Injectivity almost everywhere and mappings with finite distortion ... - arXiv

arXiv:1704.08022v4 [math.FA] 3 May 2017

Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity A. O. Molchanova, S. K. Vodop0 yanov Abstract We show that a sufficient condition for the weak limit of a sequence of Wq1 -homeomorphisms with finite distortion to be almost everywhere injective for q ≥ n − 1, can be stated by means of composition operators. Applying this result, we study nonlinear elasticity problems with respect to these new classes of mappings. Furthermore, we impose loose growth conditions on the stored-energy function for the class of Wn1 -homeomorphisms with finite distortion and integrable inner as well as outer distortion coefficients. Keywords: finite distortion, functional minimisation problem, injectivity almost-everywhere, nonlinear elasticity, polyconvexity.

Introduction Some problems in nonlinear elasticity (including, for instance, those involving hyperelastic materials) reduce to that of minimising the total energy functional. In this situation, and in contrast to the case of linear elasticity, the integrand is almost always nonconvex, while the functional is nonquadratic. This renders the standard variational methods inapplicable. Nevertheless, for a sufficiently large class of applied nonlinear problems, we may replace convexity with certain weaker conditions, i.e. polyconvexity [3]. Denote by Mm×n the set of m×n matrices. Recall that a function W : Ω× 3×3 M → R, Ω ⊂ R3 , is called polyconvex if there exists a convex function G(x, ·) : M3×3 × M3×3 × R+ → R such that G(x, F, Adj F, det F ) = W (x, F ) for all F ∈ M3×3 with det F > 0, 1

A. O. Molchanova, S. K. Vodop0 yanov

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almost everywhere (a.e.) in Ω. Let Ω be a bounded domain in R3 , i.e. a bounded connected open set, which satisfies a strong Lipschitz condition. Ball’s method [3] is to consider a sequence {ϕk }k∈N minimising the total energy functional Z (0.1) I(ϕ) = W (x, Dϕ) + Θ(x, ϕ) dx. Ω

over the set of admissible deformations AB = {ϕ ∈ W11 (Ω), I(ϕ) < ∞, J(x, ϕ) > 0 a.e. in Ω, ϕ|∂Ω = ϕ|∂Ω }, (0.2) where ϕ are Dirichlet boundary conditions and J(x, ϕ) stands for the Jacobian of ϕ, J(x, ϕ) = det Dϕ(x). Furthermore, it is assumed that the coercivity inequality W (x, F ) ≥ α(|F |p + | Adj F |q + (det F )r ) + g(x)

(0.3)

holds for almost all x ∈ Ω and all F ∈ M3×3 , det F > 0, where p > 2, p , r > 1 and g ∈ L1 (Ω), Adj F denotes the adjoint matrix, i.e. q ≥ p−1 a transposed matrix of (2 × 2)-subdeterminants of F . Moreover, the storedenergy function W is polyconvex. By coercivity, it follows that the sequence (ϕk , Adj Dϕk , det Dϕk ) is bounded in the reflexive Banach space Wp1 (Ω) × Lq (Ω) × Lr (Ω). Hence, there exists a subsequence converging weakly to an element (ϕ0 , Adj Dϕ0 , det Dϕ0 ). For the limit ϕ0 to belong to the class AB of admissible deformations, we need to impose the additional condition: W (x, F ) → ∞ as det F → 0+

(0.4)

(see [7] for more details). This condition is quite reasonable since it fits in with the principle that “extreme stress must accompany extreme strains”. Another important property of this approach is the sequentially weakly lower semicontinuity of the total energy functional, I(ϕ) ≤ lim I(ϕk ), k→∞

which holds because the stored-energy function is polyconvex. It is also worth noting that Ball’s approach admits the nonuniqueness of solutions observed experimentally (see [3] for more details).

Injectivity almost everywhere and mappings with finite distortion

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One of the most important requirements of continuum mechanics is that interpenetration of matter does not occur, from which it follows that any deformation has to be injective. Global injectivity of deformations has been established by J. Ball [4] for hyperelastic bodies whose energy “grows sufficiently fast for extreme strains”, within the existence theory based on minimisation of the energy [3]. More precisely, if ϕ : Ω → Rn , Ω ⊂ Rn , is a mapping in Wp1 (Ω), p > n, coinciding on the boundary ∂Ω with a homeomorphism ϕ and J(x, ϕ) > 0 a.e. in Ω, ϕ(Ω) is strongly Lipschitz, and if for some σ > n Z Z | Adj Dϕ(x)|σ −1 σ |(Dϕ(x)) | J(x, ϕ) dx = dx < ∞, (0.5) J(x, ϕ)σ−1 Ω

Ω

then ϕ is a homeomorphism of Ω on ϕ(Ω) and ϕ−1 ∈ Wσ1 (ϕ(Ω)). To apply this result to nonlinear elasticity it is required that some additional conditions on the stored-energy function be imposed in order to obtain invertibility of deformations. Thus, in [4] (see also Exercise 7.13 of [12]), it is considered a domain Ω ⊂ R3 with a strongly Lipschitz boundary ∂Ω and a polyconvex stored-energy function W . Suppose that there exist constants 2q , as well as a function g ∈ L1 (Ω) α > 0, p > 3, q > 3, r > 1, and m > q−3 such that W (x, F ) ≥ α(|F |p + | Adj F |q + (det F )r + (det F )−m ) + g(x)

(0.6)

for almost all x ∈ Ω and all F ∈ M3×3 , det F > 0. Take a homeomorphism ϕ : Ω → Ω0 in Wp1 (Ω) with J(x, ϕ) > 0 a.e. in Ω. Then there exists a mapping ϕ : Ω → Ω0 minimising the total energy functional (0.1) over the set of admissible deformations (0.2), which is a homeomorphism due to (0.5) with > 3. ϕ−1 ∈ Wσ1 (Ω), σ = q(1+m) q+m In this article we obtain the injectivity property (Theorem 3.1) based on the boundedness of the composition operator ϕ∗ : L1p (Ω0 ) → L1q (Ω). Boundedness of these operators is intimately related to a condition of finite distor1 tion. Recall that a W1,loc -mapping f : Ω → Rn with nonnegative Jacobian, J(x, f ) ≥ 0 a.e., is called a mapping with finite distortion (f ∈ F D(Ω)) if |Df (x)|n ≤ K(x)J(x, f ) for almost all x ∈ Ω, where 1 ≤ K(x) < ∞ a.e. in Ω. A function |Dϕ(x)|n KO (x, ϕ) = J(x, ϕ)

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is called the outer distortion coefficient 1 . It is worth noting that mappings with finite distortion arise in nonlinear elasticity from geometric considerations: it would be desirable that the deformation is continuous, maps sets of measure zero to sets of measure zero, is a one-toone mapping and that the inverse map has “good” properties. Hence, many research groups all over the word have worked on this issue (see [2, 10, 11, 20, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 35, 43, 61] and a lot more). It is known that in the planar case (Ω, Ω0 ⊂ R2 ) a homeomorphism 1 1 ϕ ∈ W1,loc (Ω) has an inverse homeomorphism ϕ ∈ W1,loc (Ω0 ) if and only if ϕ is 1 -regularity a mapping with finite distortion [24, 25]. In the spatial case Wn,loc 1 of the inverse mapping was shown for Wq,loc -homeomorphism, q > n−1, with the integrable inner distortion 2 KI (x, ϕ) =

| Adj Dϕ(x)|n . J(x, ϕ)n−1

Moreover, the relaxation of (0.5) on the case σ = n, Z Z Z | Adj Dϕ(x)|n −1 n |Dϕ (y)| dy = dx = KI (x, ϕ) dx, J(x, ϕ)n−1 Ω0

Ω

Ω

holds (see [43] for more details). In [29, 30, 31, 32] the authors study Wn1 -homeomorphisms ϕ : Ω → Ω0 between two bounded domains in Rn with finite energy and consider the behaviour of such mappings. In general, the weak Wn1 -limit of a sequence of homeomorphisms may lose injectivity. However, if there is a requirement on totally boundedness of norms of the inner distortion kKI (·, ϕ) | L1 (Ω)k and some additional requirements, then the limit map is a homeomorphism. The main idea behind the proof of existence and global invertibility is to investigate admissible deformations ϕk in parallel with its inverse ϕ−1 k along 3 to a minimising sequence {ϕk }. This is possible due to integrability of the inner distortion as this ensures the existence and regularity of an inverse map belonging to Wn1 . Note that the authors of these papers include requirements of integrability of the inner distortion coefficient in the coercive inequality. 1

It is assumed that KO (x, ϕ) = 1 if J(x, ϕ) = 0. Here KI (x, ϕ) = 1 if | Adj Dϕ(x)| = 0, and KI (x, ϕ) = ∞ if | Adj Dϕ(x)| = 6 0 and J(x, ϕ) = 0. 3 See [57] for another proof of this property under weaker assumption 2

Injectivity almost everywhere and mappings with finite distortion

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The authors of the current paper prefer to include this condition to the class of admissible deformation, so as to obtain more “fine graduation” of deformations. We also emphasize that the aforementioned regularity properties of an inverse homeomorphism (including the case q = n − 1) can be obtained using a technique of the theory of bounded operators of Sobolev spaces. Putting , p0 = σ, q = n − 1, q 0 = ∞, % = σ in Theorem 2.2 [57, Theorem p = σ(n−1) σ−1 3] we derive the aforementioned result from [4]. By taking p = p0 = % = n, q = n − 1, q 0 = ∞ in the same theorem one can obtain the regularity of an inverse mapping from [43]. On the other hand, the Sobolev space Wn1 is a natural candidate for elasticity deformation due to the “continuity” property of minors. It is known that rank-l minors of Dϕk are weakly converging if ϕk belongs to Wp1 with p ≥ l, 1 ≤ l < n [3, 37, 45, 47]. In the case l = n there is no weak convergence but something close to it [47, §4.5]. For achieving weak convergence of Jacobians, it is necessary to impose some additional conditions, for instance, nonnegativity of Jacobians almost everywhere [39]. Here it will be convenient for us the next formulation of this assertion, which can be found in [17]. Lemma 0.1 (Weak continuity of minors). Let Ω be a domain in Rn with locally Lipschitz boundary and a sequence fk : Ω → Rn , k = 1, 2, . . . , converge 1 weakly in Wn,loc (Ω) to a mapping f0 . For l-tuples 1 ≤ i1 < · · · < il ≤ n and 1 ≤ j1 < · · · < jl ≤ n the equality Z Z ∂(f0i1 , . . . , f0il ) ∂(fki1 , . . . , fkil ) dx = θ dx (0.7) lim θ k→∞ ∂(xj1 , . . . , xjl ) ∂(xj1 , . . . , xjl ) Ω

Ω

◦

holds for every θ in Ln/(n−l) (Ω), the space of functions in Ln/(n−l) (Ω) with compact support in Ω, and corresponding l × l minors4 of Dfk and Df0 , l = 1, 2, . . . , n − 1. Moreover, if in addition J(x, fk ) ≥ 0 a.e. in Ω, the equality (0.7) holds for l = n. Whereas we have deal with Wn1 -mappings with finite distortion in this article, we reduce coercivity conditions on the stored-energy function to W (x, F ) ≥ α|F |n + g(x). 4

(0.8)

i.e. determinants of the matrix that is formed by taking the elements of the original matrix from the rows whose indexes are in (i1 , i2 , . . . , il ) and columns whose indexes are in (j1 , j2 , . . . , jl )

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For given constants p, q ≥ 1 and M > 0 we define the class of admissible deformations H(p, q, M ) = {ϕ : Ω → Ω0 is a homeomorphism, ϕ ∈ W11 (Ω) ∩ F D(Ω), I(ϕ) < ∞, J(x, ϕ) ≥ 0 a.e. in Ω, KO (·, ϕ) ∈ Lp (Ω), kKI (·, ϕ) | Lq (Ω)k ≤ M }, where F D(Ω) is a class of mappings with finite distortion, KO (x, ϕ) and KI (x, ϕ) are the outer and the inner distortion coefficients. We prove an existence theorem in the following formulation (see precise requirements in Section 4.2). Theorem (Theorem 4.1 below). Let Ω, Ω0 ⊂ Rn be bounded domains with locally Lipschitz boundaries. Given a polyconvex function W (x, F ), satisfying the coercivity inequality (0.8), and a nonempty set H(n−1, s, M ) with M > 0, s > 1, then there exists at least one homeomorphic mapping ϕ0 ∈ H(n − 1, s, M ) such that

I(ϕ0 ) = inf{I(ϕ), ϕ ∈ H(n − 1, s, M )}.

The existence theorem is also obtained for a class H(p, q, M ; ϕ) of mappings with prescribed boundary values and a class H(p, q, M ; ϕ, hom) of deformations of the same homotopy class as a given one (it covers the case s = 1 as well). In some cases the requirements on the definition of the admissible classes are redundant. Indeed, the hypothesis of homeomorphism in classes H(n − 1, s, M ; ϕ, hom) and H(n − 1, s, M ; ϕ) follows from the integrability of KO (x, ϕ) and KI (x, ϕ) for s > 1 (see Remark 4.2); and the restriction on KO (x, ϕ) is not needed if there is boundary conditions (Remark 4.3). Furthermore, in some previous works the condition J(x, ϕ) > 0 a.e. follows from (0.4) (e.g. [3, 12]). On the other hand, for a non-constant mapping ϕ ∈ Wn1 (Ω) with the distortion coefficient KO (·, ϕ) ∈ Ls (Ω) for some s > n − 1, the set where the Jacobian vanishes is of measure zero, e.g. [33, Theorem 1.1]. Here we provide a proof of the condition J(x, ϕ) > 0 for W11 mappings, which is almost injective, has nonnegative Jacobian a.e. and has the Luzin N −1 -property (see Lemma 3.7). Moreover, the class of admissible deformations from the paper [29] is related to H(n − 1, 1, M ; ϕ, hom) (see Remark 4.3). For the same reason, the elasticity result of [4] can be derived from the result of this paper. Indeed, the integrability of the distortion coefficient follows from the H¨older inequality σr and (0.6) by s = rn+σ−n where σ = q(1+m) (see Section 5 for details). q+m

Injectivity almost everywhere and mappings with finite distortion

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Some important propreties of mappings of these classes can be found in [58]. Note also that the property of mapping to be sense preserving in the topological way follows from the property that the required deformation is a mapping with bounded (n, q)-distortion if q > n − 1 (see [9, Remark 1]). Additionally, there is a different approach to injectivity which was proposed by P. Ciarlet and I. Neˇcas in [14]. This approach rests upon the additional injectivity condition Z J(x, ϕ) dx ≤ |ϕ(Ω)|

(0.9)

Ω

on the admissible deformations if Ω ⊂ Rn is a bounded open set with C 1 smooth boundary, ϕ ∈ Wp1 (Ω), p > n, and J(x, ϕ) > 0 a.e. in Ω. Under these assumptions, there exists an almost everywhere injective minimiser of the total energy functional. In the three-dimensional case the relation (0.9) under the weaker hypothesis p > n − 1 was studied in [52]. In this case, ϕ may no longer be continuous and the inverse mapping ϕ−1 has only regularity BVloc (ϕ(Ω), Rn ). Local invertibility properties of the mapping ϕ ∈ Wp1 (Ω), p ≥ n, under the condition J(x, ϕ) > 0 a.e., can be found in [15]. Some other studies of local and global invertibility in the context of elasticity can be found in [8, 13, 21, 22, 32, 40, 41, 49, 50, 51]. Also, see [5, 6] for a general review of research in the elasticity theory. We will now give an outline of the paper. The first and the second sections contain general auxiliary facts and some facts about mappings with finite distortion. The third section is devoted to the injectivity almost everywhere property (Theorem 3.1). This property follows from jointly boundedness of pullback operators defined by a sequence of homeomorphisms ϕk and the uniform convergence of inverse homeomorphisms ψk (Lemma 3.2). Moreover, as a consequence, we obtain the strict inequality J(x, ϕ0 ) > 0 a.e. (Lemma 3.7). The fourth section is dedicated to the existence theorem. In the fifth section we give two examples. In the first example we consider a stored-energy function W (F ) and the Dirichlet data for which minimisation problems for AB and A(s, M ; ϕ) (which is a simplification of H(n−1, s, M ; ϕ)) have solutions. The second example exhibits a function W (F ) violating the coercivity (0.3) and the asymptotic condition (0.4); nevertheless, there exists a solution to the minimisation problem in H(n − 1, s, M ). Some ideas of this article were announced in the note [61].

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1

Geometry of domains

In this section we present some important concepts and statements necessary to proceed. On a domain Ω ⊂ Rn , i.e. a nonempty, connected, open set, we define in the standard way (see [36] for instance) the space C0∞ (Ω) of smooth functions with compact support, the Lebesgue spaces Lp (Ω) and Lp,loc (Ω) of 1 integrable functions, and Sobolev spaces Wp1 (Ω) and Wp,loc (Ω), 1 ≤ p ≤ ∞. A mapping f ∈ L1,loc (Ω) belongs to homogeneous Sobolev class L1p (Ω), p ≥ 1, if it has the weak derivatives of the first order and its differential Df (x) belongs to Lp (Ω). For working with geometry of domains we need the following definition. Definition 1. A homeomorphism ϕ : U → U 0 of two open sets U , U 0 ⊂ Rn is called a quasi-isometric mapping if the following inequalities |ϕ(y) − ϕ(x)| ≤M y→x |y − x| lim

and

|ϕ−1 (y) − ϕ−1 (z)| ≤M y→z |y − z| lim

hold for all x ∈ U and z ∈ U 0 where M is some constant independent of the choice of points x ∈ U and z ∈ U 0 . Definition 2. A mapping ϕ : U → U 0 of two open sets U , U 0 ⊂ Rn is a biLipschitz mapping if the following inequality l|y − x| ≤ |ϕ(y) − ϕ(x)| ≤ L|y − x| holds for all x, y ∈ U where l and L are some constants independent of the choice of points x, y ∈ U . Recall the following definition. Definition 3. A domain Ω ⊂ Rn is called a domain with locally quasiisometric boundary whenever for every point x ∈ ∂Ω there are a neighbourhood Ux ⊂ Rn and a quasi-isometric mapping νx : Ux → B(0, rx ) ⊂ Rn , where the number rx > 0 depends on Ux , such that νx (Ux ∩ ∂Ω) ⊂ {y ∈ B(0, rx ) | yn = 0} and νx (Ux ∩ Ω) ⊂ {y ∈ B(0, rx ) | yn > 0}. Remark 1.1. In some papers, a bi-Lipschitz mapping (with the same constants l and L for all points x ∈ ∂Ω) rather than a quasi-isometric mapping, as in this description, is used. It is evident that the bi-Lipschitz mapping is

Injectivity almost everywhere and mappings with finite distortion

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also a quasi-isometric one. The inverse implication is not valid but the following assertion is true: every quasi-isometric mapping is locally bi-Lipschitz one (see a lemma below). Hence Ω is a domain with locally Lipschitz boundary if and only if it is a domain with quasi-isometric boundary. Lemma 1.1. Let ϕ : Ω → Ω0 be a quasi-isometric mapping then for any fixed ball B b Ω the inequality5 dϕ(B) (ϕ(x), ϕ(y)) ≤ L|ϕ(x) − ϕ(y)| holds for all points x, y ∈ B with some constant L depending on the choice of B only6 . 1 Proof. Take an arbitrary function g ∈ W∞ (ϕ(B)). Then ϕ∗ (g) = g ◦ ϕ ∈ 1 W∞ (B) and, by Whitney type extension theorem (see for instance [54, 55]), 1 1 there is a bounded extension operator extB : W∞ (B) → W∞ (Rn ). Multiply ∗ ∞ extB (ϕ (g)) by a cut-off-function η ∈ C0 (Ω) such that η(x) = 1 for all 1 (Ω), equals 0 points x ∈ B. Then the product η · extB (ϕ∗ (g)) belongs to W∞ 1 near the boundary ∂Ω and its norm in W∞ (Ω) is controlled by the norm 1 (ϕ(B))k. kg | W∞ 1 (Ω0 ), equals 0 near It is clear that ϕ−1∗ (η · extB (ϕ∗ (g))) belongs to W∞ 1 (Ω0 ) is controlled by the norm kg | the boundary ∂Ω0 and its norm in W∞ −1∗ 1 W∞ (ϕ(B))k. Extending ϕ (η · extB (ϕ∗ (g))) by 0 outside Ω0 we obtain a bounded extension operator 1 1 extϕ(B) : W∞ (ϕ(B)) → W∞ (Rn ).

It is well-known (see for example [54, 55]) that a necessary and sufficient condition for the existence of such an operator is an equivalence of the interior metric in ϕ(B) to the Euclidean one: the inequality dϕ(B) (u, v) ≤ L|u − v| holds for all points u, v ∈ ϕ(B) with some constant L. 5

Here dϕ(B) (u, v) is the intrinsic metric in the domain ϕ(B) defined as the infimum over the lengths of all rectifiable curves in ϕ(B) with endpoints u and v. 6 It is well-known that a mapping is quasi-isometric iff the lengths of a rectifiable curve in the domain and of its image are comparable. The last property means the following one: given mapping ϕ : Ω → Ω0 is quasi-isometric iff L−1 dB (x, y) ≤ dϕ(B) (ϕ(x), ϕ(y)) ≤ LdB (x, y) for all x, y ∈ B.

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2

Mappings with finite distortion

Mappings with finite distortion is a natural generalization of mappings with bounded distortion. The reader not familiar with mappings with bounded distortion may look at [47, 48]. To take a close look at the theory of mapping with bounded distortion, the reader can study monographs [25, 27]. Recall that for topological spaces X and Y , a continuous mapping f : X → Y is discrete if f −1 (y) is a discrete set for all y ∈ Y and f is open if it takes open sets onto open sets. Definition 4 ([28, 59]). Given an open set Ω ⊂ Rn and a mapping f : Ω → 1 Rn with f ∈ W1,loc (Ω) is called a mapping with finite distortion, written f ∈ F D(Ω), whenever and |Df (x)|n ≤ K(x)|J(x, f )| for almost all x ∈ Ω, where 1 ≤ K(x) < ∞ a.e. in Ω. In other words, the finite distortion condition amounts to the vanishing of 1 the partial derivatives of f ∈ W1,loc (Ω) almost everywhere on the zero set of the Jacobian. If K ∈ L∞ (Ω), f is called a mapping with bounded distortion (or a quasiregular mapping). For a mapping with finite distortion with J(x, f ) ≥ 0 a.e. the functions KO (x, f ) =

|Df (x)|n J(x, f )

and KI (x, f ) =

| Adj Df (x)|n J(x, f )n−1

(2.1)

when 0 < J(x, f ) < ∞ and KO (x, f ) = KI (x, f ) = 1 otherwise are called the outer and the inner distortion coefficients of f at the point x. It is easy to see that 1 KIn−1 (x, f ) ≤ KO (x, f ) ≤ KIn−1 (x, f ) a.e. x ∈ Ω. In 1967 Yu. Reshetnyak proved strong topological properties of mappings with bounded distortion: continuity, openness and discreteness [46]. 1 Theorem 2.3 of [59] shows that Wn,loc -mapping with finite distortion and nonnegative Jacobian, J(x, f ) ≥ 0 a.e., is continuous. In recent years, a lot of research has been done in order to find the sharp assumptions for these topological properties in the class of mappings with finite distortion, for example, [20, 23, 26, 28, 34].

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Theorem 2.1 ([44]). Let f : Ω → Rn , n ≥ 2, be a non-constant mapping 1 with finite distortion satisfying J(x, f ) ≥ 0 a.e., f ∈ Wn,loc (Ω), KO (·, f ) ∈ Ln−1,loc (Ω) and KI (·, f ) ∈ Ls,loc (Ω) for some s > 1. Then f is discrete and open. On the other hand, mappings with finite distortion are closely related to boundedness of composition operators of Sobolev spaces. Recall that a measurable mapping ϕ : Ω → Ω0 induces a bounded operator ϕ∗ : L1p (Ω0 ) → L1q (Ω) by the composition rule, 1 ≤ q ≤ p < ∞, if the operator ϕ∗ : L1p (Ω0 ) ∩ Liploc (Ω0 ) → L1q (Ω) with ϕ∗ (f ) = f ◦ ϕ, f ∈ L1p (Ω0 ) ∩ Liploc (Ω0 ), is bounded. Lemma 2.1 ([64]). If a measurable mapping ϕ induces a bounded composition operator ϕ∗ : L1p (Ω0 ) → L1q (Ω), 1 ≤ q ≤ p ≤ ∞, then ϕ has finite distortion. Now we consider a generalization of inner and outer distortion functions, which is more conducive to dealing with composition and pullback operators. 1 Following [57], for a mapping f : Ω → Ω0 of class W1,loc (Ω) define the (outer ) distortion operator function   |Df (x)| for x ∈ Ω \ Z, Kf,p (x) = |J(x, f )|1/p  0 otherwise, and the (inner ) distortion operator function   | Adj Df (x)| for x ∈ Ω \ Z, Kf,p (x) = |J(x, f )|(n−1)/p  0 otherwise, where Z is a zero set of the Jacobian J(x, f ). n n Remark 2.1. Note that KO (x, f ) = Kf,n (x) and KI (x, f ) = Kf,n (x) if x ∈ Ω \ Z. Hence KO (·, f ) ∈ Ln−1 (Ω) results in Kf,n (·) ∈ Ln(n−1) (Ω), and kKI (·, f ) | Ls (Ω)k ≤ M implies kKf,n (·) | L% k ≤ M 1/n for % = ns.

The following theorem shows the regularity properties which ensure that the direct and the inverse homeomorphisms belong to corresponding Sobolev classes.

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Theorem 2.2 ([57, Theorem 3]). Let ϕ : Ω → Ω0 be a homeomorphism with the following properties: 1 (Ω), n − 1 ≤ q ≤ ∞; 1. ϕ ∈ Wq,loc

2. the mapping ϕ has finite codistortion: Adj Dϕ(x) = 0 a.e. on the set Z; 3. Kϕ,p ∈ L% (Ω), where q = p).

1 %

=

n−1 q

−

n−1 , p

n − 1 ≤ q ≤ p ≤ ∞ (% = ∞ for

Then the inverse homeomorphism ϕ−1 has the following properties: 1. ϕ−1 ∈ Wp10 ,loc (Ω0 ), where p0 =

p , p−n+1

(p0 = 1 for p = ∞);

2. ϕ−1 has finite distortion (J(y, ϕ−1 ) > 0 a.e. for n ≤ q); 3. Kϕ−1 ,q0 ∈ L% (Ω0 ), where q 0 =

q q−n+1

(q 0 = ∞ for q = n − 1).

Moreover, kKϕ−1 ,q0 (·) | L% (Ω0 )k = kKϕ,p (·) | L% (Ω)k. Remark 2.2. In the place of condition 3 the outer distortion operator function can be used: Kϕ,p ∈ Lκ (Ω) where κ1 = 1q − p1 , n − 1 ≤ q ≤ p ≤ ∞ (κ = ∞ for q = p). Then the next estimate kKϕ−1 ,q0 (·) | L% (Ω0 )k ≤ kKϕ,p (·) | Lκ (Ω)kn−1 is valid (see [57, Theorem 4]). Theorem 2.3 ([57, 63, 64]). A homeomorphism ϕ : Ω → Ω0 induces a bounded composition operator ϕ∗ : L1p (Ω0 ) → L1q (Ω),

1 ≤ q ≤ p < ∞,

where ϕ∗ (f ) = f ◦ ϕ for f ∈ L1p (Ω0 ), if and only if the following conditions hold: 1 1. ϕ ∈ Wq,loc (Ω);

2. the mapping ϕ has finite distortion;

Injectivity almost everywhere and mappings with finite distortion 3. Kϕ,p (·) ∈ Lκ (Ω), where q = p).

1 κ

=

1 q

13

− p1 , 1 ≤ q ≤ p < ∞ (and κ = ∞ for

Moreover, kϕ∗ k ≤ kKϕ,p (·) | Lκ (Ω)k ≤ Ckϕ∗ k for some constant C. Remark 2.3. Necessity is proved in [63, 64] (see also earlier work [53]), and sufficiency, in Theorem 6 of [57]. Theorem 2.4 ([57]). Assume that a homeomorphism ϕ : Ω → Ω0 1. induces a bounded composition operator ϕ∗ : L1p (Ω0 ) → L1q (Ω) for n−1 ≤ q ≤ p ≤ ∞, where ϕ∗ (f ) = f ◦ ϕ for f ∈ L1p (Ω0 ), 2. has finite distortion for n − 1 ≤ q ≤ p = ∞. Then the inverse mapping ϕ−1 induces a bounded composition operator q p and p0 = p−n+1 , and has finite ϕ : L1q0 (Ω) → L1p0 (Ω0 ), where q 0 = q−n+1 distortion. Moreover, −1∗

kϕ−1∗ k ≤ kKϕ−1 ,q0 (·) | Lρ (Ω0 )k ≤ kKϕ,p (·) | Lκ (Ω)kn−1 , where

1 1 1 = 0 − 0. ρ p q

Recall that a differential (n − 1)-form ω on Ω0 is defined as n X

ω(y) =

ck ∧ . . . ∧ dyn . ak (y) dy1 ∧ dy2 ∧ . . . ∧ dy

k=1

A form ω, with measurable coefficients ak , belongs to Lp (Ω0 , Λn−1 ) if 0

kω | Lp (Ω , Λ

n−1

)k =

Z X n Ω

p/2

a2k (y)

1/p dy

< ∞.

k=1

1 Let f = (f1 , . . . , fn ) : Ω → Ω0 belongs to Wq(n−1),loc (Ω) and ω be a smooth ∗ n − 1-form. Then the pullback ϕ ω can be written as

∗

f ω(x) =

n X k=1

ck ∧ . . . ∧ dfn . ak (f (x)) df1 ∧ . . . ∧ df

A. O. Molchanova, S. K. Vodop0 yanov

14

For any ω ∈ Lp (Ω0 , Λn−1 ) the pullback operator f˜∗ ω(x) is defined by continuity [56, Corollary 1.1]: ( f ∗ ω(x), if x ∈ Ω \ (Z ∪ Σ), ∗ ˜ f ω(x) = (2.2) 0, otherwise. As consequence of [56, Theorem 1.1] we can obtain Theorem 2.5. A homeomorphism f : Ω → Ω0 induces a bounded pullback operator f˜∗ : Lp (Ω0 , Λn−1 ) → Lq (Ω, Λn−1 ), 1 ≤ q ≤ p ≤ ∞, if and only if: 1. f : Ω → Ω0 has finite codistortion; 2. Kf,p(n−1) ∈ Lκ (Ω) where

1 κ

=

1 q

− p1 .

Moreover, the norm of the operator f˜∗ is comparable with kKf,p | Lκ (Ω)k. Theorem 2.6 ([56, Theorem 1.3]). Assume that a homeomorphism 1 ϕ : Ω → Ω0 belongs to Wn−1,loc (Ω) and induces a bounded pullback op∗ 0 n−1 erator ϕ˜ : Lp (Ω , Λ ) → Lq (Ω, Λn−1 ), for 1 ≤ q ≤ p ≤ ∞. Then 1 the inverse mapping ϕ−1 ∈ W1,loc (Ω) induces a bounded pullback operator ∗ 1 1 −1 g ϕ : Lq0 (Ω, Λ ) → Lp0 (Ω, Λ ), where q 0 = q and p0 = p . Moreover, the q−1

p−1

−1 ∗ is comparable with the norm of ϕ g norm of the operator ϕ ˜∗ .

3

Almost-everywhere injectivity

It is well known that the limit of homeomorphisms need not be homeomorphism or even an injective mapping. It is illustrated by the simple example of mappings ϕk (x) = |x|k−1 x on the punctured unit ball. Here we have the limit mapping ϕ0 (x) ≡ 0 and injectivity is lost. Recall that a mapping ϕ : Ω → Rn is called injective almost everywhere whenever there exists a negligible set S outside which ϕ is injective. The sequence of homeomorphisms ϕk = (ϕk,1 , ϕk,2 ) : [−1, 1]2 → [−1, 1]2 of the class W21 ([−1, 1]2 ) with integrable distortion, such that ( 2x1 ξk (x2 ) if x1 ∈ [0, 21 ], ϕk,1 (x1 , x2 ) = 2(1 − ξk (x2 ))x1 − (1 − 2ξk (x2 )) if x1 ∈ ( 21 , 1], ϕk,1 (−x1 , x2 ) = −ϕk,1 (x1 , x2 ),

ϕk,1 (x1 , −x2 ) = ϕk,1 (x1 , x2 ),

Injectivity almost everywhere and mappings with finite distortion

15

Figure 1: The sequence of homeomorphisms with an almost everywhere injective limit and ϕk,2 (x1 , x2 ) = x2 , with ξk (t) = everywhere can be lost either.

1+(k−1)t , 2k

shows that injectivity almost

Theorem 3.1. Let Ω, Ω0 ⊂ Rn be bounded domains with locally Lipschitz 1 (Ω), n−1 ≤ boundaries. Consider a sequence of homeomorphisms ϕk ∈ Wr,loc r ≤ n, with J(x, ϕk ) ≥ 0 a.e., such that: 1 1. ϕk → ϕ0 weakly in Wr,loc (Ω) with J(x, ϕ0 ) ≥ 0 a.e. in Ω;

2. every mapping ϕk induces ϕ∗k : L1n (Ω0 ) → L1r? (Ω), r? ≥ 1;

a

bounded

composition

operator

3. the norms of the operators kϕ∗k k are totally bounded. Then the mapping ϕ0 is injective almost everywhere. At first, we verify that the limit mapping ϕ0 induces a bounded composition operator ϕ∗0 : L1n (Ω0 ) ∩ Lip(Ω0 ) → L1r? (Ω). Lemma 3.1. If conditions of Theorem 3.1 are fulfilled, then the mapping ϕ0 induces a bounded composition operator ϕ∗0 : L1n (Ω0 ) ∩ Lip(Ω0 ) → L1r? (Ω). Proof. Consider u ∈ L1n (Ω0 ) ∩ Lip(Ω0 ). Since kϕ∗k k ≤ C, the sequence wk = ϕ∗k u = u ◦ ϕk is bounded in L1r? (Ω). Using a compact imbedding of Sobolev

A. O. Molchanova, S. K. Vodop0 yanov

16

spaces (see [1, Theorem 6.3] for instance), we obtain a subsequence with nr? wk → w0 in Lt (Ω) where 1 < t < n−r ? . From this sequence, in turn, we can extract a subsequence which converges almost everywhere in Ω. If u ∈ L1n (Ω0 ) ∩ Lip(Ω0 ) then w0 (x) = u ◦ ϕ0 (x) for almost all x ∈ Ω. On the other hand, since wk converges weakly to w0 in L1r? (Ω), we have ku ◦ ϕ0 | L1r? (Ω)k = kw0 | L1r? (Ω)k ≤ lim kwk | L1r? (Ω)k k→∞

= lim k→∞

kϕ∗k (u)

|

L1r? (Ω)k

≤ lim kϕ∗k k · ku | L1n (Ω0 )k k→∞

≤ C · ku | L1n (Ω0 )k. Thus, ϕ0 induces a bounded composition operator ϕ∗0 : L1n (Ω0 )∩Lip(Ω0 ) → L1r? (Ω), and moreover, kϕ∗0 k ≤ C.

Consider some regularity properties of the sequence {ϕk }k∈N which meet the requirements of Theorem 3.1. Lemma 3.2. If conditions of Theorem 3.1 are fulfilled and ψk = ϕ−1 k , then there exists a subsequence {ψkl }l∈N such that ψkl → ψ0 locally uniformly. Proof. Notice that the sequence ψk is uniformly bounded since the domain Ω is bounded. 1 On the other hand, since ψk ∈ Wn,loc (Ω0 ) by Theorem 2.2 we obtain the estimate (corollary of Lemma 4.1 of [38])  − n1 r0 0 osc(ψk , S(y , r)) ≤ L ln r

Z

! n1 |Dψk (y)|n dy

,

B(y 0 ,r0 )

r0 centered at y 0 and B(y 0 , r0 ) ⊂ Ω0 2 is the ball of radius r0 centered at y 0 . It follows the equicontinuity of the family of functions {ψk }k∈N on any compact part of Ω0 . where S(y 0 , r) is the sphere of radius r
0 (here 2Q is a cube with the same center as Q and the edges stretched by a factor of two compared to Q). Since ϕ is a measurable mapping, by Luzin’s theorem there is a compact set T ⊂ Q\ϕ−1 (E) of positive measure such that ϕ : T → Ω0 is continuous. Then, the image ϕ(T ) ⊂ Ω0 is compact and ϕ(T ) ∩ E = ∅. Consider an open set U ⊃ E with ϕ(T ) ∩ U = ∅ and U ∩ Ω0 6= ∅. Choose a tuple {B(yi , ri )}i∈N of balls in accordance with Lemma 3.4: {B(yi , ri )}i∈N and {B(yi , 2ri )}i∈N are coverings of U , and the multiplicity of the covering {B(yi , 2ri )}i∈N is finite (B(yi , 2ri ) ⊂ U for all i ∈ N). Then the function fi associated to the ball B(yi , ri ) enjoys ϕ∗ fi = 1 on ϕ−1 (B(yi , ri )) and ϕ∗ f = 0 outside ϕ−1 (B(yi , 2ri )), in particular ϕ∗ fi = 0 on T . In addition, we have the estimate 1

1

1

kϕ∗ fi | L1q (2Q)k ≤ kϕ∗ fi | L1q (Ω)k ≤ C1 Φ(B(yi , 2ri )) σ |B(yi , ri )| p − n . By the Poincar´e inequality (see [36] for instance), for every function g ∈ 1 Wq,loc (Q) with q < n vanishing on T , we have Z

q∗

1/q∗

|g| dx

n/q ∗

Z

≤ C2 l(Q)

Q

1/q

q

|∇g| dx 2Q

nq and l(Q) is the edge length of Q. where q ∗ = n−q Applying the Poincar´e inequality to the function ϕ∗ fi and using the last two estimates, we obtain 1

1

1

1

1

|ϕ−1 (B(yi , ri )) ∩ Q| q − n ≤ C3 Φ(B(yi , 2ri )) σ |B(yi , ri )| p − n . In turn, H¨older’s inequality guarantees that ∞ X

! 1q − n1 |ϕ−1 (B(yi , ri )) ∩ Q|

i=1

≤ C3

∞ X i=1

! σ1 Φ(B(yi , 2ri ))

∞ X

! p1 − n1 |B(yi , ri )|

.

i=1

As the open set U is arbitrary, this estimate yields |ϕ−1 (E) ∩ Q| = 0. Since the cube Q ⊂ Ω is arbitrary, it follows that |ϕ−1 (E)| = 0.

A. O. Molchanova, S. K. Vodop0 yanov

22

1 Due to the sequence {ϕk }k∈N converges weakly in Wr,loc (Ω), by embedding theorem (see [1, Theorem 6.3] for instance), picking up the subsequence, it is reputed that ϕ0 is an almost everywhere pointwise limit of the homeomorphisms ϕk : Ω → Ω0 . In this case the images of some points x ∈ Ω may belong to the boundary ∂Ω0 . Denote by S ⊂ Ω a negligible set on which the convergence ϕk (x) → ϕ0 (x) as k → ∞ fails. If x ∈ Ω \ S with ϕ(x) ∈ Ω0 then the injectivity follows from 0 the uniform convergence of ψk = ϕ−1 k on Ω (see Lemma 3.2) and the identity

ψk ◦ ϕk (x) = x,

x ∈ Ω \ S,

Passing to the limit as k → ∞, we infer that ψ0 ◦ ϕ0 (x) = x, x ∈ Ω \ S. Hence, we deduce that if ϕ0 (x1 ) = ϕ0 (x2 ) ∈ Ω0 for two points x1 , x2 ∈ Ω \ S then x1 = x2 . Since for the domain Ω0 with locally Lipschitz boundary we have |∂Ω0 | = 0, Lemmas 3.1 and 3.6 imply Theorem 3.1. Let us mention another interesting corollary of Theorem 3.2. Recall that a mapping f : Ω → Ω0 is said to be approximative differentiable at x ∈ Ω with approximative derivative Df (x) if there is a set A ⊂ Ω of density one at x7 such that f (y) − f (x) − Df (x)(y − x) = 0. y→x, y∈A ky − xk lim

It is well known that Sobolev functions are approximative differentiable a.e. (see [16, 25] for more details). Lemma 3.7. If an almost everywhere injective mapping ϕ : Ω → Ω0 with ϕ ∈ W11 (Ω) and J(x, ϕ) ≥ 0 a.e. in Ω has the Luzin N −1 -property then J(x, ϕ) > 0 for almost all x ∈ Ω. Proof. Let E be a set outside which the mapping ϕ is approximatively differentiable and has the Luzin N −1 -property. Since ϕ ∈ W11 (Ω), then |E| = 0 (see [19, 65]). In addition, we may assume that {x ∈ Ω \ E | J(x, ϕ) = 0} is 7

i.e. limr→0

|A∩B(x,r)| |B(x,r)|

=1

Injectivity almost everywhere and mappings with finite distortion

23

contained in a Borel set Z of measure zero. Put σ = ϕ(Z). By the changeof-variable formula [19, Theorem 2], taking the injectivity of ϕ into account, we obtain Z Z Z χZ (x)J(x, ϕ) dx = (χσ ◦ ϕ)(x)J(x, ϕ) dx = χσ (y) dy. Ω\Σ

Ω\Σ

Ω0

By construction, the integral in the left-hand side vanishes; consequently, |σ| = 0. On the other hand, since ϕ has the Luzin N −1 -property, we have |Z| = 0.

4

Elasticity

The goal of this section is to prove the existence theorem for minimizing problem of energy functional in the classes H(n − 1, s, M ; ϕ) where s ∈ [1, ∞]. Our prove works for all values of parameter s. It is worth to note that at s = 1 some results of this section look like some statements of paper [29]. In our proof we use different arguments, such as the boundedness of composition operators. It gives an opportunity to apply them to new classes of deformations. Naturally, the proof of our main result differs substantially from previous works and is based crucially on the results and methods of [57]. Comparison of our results with those in another papers see in Remark 4.3 and Section 5.

4.1

Polyconvexity

Let F = [fij ]i,j=1,...,n be a (n × n)-matrix. For every pair of ordered tuples I = (i1 , i2 , . . . il ), 1 ≤ i1 < · · · < il ≤ n, and J = (j1 , j2 , . . . jl ), 1 ≤ j1 < · · · < jl ≤ n, define l × l-minor of the matrix F fi j · · · fi j 1 l 11 .. . ... FIJ = ... . f i l j1 · · · f il jl Notice that n × n-minor is the determinant of F . Let F# be an ordered list
of all minors of F . Let F# ∈ D ⊂ RN for sufficiently large N (N = 2n ), n where D be a convex set with nonnegative n × n-minor.

A. O. Molchanova, S. K. Vodop0 yanov

24

Definition 5 ([3]). A function W : Mn×n → R is polyconvex if there exists a convex function G : D → R, such that G(F# ) = W (F ). Examples of polyconvex but not convex functions are W (F ) = det F and W (F ) = tr Adj F T F = k Adj F T F k2 (see, for example, [12]). It is known that for a hyperelastic material with experimentally known Lam´e coefficients it can be constructed a stored-energy function of an Ogden material (see [42, 12] for more details). On the other hand, a well-known Saint-Venant–Kirchhoff material, is not polyconvex [12, Theorem 4.10].

4.2

Existence theorem

Let Ω, Ω0 ⊂ Rn be two bounded domains with locally Lipschitz boundaries. Recall that a mapping G : Ω × Rm → R enjoys the Carath´eodory conditions whenever G(x, ·) is continuous on Rm for almost all x ∈ Ω; and G(·, a) is measurable on Ω for all a ∈ Rm . Consider a functional Z I(ϕ) = W (x, Dϕ(x)) dx, Ω

where W : Ω × Mn×n → R is a stored-energy function with the following properties: (a) polyconvexity: there exists a convex function G : Ω × D → R, D ⊂ RN , meeting Carath´eodory conditions such that for all F ∈ Mn×n , det F ≥ 0, the equality G(x, F# ) = W (x, F ) holds almost everywhere in Ω; (b) coercivity: there exists a constant α > 0 and a function g ∈ L1 (Ω) such that W (x, F ) ≥ α|F |n + g(x) (4.1)

Injectivity almost everywhere and mappings with finite distortion

25

for almost all x ∈ Ω and all F ∈ Mn×n , det F ≥ 0. Given constants p, q ≥ 1, M > 0 define the class of admissible deformations H(p, q, M ) = {ϕ : Ω → Ω0 is a homeomorphism, ϕ ∈ W11 (Ω) ∩ F D(Ω), I(ϕ) < ∞, J(x, ϕ) ≥ 0 a.e. in Ω, KO (·, ϕ) ∈ Lp (Ω), kKI (·, ϕ) | Lq (Ω)k ≤ M }, where KO (x, ϕ) and KI (x, ϕ) are the outer and the inner distortion functions defined by (2.1). Theorem 4.1 (Existence theorem). Suppose that conditions (a) and (b) on the function W (x, F ) are fulfilled and the set H(n − 1, s, M ) is nonempty, M > 0, s > 1. Then there exists at least one homeomorphic mapping ϕ0 ∈ H(n − 1, s, M )

such that

I(ϕ0 ) = inf{I(ϕ), ϕ ∈ H(n − 1, s, M )}. 0

If there is a homeomorphic Dirichlet data ϕ : Ω → Ω , ϕ ∈ Wn1 (Ω), J(x, ϕ) > 0 a.e. in Ω, kKI (·, ϕ) | Lq (Ω)k ≤ M , and I(ϕ) < ∞, than we can define the classes of admissible deformations H(p, q, M ; ϕ) = {ϕ ∈ H(p, q, M ), ϕ|∂Ω = ϕ|∂Ω a.e. on ∂Ω}. Because of Theorem 4.1 and compactness of the trace operator (see [36, Sect. 1.4.5–1.4.6] for instance) it can be easily obtained the next existence theorem with respect to a Dirichlet boundary condition ϕ|∂Ω = ϕ|∂Ω a.e. on ∂Ω. Corollary 4.1. Suppose that conditions (a) and (b) on the function W (x, F ) are fulfilled and the set H(n − 1, s, M ; ϕ) is nonempty, M > 0, s ≥ 1. Then there exists at least one mapping ϕ0 ∈ H(n − 1, s, M ; ϕ) such that I(ϕ0 ) = inf{I(ϕ), ϕ ∈ H(n − 1, s, M ; ϕ)}. In some cases it is more convenient to consider deformations of the same homotopy class as a given homeomorphism ϕ instead of deformations with prescribed boundary values. In this case we can define the next class of admissible deformations H(p, q, M ; ϕ, hom) = {ϕ ∈ H(p, q, M ), ϕ belongs to the same homotopy class as ϕ}.

A. O. Molchanova, S. K. Vodop0 yanov

26

Corollary 4.2. Suppose that conditions (a) and (b) on the function W (x, F ) are fulfilled and the set H(n − 1, s, M ; ϕ, hom) is nonempty, M > 0, s ≥ 1. Then there exists at least one mapping ϕ0 ∈ H(n − 1, s, M ; ϕ, hom) such that I(ϕ0 ) = inf{I(ϕ), ϕ ∈ H(n − 1, s, M ; ϕ, hom)}. 1

−

1

Remark 4.1. If q1 ≤ q2 and M2 |Ω| q1 q2 ≤ M1 we have natural embeddings of families of the admissible deformations H(p, q2 , M2 ) ⊂ H(p, q1 , M1 ). If p1 ≤ p2 then H(p2 , q, M ) ⊂ H(p1 , q, M ) holds. Remark 4.2. Note that we can omit the condition that ϕ is a homeomorphism in the definition of H(n − 1, s, M ; ϕ, hom) (and H(n − 1, s, M ; ϕ)) if s > 1. Since ϕ ∈ H(n − 1, s, M ; ϕ, hom) belongs to Wn1 (Ω), KO (·, ϕ) ∈ Ln−1 (Ω) and KI (·, ϕ) ∈ Ls (Ω), s > 1, the mapping ϕ is continuous, open and discrete (Theorem 2.1 and [44]). Also, it is known that continuous open discrete mapping ϕ, with the same homotopy class as a given homeomorphism ϕ ∈ Wn1 (Ω), is also a homeomorphism of Ω onto Ω0 (see [29] for instance). Remark 4.3. Added to this is the fact that if we have boundary conditions, we do not need restriction on KO (x, ϕ) (see Remark 4.5 for details). Thereafter for s ≥ 1 instead of H(n − 1, s, M ; ϕ) and H(n − 1, s, M ; ϕ, hom) we can consider classes A(s, M ; ϕ) ={ϕ ∈ A(s, M ), ϕ|∂Ω = ϕ|∂Ω a.e. on ∂Ω} and (4.2) A(s, M ; ϕ, hom) ={ϕ ∈ A(s, M ), ϕ belongs to the same homotopy class as ϕ}, (4.3) where A(s, M ) = {ϕ : Ω → Ω0 is a homeomorphism, ϕ ∈ W11 (Ω) ∩ F D(Ω), I(ϕ) < ∞, J(x, ϕ) ≥ 0 a.e. on Ω, kKI (·, ϕ) | Ls (Ω)k ≤ M }. Note that, for a mapping being of the class A(1, M ) we ask the same requirenments as those in the paper [29].

Injectivity almost everywhere and mappings with finite distortion

4.3

27

Proof of the existence theorem

In this section we prove the existence of a minimising mapping for the functional Z I(ϕ) = I(ϕ) − g(x) dx. Ω

Observe now that the coercivity (4.1) of the function W and the corollary of the Poincar´e inequality (see [12, Theorem 6.1-8] for instance) ensure the existence of constants c > 0 and d ∈ R such that Z (4.4) I(ϕ) = I(ϕ) − g(x) dx ≥ ckϕ | Wn1 (Ω)kn + d Ω

for every mapping ϕ ∈ H = H(n − 1, s, M ). Take a minimising sequence {ϕk } for the functional I. Then lim I(ϕk ) = inf I(ϕ).

k→∞

ϕ∈H

By (4.4) and the assumption inf I(ϕ) < ∞, the sequence {ϕk }k∈N is bounded ϕ∈H

in Wn1 (Ω). Hence, by Lemma 0.1, there exists a minimising sequence fulfilling the conditions   ϕk −→ ϕ0 weakly in Wn1 (Ω),    Adj Dϕ −→ Adj Dϕ n weakly in L n−1 k 0 ,loc (Ω),  ...    J(·, ϕ ) −→ J(·, ϕ ) weakly in L1,loc (Ω) k 0 as k → ∞, where ϕ0 guarantees the sharp lower bound I(ϕ0 ) = inf I(ϕ). ϕ∈H

It remains to verify that ϕ0 ∈ H. To this end, we need the properties of mappings of H. Remark 4.4. If a homeomorphism ϕ ∈ H then by Remark 2.1 and Theorem 2.5 it induces a bounded pullback operator ϕ˜∗ : Ln/(n−1) (Ω0 , Λn−1 ) → n(n−1)% ≥ n − 1. Furthermore, we have the estiLr/n−1 (Ω, Λn−1 ) where r = n%+n−% mate kϕ˜∗ k ≤ kKϕ,n | L% (Ω)k ≤ Ckϕ˜∗ k with some constant C.

A. O. Molchanova, S. K. Vodop0 yanov

28

If a homeomorphism ϕ induces a bounded pullback operator ϕ˜ : Ln/(n−1) (Ω0 , Λn−1 ) → Lr/(n−1) (Ω, Λn−1 ) then by Theorem 2.6 the inverse mapping ψ = ϕ−1 has finite distortion and induces a bounded pullback operar . Moreover, kψ˜∗ k ∼ kϕ˜∗ k8 . As tor ψ˜∗ : Lr0 (Ω, Λ1 ) → Ln (Ω0 , Λ1 ) for r0 = r−n+1 there is a case of 1-forms, it is equivalent to a bounded composition operator ψ ∗ : L1r0 (Ω) → L1n (Ω0 ) and kψ ∗ k = kψ˜∗ k. Further, in accordance with Theorem 2.4 an inverse homeomorphism ϕ = ψ −1 has finite distortion and induces a bounded composition operator r ϕ∗ : L1n (Ω0 ) → L1r? (Ω) for r? = (n−1)2 −r(n−2) and kϕ∗ k ∼ kψ ∗ kn−1 ∼ kϕ˜∗ kn−1 . ∗

Now we prove that J(x, ϕ0 ) ≥ 0 a.e. in Ω. Lemma 4.1. The limit mapping ϕ0 satisfies J(·, ϕ0 ) ≥ 0 a.e. in Ω. The inequality J(·, ϕ0 ) ≥ 0 follows directly from the weak convergence of J(·, ϕk ) in L1 (K), for every K b Ω. Among other things, we can establish the nonnegativity of the Jacobian by using weak convergence ([47, §4.5]). Further, Lemma 3.7 implies the limit mapping ϕ0 satisfies the strict inequality J(x, ϕ0 ) > 0 a.e. in Ω. Now by Theorem 3.1 the mapping ϕ0 is almost-everywhere injective (moreover, injectivety can be lost only if points go to the boundary) and J(x, ϕ0 ) > 0 a.e.; furthermore, since ϕ0 ∈ Wn1 (Ω) has finite distortion (by Remark 4.4, Lemma 3.1 and [64, Theorem 1]) and if KO (·, ϕ0 ) ∈ Ln−1 (Ω) and KI (·, ϕ0 ) ∈ Ls (Ω), s > 1, the mapping ϕ0 is continuous, discrete and open by Theorem 2.1. Therefore so ϕ0 is a homeomorphism. Remark 4.5. Theorem 2.1 is not known if s = 1. However, we include the case s = 1 for classes H(n−1, s, M ; ϕ) and H(n−1, s, M ; ϕ, hom) (A(s, M ; ϕ) and A(s, M ; ϕ, hom)). Indeed, in the proof of Theorem 3.1 there are a sequence of homeomorphisms {ϕk }k∈N and a sequence of inverse homeomorphisms {ψk }k∈N , which converge locally uniformly to ϕ0 and ψ0 respectively. Then ϕ0 and ψ0 are continuous and ψ0 ◦ ϕ0 (x) = x,

ϕ0 ◦ ψ0 (y) = y,

if ϕ0 (x) 6∈ ∂Ω0 and ψ0 (y) 6∈ ∂Ω. Since ϕ0 coincides with given homeomorphism ϕ on the boundary (or is in the same homotopy class), µ(y, Ω, ϕ0 ) = 1 for y 6∈ ϕ0 (∂Ω). Therefore for 8

a ∼ b means there exist constants C1 , C2 > 0, such that C1 a ≤ b ≤ C2 a

Injectivity almost everywhere and mappings with finite distortion

29

y ∈ Ω0 there is x ∈ Ω such that ϕ0 (x) = y ∈ Ω0 . Passing to the limit in ψk ◦ ϕk (x) = x, we obtain ψ0 (y) = x ∈ Ω. Similar we obtain ϕ0 (x) = y ∈ Ω0 for x ∈ Ω. In order to make sure that ϕ0 ∈ H it remains to verify KO (·, ϕ0 ) ∈ Ln−1 (Ω) and kKI (·, ϕ0 ) | Ls (Ω)k ≤ M. It follows from the semicontinuity property of distortion coefficient [17], [27, Theorem 8.10.1] (see this property under weaker assumption and some generalisation in [62, 60]). In order to complete the proof, it remains to verify lower semicontinuity of the functional Z Z W (x, Dϕk ) dx, W (x, Dϕ0 ) dx ≤ lim k→∞ Ω

Ω

using conventional technique for polyconvex case (see, for example, [39, §5]).

5

Examples

As our first example consider an Ogden material with the stored-energy function W1 of the form p

q

W1 (F ) = a tr(F T F ) 2 + b tr Adj(F T F ) 2 + c(det F )r + d(det F )−m ,

(5.1)

2q where a > 0, b > 0, c > 0, d > 0, p > 3, q > 3, r > 1, and m > q−3 . Then W1 (F ) is polyconvex and the coercivity inequality holds [12, Theorem 4.9-2]:
W1 (F ) ≥ α |F |p + | Adj F |q + c(det F )r + d(det F )−m .

We have to solve the minimisation problem I1 (ϕB ) = inf{I1 (ϕ) : ϕ ∈ AB }, where I1 (ϕ) =

R

(5.2)

W1 (Dϕ(x)) dx and the class of admissible deforma-

Ω

tions AB = {ϕ ∈ W11 (Ω), I1 (ϕ) < ∞, J(x, ϕ) > 0 a.e. in Ω, ϕ|∂Ω = ϕ|∂Ω a.e. on ∂Ω} is defined by (0.2) for a homeomorphic boundary conditions ϕ : Ω → Ω0 , ϕ ∈ Wp1 (Ω), J(x, ϕ) > 0 a.e. in Ω and I1 (ϕ) < ∞. The

A. O. Molchanova, S. K. Vodop0 yanov

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result of John Ball [4] ensures that there exists at least one solution ϕB ∈ AB to this problem, which is a homeomorphism in addition. Denote inf I1 (ϕ) + m = M for any m > 0 and consider a class, defined ϕ∈AB

by (4.2), A(s, M ; ϕ) = {ϕ : Ω → Ω0 is a homeomorphism, ϕ ∈ W11 (Ω) ∩ F D(Ω), I1 (ϕ) < ∞, J(x, ϕ) ≥ 0 a.e. in Ω, kKI (·, ϕ) | Ls (Ω)k ≤ M, ϕ|∂Ω = ϕ|∂Ω a.e. on ∂Ω}. It is easy to check that ϕ ∈ AB is a homeomorphism (by [4, Theorem 2]), has finite distortion (as J(x, ϕ) ≥ 0 a.e.) and kKI (·, ϕ) | Ls (Ω)k ≤ M by σr > 1 where σ = q(1+m) > n. It means that H¨older inequality for s = rn+σ−n q+m AB ∩ A(s, M ; ϕ) 6= ∅. Moreover, a minimising sequence {ϕk } ⊂ AB of the problem (5.2) belongs to A(s, M ; ϕ) as well. On the other hand, for the functions of the form (5.1) Theorem 4.1 holds. Indeed, W1 (F ) is polyconvex and satisfies W1 (F ) ≥ α|F |3 − α, where α plays the role of the function h(x) of (4.1). When we consider the same boundary conditions ϕ : Ω → Ω0 and solve the minimisation problem I1 (ϕ0 ) = inf{I1 (ϕ) : ϕ ∈ A(s, M ; ϕ)} Lemma 4.1 and Remark 4.3 yields a solution ϕ0 ∈ A(s, M ; ϕ) which is a homeomorphism. Let us discuss another example. Here the stored-energy function is of the form 3 W2 (F ) = a tr(F T F ) 2 . This function is polyconvex and satisfies W2 (F ) ≥ αkF k3 , but violates the inequality of the form (0.3). Moreover, W2 (F ) violates the asymptotic condition W2 (x, F ) → ∞ as det F → 0+ , which plays an important role in [4, 7] and other articles. Nevertheless, for the stored-energy function W2 there exists a solution to the minimisation problem I2 (ϕ0 ) = inf I2 (ϕ) R in the class of homeomorhisms ϕ ∈ H(n − 1, s, M ), s > 1, where I2 (ϕ) = W2 (Dϕ(x)) dx. Ω

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Disclosure of potential conflict of interest Funding: This work was partially supported by a Grant of the Russian Foundation of Basic Research (Project 14-01-00552) and the Russian Science Foundation (Agreement No. 16-41-02004). Conflict of Interest: The authors declare that they have no conflict of interest.

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