A new example for the Lavrentiev phenomenon in nonlinear elasticity

We present a new example for the Lavrentiev phenomenon in context of nonlinear elasticity, caused by an interplay of the elastic energy’s resistance to infinite compression and the Ciarlet–Nečas condition, a constraint preventing global interpenetration of matter on sets of full measure.


Introduction
Following the by-now classical theory of nonlinear elasticity [1,2,12,35], we consider an elastic body occupying in its reference configuration an open bounded set Ω ⊆ R d with Lipschitz boundary ∂Ω, subject to a prescribed boundary condition on a part Γ ⊂ ∂Ω with positive surface measure, i.e., H d−1 (Γ) > 0. A possible deformation of the body is described by a mapping y : Ω → R d such that y = y 0 on Γ, where y 0 is the imposed boundary data.Its associated internally stored elastic energy is given by the functional with a function W representing material properties: the local energy density, which is assumed to be a function of the deformation gradient.A crucial aspect of this mathematical model [6] is to define a suitable class of admissible deformations that capture relevant features, such as non-interpenetration of matter, which mathematically translates into injectivity of y.However, considering different admissible classes can lead to a Lavrentiev phenomenon, i.e., the functional infima differ when restricting the minimization of (1.1) to more regular deformations, such as W 1,∞ in place of W 1,p .Functionals demonstrating this behavior were first discovered in the early 20th century [29,31].There the minimum value over W 1,1 is strictly less than the infimum over W 1,∞ .For an extensive survey on the Lavrentiev phenomenon in a broader context, we refer the interested reader to [11].
In the context of nonlinear elasticity, the Lavrentiev phenomenon was first observed with admissible deformations that allow cavitations, i.e., the formation of voids in the material [4].For the study of cavitations, we refer to [9,23] and references therein.
A natural question raised in [7] and [5] is: Can the Lavrentiev phenomenon occur for elastostatics under growth conditions on the stored-energy function, ensuring that all finite-energy deformations are continuous?This is indeed the case, and the first example of this kind has been given in two dimensions [17,18,19].It features an energy density with desirable properties: W is smooth, polyconvex, frame-indifferent, isotropic, W (F ) |F | p with p > 2, and W (F ) → ∞ as det F → 0+.Moreover, admissible deformations are almost everywhere (a.e.) injective.In these examples, the reference configuration is represented by a disk sector Ω α := {r(cos θ, sin θ) : 0 < r < 1, 0 < θ < α}.A crucial aspect for the emergence of the Lavrentiev phenomenon in that example is the local behavior of (almost) minimizers near the tip at r = 0, interacting with a particular choice of boundary conditions.The latter fix the origin y(0, 0) = (0, 0), y(1, θ) = (1, β α θ) and y(Ω α ) ⊂ Ω β , where 0 < β < 3  4 α.In the current paper, we provide examples of the Lavrentiev phenomenon in elasticity both in two and three dimensions.The elastic energy is of a simple neo-Hookean form with physically reasonable properties as described above, and admissible deformations are continuous and a.e.-injective.Differently from [17,18,19], the Lavrentiev phenomenon in our example is not related to the local behavior of almost minimizers near prescribed boundary data, but to a possible global self-intersection of the material that still maintains a.e.injectivity by compressing two different material cross-sections to a single point (or line in 3D) of self-contact in deformed configuration.It turns out to be energetically favorable due to our particular choice of boundary conditions but is no longer possible if we restrict to a sufficiently smooth class of admissible deformations.This then leads to a higher energy infimum.
Throughout the paper, we consider locally orientation preserving deformations with p-Sobolev regularity If p > d, the Sobolev embedding theorems ensure the continuity of W 1,p -mappings.The question of injectivity of deformations, i.e., non-interpenetration of matter, is more delicate and it has been extensively studied.Let us mention just a few references.For local invertibility conditions, see [8,16,25].As for global injectivity one may ask some coercivity with respect to specific ratios of powers of a matrix F , its cofactor matrix cof F , and its determinant det F combined with global topological information from boundary values [3,24,27,28,33,38] or second gradient [21], as well as other regularity [13,37,38] and topological restrictions such as (INV)-condition [8,14,22,34,36] and considering limits of homeomorphisms [10,15,27,33].In this paper, we adopt the approach from [13], where the authors investigate a class of mappings y ∈ W 1,p + (Ω; R d ) satisfying the Ciarlet-Nečas condition: and prove that the mappings of this class are a.e.-injective.
In the examples we consider W (F ) |F | p + (det F ) −q , the reference configuration Ω and the boundary data y 0 are chosen in such a way that the energy E favors deformations that have nonempty sets of non-injectivity.In particular, we construct in Section 3 (resp.Section 4) a competitor y ∈ W 1,p + (Ω; R 2 ) (resp.y ∈ W 1,p + (Ω; R 3 )) satisfying the Ciarlet-Nečas condition (CN) and having a line (resp.a plane) of non-injectivity.The energy of such deformation is shown to be strictly less than that of Lipschitz deformations, for which injectivity is ensured everywhere.The global injectivity in this case follows from the Reshetnyak theorem for mappings of finite distortion [30].Specifically, a mapping y ∈ W  [20,Lemma 3.3].For a general theory of mappings of finite distortion the reader is referred to [26].
Our example also shows that, depending on the precise properties of the energy density W , there can be an energy gap between the class of orientation preserving a.e.injective deformations (i.e., satisfying the Ciarlet-Nečas condition) on the one hand and the strong (or weak) closure of Sobolev homeomorphisms in the ambient Sobolev space on the other hand.If these classes do not coincide (which can certainly happen if there is not enough control of the distortion via the energy to apply the Reshetnyak theorem [30] as above), one has to carefully choose which constraint to use to enforce non-interpenetration of matter, even if p > d.In our example, the Ciarlet-Nečas condition does allow a "deep" self-interpenetration in such a scenario.As a matter of fact, this self-interpenetration is also topologically stable in the sense that all C 0 -close deformations still self-intersect (see Figures 1 and 2 for reference and deformed configurations in the 2D case).To us, it seems doubtful that such a deformation corresponds to a physically meaningful state.This strongly speaks for preferring a closure of homeomorphisms as the admissible class in such cases.An open problem in this context is to find sharp conditions for the energy density so that all a.e.-injective orientation preserving Sobolev maps can be found as strong (or weak) limits of Sobolev homeomorphisms in W 1,p .In case p ≥ d, having K y ∈ L κ with κ > d − 1 as above is clearly sufficient, but probably not necessary, at least not in dimension d ≥ 3.
The plan of the paper is the following.Section 2 is dedicated to the general setting of the problem and a few basic auxiliary results.In Sections 3 and 4, we discuss the Lavrentiev phenomenon in dimensions two and three for the energy E in the class of deformations y ∈ W 1,p + (Ω, R d ) satisfying the Ciarlet-Nečas condition (CN) as well as suitable Dirichlet boundary conditions on selected parts of ∂Ω.
Proposition 2.1.For W given by (2.1) and (2.2), we have that For later use, we point out the following proposition, which express the minimality of the identity map in a quantitative form.We denote from now on F 2 the operator norm of F ∈ R d×d , i.e., F 2 := sup{|F e| : |e| = 1}.Proposition 2.2.For W given by (2.1) and (2.2), we have the following lower bound with some constant c = c(d, p, q) > 0.
In both the examples we present in this paper, we fix as reference configuration an open bounded set Ω s ⊆ R d with Lipschitz boundary ∂Ω s .Our set will always have two connected components whose precise shape will be chosen depending on the dimension d and will further depend on a parameter s > 0. For every y ∈ W 1,p + (Ω s , R d ) we define the energy functional In particular, notice that the energy E s is normalized to 0 at y = id, since W attains minimum value on SO(d).Let Γ s be a subset of ∂Ω s , The existence of minimizers is nowadays classic and follows, e.g., from [13, Theorem 5] due to Proposition 2.1 since p > d.

The Lavrentiev phenomenon in dimension two
In dimension d = 2 we consider a reference configuration Ω s consisting of two stripes of width 0 < s < 1, given by (see also Fig. 1) We denote by Γ s the subset of ∂Ω s given by On Γ s we impose the following Dirichlet boundary condition Notice that on both pieces of Ω s the function y 0 is such that ∇y 0 = 1l, which minimizes W pointwise.However, y 0 (Ω s ) is cross-shaped with y 0 doubly-covering the center.Hence, y 0 is not globally injective and does not satisfy (CN).It is not hard to see that Y s , defined by (2.7), still contains many admissible functions as long as s < 1.
We start with the proof of Proposition 3.2.
Proof of Proposition 3.2.In order to prove (3.3) we explicitly construct a deformation y α,β forming a cross with self-intersection.By squeezing with suitable rate two central crosssections to a point, which will be the only point of intersection in y α,β (Ω), we produce an almost-minimizer of E s in Y s .
We now estimate the behavior of the energy E s (y α,β ) as s → 0. Below, the symbol stands for an inequality up to a positive multiplicative constant independent of s ∈ (0, 1] and x ∈ S 1 .We further write ≈ if such inequalities hold in both directions.By minimality of the identity matrix, by definition of W , and by construction of y α,β on S ′ 1 , we have that Moreover, since 0 < α < 1, the mean value theorem gives This means that on S ′′ 1 it holds that |∇y α,β − 1l| s α uniformly in x.By Taylor expansion of W at 1l (where DW (1l) = 0 by definition of γ), we infer that Combining (3.6) and (3.7), we obtain the following upper bound for the energy for all sufficiently small s as long as (3.5) holds: (3.9) For x ∈ S 2 , we extend y α,β with a suitable shifted copy.With a slight abuse of notation, we set where Q and ξ are given by (3.1).It is straightforward that By the change-of-variables formula for Sobolev mappings, y α,β satisfies (CN).Clearly, the estimate (3.9) holds true also on S 2 .This concludes the proof of (3.3) for q ∈ (1, p p−2 ).
To cover q = p p−2 we need to consider a slightly different example.With the same notation introduced above for S ′ 1 and S ′′ 1 , we set ŷα,β (x and for x ∈ S ′′ and On S ′′ 1 we can repeat the argument of (3.7).This concludes the proof of the proposition.

Proof of Proposition 3.3. With an explicit construction of a competitor y
one can show that there exists M > 0 such that min Let us now fix y ∈ W 1,∞ (Ω s ; R 2 ) ∩ Y s with finite energy E s (y).We claim that for a.e.σ ∈ (−s, s) one of the following inequality is satisfied: has to intersect {z ∈ R 2 | z 2 = ζ} (see also Fig. 4).For σ ∈ (−s, s), we distinguish two cases: 1 ) and y(T σ 2 ) only intersect {σ} × (−1, 1) and (−1, 1) × {σ}, respectively.Denoting by K(x, y(x)) the distorsion of we notice that y satisfies with q > d − 1 = 1.Since y is nonconstant, due to the boundary data, by the Reshetnyak theorem [30] for mappings of finite distortion, y is open and discrete.Moreover, any open map that is injective almost everywhere is indeed injective everywhere (as pointed out in [20,Lemma 3.3]).Hence, the case (ii) is impossible, and the general deofrmation is pictured in Fig. 3. Therefore, for every σ ∈ (−s, s) we are in the case (i).For every σ ∈ (−s, s) such that the integrals in (3.10) are well defined, we may assume without loss of generality that y(T σ 1 ) ∩ [{σ} × [1, +∞)] = ∅ (the other cases can be treated similarly), and let x 1 ∈ (−1, 1) be such that y(x 1 , σ) ∈ y(T σ 1 ) ∩ [{σ} × [1, +∞)].Since the shortest path connecting (−1, σ) to the point y(x 1 , σ) is the segment, by the boundary conditions of y we have that With the same argument, we deduce that We are now in a position to conclude for (3.4).We define the sets In view of (3.10), we have that A∪B = (−s, s), up to a set of L 1 -measure zero.Moreover, A ∩ B = ∅.By (2.4) we estimate (recall that • 2 denotes the operator norm) Thanks to the Jensen inequality, to (3.10), and to the definition of A and B, we continue in (3.14) with for some positive constant m independent of y and of s.This concludes the proof of (3.4).
Remark 3.5.The argument in Remark 3.4 also shows that the two-dimensional example in Proposition 3.2 is optimal in the following sense: if p > 2 and q > p p−2 , then every y ∈ W 1,p (Ω s ; R 2 ) ∩ Y s with finite energy satisfies K y ∈ L η (Ω s ) for η = pq 2q+p > 1 (see (3.15)).Hence, y has to be injective.This would rule out the example constructed in the proof of Proposition 3.2.

The Lavrentiev phenomenon in dimension three
In this section, we show a three-dimensional generalization of the Lavrentiev phenomenon proven in Theorem 3.1.The example is created by simply thickening the twodimensional version in another direction, corresponding to the variable x 1 below, while (x 2 , x 3 ) correspond to the two variables of the 2D example.
For s ∈ (0, 1), the reference configuration Ω s consists now of the union of two thin cuboids of width s.Namely, we write We consider the Dirichlet datum and the set of admissible deformations Similar to Theorem 3.1, we have the Lavrentiev phenomenon in the following form.Theorem 4.1.For every p ∈ (3, 4) and every q ∈ (2, p p−2 ) there exists s ∈ (0, 1] such that for every s ∈ (0, s] the following holds: The proof of Theorem 4.1 is subdivided into two propositions given below.Compared to the two-dimension case, we now have to face an additional difficulty, because "fully going around" (case (i) in the proof of Proposition 3.3) is no longer the only way the two pieces can avoid each other after deformation.In principle, it should be possible to generalize our three-dimensional example to any dimension d ≥ 3, but for the sake of simplicity, we will stick to d = 3, the practically most relevant case.For every p ∈ (3, 4) and every q ∈ (2, p p−2 ), there exists s ∈ (0, 1) such that for every s ∈ (0, s] with constants 0 < m < M < +∞ independent of s.
We start with the proof of Proposition 4.2.
Proof of Proposition 4.2.As in the proof of Proposition 3.2, it is enough to construct a sequence of competitors y s ∈ Y s satisfying E s (y s ) = o(s) as s ց 0. To this purpose, let us fix α, β ∈ (0, 1) (to be determined later on) and let us define In order to prove the asymptotic (4.3) we define the map y α,β : Ω s → R 3 as To show that y α,β ∈ Y s for s small, we have to show that ∇y α,β ∈ L p (Ω s ; R 3×3 ).We focus on S 1 , as the definition of y α,β leads to the same computations on S 2 .By construction of y α,β , on S 1 we have that We notice that Imposing the integrability of (det ∇y α,β ) −q on S 1 we deduce that it must be Combining (4.5) and (4.6), we infer that for any choice of p ∈ (3, 4) and of q ∈ (2, p p−2 ), we can find α, β ∈ (0, 1) such that y α,β ∈ Y s with (det ∇y α,β From (4.5)-(4.7)we deduce that there exists ρ ∈ (0, 1) (depending on α, β but not on s) such that 0 ≤ Arguing in the same way, estimate (4.10) can be obtained on S 2 , leading to (4.3).This concludes the proof of the proposition.
The following two lemmas show some useful properties of deformations y ∈ W 1,∞ (Ω s ; R 3 )∩ Y s with low energy, which will be useful to conclude for (4.4).In the sequel, we denote by π : R → [−1, 1] the projection of R to the interval [−1, 1], defined as Lemma 4.4.There exists M > 0 such that for every s ∈ (0, 1) min y∈W Proof.The thesis follows easily by a direct construction of a competitor y ∈ W 1,∞ (Ω s ; R 3 )∩ Y s .For instance, we define Then, it is clear that E s (y) ≤ Ms for some M > 0 independent of s.
In particular, we notice that, due to the boundary condition on Γ s , we have that Then, by (4.13) and by the Chebyshev inequality, Let us now fix x 1 ∈ D ε and x 2 ∈ [−1, 1], let us denote by θ x 1 : [−1, 1] → R 3 the curve θ x 1 (t) := y(x 1 , t, σ), and let us write In particular, we notice one of the two cases must hold: In the case (i), by definition of D ε and by the boundary conditions on y we have that we may repeat the argument of the first two lines of (4.20) and obtain that |v| ≤ ε 1 p+1 .All in all, we have shown that for every x 1 ∈ D ε and every To achieve (4.14) it remains to consider x 1 / ∈ D ε .In this case, by (4.19) we may find . Then, by triangle inequality, by the Hölder continuity (4.18) of y, and by the previous step we have that for every for a suitable constant c N,p > 0 depending only on γ and c N,p , and therefore only on p and N.This concludes the proof of (4.14).
The same argument can be used to infer (4.17) taking into account the boundary conditions on ∂S 2 .
We are now in a position to prove Proposition 4.3.
Proof of Proposition 4.3.Since Lemma 4.4 holds, we are left to provide a lower bound for the minimum problem (4.4).To this purpose, let M > 0 be the constant determined in Lemma 4.4 and fix a deformation y ∈ W 1,∞ (Ω s ; R 3 ) ∩ Y s such that E s (y) ≤ (M + 1)s.We further set γ := 1 − 2 p and fix ε > 0 such that c N,p ε γ p+1 ≤ 1 3 , where c N,p > 0 is the constant defined in Proposition 4.5.We claim that for every σ ∈ A N ∩ B N , at least one of the following inequalities must hold: We now show that given (4.28a) and (4.28b), y ∈ Y s cannot be injective in Ω s .This immediately yields a contradiction, as we already know that any y ∈ W 1,∞ (Ω s ; R 3 ) ∩ Y s with finite energy must be a homeomorphism (as a consequence of the theory mappings of finite distortion [30], as already outlined in the introduction).
Notice that if g(x 1 , τ1 , τ2 ) = 0 for some (x 1 , τ1 , τ2 ) ∈ (−1, 1) 3 , then y is not injective, since (x 1 , τ1 , σ) ∈ S 1 , (0, σ + 4, −τ 2 ) ∈ S 2 and the values of y on these two points coincide.As a consequence of (4.28) and of the boundary conditions of y ∈ Y s on Γ s , the vector field All in all, we have shown that any deformation y ∈ W 1,∞ (Ω s ; R 3 ) ∩ Y s satisfying (4.23) has energy E s (y) ≥ δs for some positive constant δ independent of s.Thus, (4.4) holds and the proof of the proposition is concluded.