses of [13,Lemma 8.1] in the case that n = 2. Then if The rest of this section handles the case h ′ (λ 2 ) > 0 and m(Ω) > 0, where m is given by (2.6), which is the situation not covered by Proposition 2.5. The following is a slightly improved version of a lemma by Zhang which, although stated here for general n, will only be needed in the case n = 2.
The constants C 1 (M, q) and C 2 (q) are given by .
Next, define f M : where C 1 (M, q) and C 2 (q) are as in (2.8) and (2.9), respectively. We remark that the continuity is a consequence of the improved (i.e. increased) value for C 2 (q) provided in Lemma 2.6. More importantly, a larger value for C 2 (q) makes our estimate of the critical load more accurate: see (2.32), for example.
Then, by combining (2.11) and (2.12) with the definition of F M , we obtain Therefore, by (2.10), Integrating this, applying the definition of the stored-energy function W , using Ω (∇u − λ1) dx = 0, and recalling that det ∇u = λ 1 λ 2 , gives Then in view of the convexity of h we get As has already observed, we need only consider h ′ (λ 2 ) > 0, since Proposition 2.5 covers the case The rest of this section is devoted to finding conditions on λ which ensure that The following result, in which inequality (2.16) is part of [2, Lemma 5.3], allows us to deal with the term involving G λ 2 . We give a short elementary proof here to keep the paper self-contained; we also give a refined version of the estimate (2.16) which provides an 'excess term' (an estimate of the difference between the two sides of the inequality (2.16)): see (2.17) below.
We first give a direct proof of (2.16).
Inserting this into (2.20), recalling that and carrying out what becomes a trivial integration yields (2.17).
We now return to the estimate of G λ 2 . Indeed, since we are working under the assumption λh ′ (λ 2 ) > 0 for every λ > 0, applying Lemma 2.8 gives To deal with the term involving G λ 1 we find an explicit condition on λ which ensures that Lemma 2.9. The function
Proof. We divide the proof into two parts, the first of which is devoted to proving the sufficiency of (2.24) and (2.25).
We now draw the preceding discussions and results together. (2.32) Then any u ∈ A λ satisfies I(u) ≥ I(u λ ).
2.1. Error estimates. In this section we are interested in understanding the properties of those u ∈ A λ such that I(u) − I(u λ ) is small and positive. Hence we focus on the case h ′ (λ 2 ) > 0 to which the results of the previous section apply. Accordingly, we impose the hypotheses of Theorem 2.10 and strengthen inequality (2.32) to read 1 The main result of this subsection is the following.
Theorem 2.11. Assume that (2.33) holds. Then there is a constant c = c(Ω, λ, q) > 0 such that for every u ∈ A λ

34)
where δ(u) := I(u) − I(u λ ). Moreover, The proof of Theorem 2.11 is given in stages below. In view of the idea is that if δ(u) is small then the same must be true of the two (necessarily nonnegative) terms in the right-hand side of (2.36). The first inequality, (2.34), follows from a smallness assumption on Ω G λ 1 (Λ) dx: see Proposition 2.14 below, while inequality (2.35) is a consequence of small Ω G λ 2 (Λ) dx and follows in a straightforward way from (2.17). We remark that an inequality like (2.35) is not available in the three dimensional case, or at least we could not derive it. The chief difficulty is the lack of an explicit expression for λ 1 (ξ) + λ 2 (ξ) + λ 3 (ξ) for ξ ∈ R 3×3 : cf. (2.18) and (2.19).
We now turn to inequality (2.34). To this end we introduce the function g : [0, +∞) → [0, +∞) defined by (2.37) For later use we notice that g is convex.
Proof. It is clear from the last part of the proof of Lemma 2.9 that inequality (2.33) implies that (2.24) holds with strict inequality. Thus for some constant c > 0. Reusing the notation Λ − Λ 0 = ρ(cos µ, sin µ) and G(ρ, µ) := G λ 1 (Λ), the case ρ ≥ √ 2λ can be handled as follows. Let ǫ > 0 and write where Y := h ′ (λ 2 ). By applying the reasoning in the proof of Lemma 2.9 to the functioñ we see thatG(ρ, µ) ≥ 0 provided (2.40) Inequality (2.33) clearly implies that C 2 exceeds the right-hand side of (2.40) by a fixed amount; thus, if ǫ > 0 is sufficiently small, inequality (2.40) holds. Hence We will see that inequality (2.34) is a consequence of the L 2 +L q rigidity estimate [5, Theorem 1.1]. We recall here the following variant (see [1, Lemma 3.1]) which is suitable for our purposes. Lemma 2.13. Let U ⊂ R n be a bounded domain with Lipschitz boundary. Let λ > 0 and g be as in (2.37). There exists a constant c = c(U, λ, q) > 0 with the following property: for every v ∈ W 1,q (U ; R n ) there is a constant rotation R ∈ SO(n) satisfying U g(|∇v − λR|) dx ≤ c U g(dist(∇v, λSO(n))) dx.
(2.44) By virtue of the convexity of g, combining Jensen's inequality with (2.43) gives and therefore (2.48) Then to prove (2.44) we need to distinguish two cases.
By definition g(t) = t 2 /2 for t ≤ 1, so that (2.45) and (2.48) immediately yield When t > 1 we have g(t) > 1/2, then hence the claim is proved. We now notice that the convexity of g together with its definition entails g(s + t) ≤ c g(s) + t 2 for every s, t ≥ 0 and for some c > 0. Indeed we have Then choosing R as in (2.43) and combining the latter with (2.44) implies Finally, since we can find c > 0 such that min{t 2 , t q } ≤ c g(t) for every t ≥ 0, we obtain which is the thesis.
Remark 2.15. Using (2.49) and the definition of g we obtain Then recalling that q < 2, Hölder's inequality combined with (2.50) yields On the other hand we clearly have (2.52) Therfore (2.51) and (2.52) together give which on applying Poincaré's inequality finally implies If λ satisfies (2.33) then from (2.53) we can conclude that u λ is the unique global minimiser of I among all maps u in A λ and, moreover, that u λ lies in a potential well.

The three dimensional case
In this section we seek conditions analagous to those obtained in the two dimensional case ensuring that u λ is the unique global minimizer of an appropriately defined stored-energy function. For simplicity we focus on the following W : R 3×3 → [0, +∞] given by where 2 < q < 3, γ > 0 is a fixed constant, Z : R 3×3 → [0, +∞) is convex and C 1 , and h has properties (H1)-(H3). Applying [12, Lemma A.1] to A → |A| q gives where Moreover, we clearly have Therefore, by gathering (3.2) and (3.4) and appealing to the convexity of Z and h, we obtain for any u ∈ A λ , where A λ is the class of admissible maps given by (1.4) with n = 3. Integrating (3.5) and using the facts that both ∇u and cof ∇u are null Lagrangians in W 1,q (Ω, R 3 ) for q ≥ 2, we obtain By analogy with Proposition 2.5 we can deal with the case h ′ (λ 3 ) ≤ 0 by imposing condition (INV) on a suitably defined extension of u, as follows.
Assume that u e satisfies the hypotheses of [13,Lemma 8.1] in the case that n = 3. Then if The argument which precedes Proposition 2.5 implies that the integral term is not greater than zero, which when coupled with the assumption h ′ (λ 3 ) ≤ 0 easily gives the desired inequality.
Lemma 3.2. Let W be as in (3.1) and let u ∈ A λ . Then Proof. For brevity we writeλ i := λ i − λ for i = 1, 2, 3. It follows that Inserting this into (3.7) gives Since the last integral may be written as we can apply [2, Lemma 5.3] again to deduce that Hence since h ′ (λ 3 ) > 0, (3.8) holds.
The foregoing results imply a three dimensional analogue of Theorem 2.10: Let λ > 0 be such that where κ is as per (3.1) and Then any u ∈ A λ satisfies I(u) ≥ I(u λ ).
Thus our result mirrors that of [12] and it produces constants which are explicit up to the inequality (3.3) obeyed by κ. In fact 5 , κ varies very nearly linearly as a function of q on the interval [2,3], the approximation κ(q) ∼ 3 − q + (2 − √ 2)(q − 2) being accurate to within 0.025 for q in (2,3) and exact at the endpoints.
3.1. Error estimates. In the three dimensional case error estimates follow an analogous pattern to those given in Section 2.1, as we now show. Let λ > 0 be such that (3.18) Theorem 3.5. Assume that (3.18) holds. Then there is a constant c = c(Ω, λ, q) > 0 such that
Proof. Throughout this proof c denotes a generic strictly positive constant possibly depending on Ω, λ, and q. The second inequality in (3.18) ensures that while the first (strict) inequality in (3.18) yields for some c > 0. To prove (3.21) we make use of the same notation as in the proof of Lemma 3.3. Let ǫ > 0 and observe that | sin 2φ sin 2θ sin θ| + ǫρ q .
Remark 3.6. If λ satisfies (3.18), from (3.19) we can conclude that also in this case u λ is the unique global minimiser of I among all maps u in A λ and moreover that u λ lies in a potential well.
We end this section by remarking that condition (3.17) can be removed from the statement of Theorem 3.4 if a certain conjecture holds, namely that the function is quasiconvex at λ1 (For i = 1, 2, 3, λ i (A) denote, as usual, the singular values of A ∈ R 3×3 .) Standard results (see, e.g., [6,Theorem 5.39 (ii) is polyconvex and hence quasiconvex, but it remains to be seen whether subtracting the term 3 i=1 λ i (A) destroys the quasiconvexity at λ1. We conjecture that it does not. To see why the quasiconvexity of P at λ1 might matter, note that from (3.9) we can write det ∇u − λ 3 =λ 1λ2λ3 + λ i<jλ iλj + λ 2 3 i=1λ i .
Recalling thatλ i := λ i − λ for i = 1, 2, 3, where each λ i is as before, the quadratic and linear terms in the last line can be expanded and recast as whose right-hand side we recognise as λP (∇u). In summary, we have shown that det ∇u − λ 3 =λ 1λ2λ3 + λh ′ (λ 3 )P (∇u).
Inserting this into (3.6) gives (on dropping the term with prefactor γ, since it will no longer be needed) If P were quasiconvex at λ1 then the second integral would by definition satisfy for any u in W 1,q (Ω) with q ≥ 2. Finally, a short calculation shows that P (λ1) = 0, so that the right-hand side of the last inequality vanishes. Thus the only condition needed in order to conclude that I(u) ≥ I(u λ ) would be (3.16), which ensures the positivity of the integral involving F λ 1 .