Paolo Piersanti
email: paolo.piersanti@uni-graz.at

A time-dependent obstacle problem in linearised elasticity

Keywords: Obstacle problem, time-dependent, penalty method, variational inequalities, linearised elasticity 2010 MSC: 35L04, 35L10, 35Q74, 35Q92

In this paper we establish the existence of solutions to a time-dependent problem for a linearly elastic body subjected to a confinement condition, expressing that all the points of the deformed reference configuration remain confined in a prescribed half space. This problem takes the form of a set of hyperbolic variational inequalities. The fact that any solution of the studied problem takes the form of a vector field instead of a real-valued function, the generality of the confinement condition under consideration, the fact that the integration domain is a subset of R 3 , and the choice of the function space where solutions are sought make the analysis substantially more complicated, thus requiring the adoption of new resolution strategies.

Introduction

Hyperbolic models are used to describe many phenomena arising in classical mechanics like, for instance, the vibration of a string under the action of an external force. In this paper, we study the existence of solutions to an obstacle problem modelling the displacement of a three-dimensional linearly elastic body confined in a half space.

Obstacle problems arise in many applicative fields: For instance, the motion of three valves of the Aorta, that can be regarded as linearly elastic shells (cf., e.g., [START_REF] Ciarlet | Mathematical Elasticity[END_REF]), is governed by a mathematical model built up in a way such that each valve remains confined in a certain portion of space without colliding with the remaining two valves.

A substantial contribution to the theory of hyperbolic obstacle problems can be found in the seminal papers [START_REF] Schatzman | A hyperbolic problem of second order with unilateral constraints: the vibrating string with a concave obstacle[END_REF] and [START_REF] Kim | A boundary thin obstacle problem for a wave equation[END_REF]. Other important contributions in this field can be found in the references [START_REF] Bock | On hyperbolic contact problems[END_REF], [START_REF] Bock | Dynamic contact problem for viscoelastic von Kármán-Donnell shells[END_REF], [START_REF] Bock | A vibrating thermoelastic plate in a contact with an obstacle[END_REF], and [START_REF] Bock | On the solutions of a dynamic contact problem for a thermoelastic von Kármán plate, Nonlinear Anal[END_REF], where the problems are set out as follows: the integration domain ω is a subset of R 2 , and the unknown function, at almost all time instants, is a real-valued function that belongs to H 2 0 (ω). It is also worth mentioning the papers [START_REF] Fundos | Existence theorems of Hartmann-Stampacchia type for hemivariational inequalities and applications[END_REF], [START_REF] Bocea | A perturbation result for a double eigenvalue hemivariational inequality and applications[END_REF], [START_REF] Rȃdulescu | Perturbations of hemivariational inequalities with constraints[END_REF], [START_REF] Papageorgiou | Nonlinear elliptic inclusions with unilateral constraint and dependence on the gradient[END_REF], [START_REF] Papageorgiou | Periodic solutions for implicit evolution equations[END_REF], [START_REF] Rȃdulescu | Global well-posedness for a class of fourth order nonlinear strongly damped wave equations[END_REF], [START_REF] Papageorgiou | Sensitivity analysis for optimal control problems governed by nonlinear evolution inclusions[END_REF] and [START_REF] Mugnai | Maximum principles for inhomogeneous elliptic inequalities on complete Riemannian manifolds[END_REF]. The nonlinear analysis tools used in this paper can be found in the recent monograph [START_REF] Papageorgiou | Nonlinear analysistheory and methods[END_REF].

The first main novelty of this paper is that the unknown, represented by displacement of the linearly elastic body under consideration, is a vector field that, at almost all time instants, belongs to a nonempty, closed , and convex subset of the Sobolev space H 1 (Ω)×H 1 (Ω)×H 1 (Ω), where Ω ⊂ R 3 is a domain.

This will require the implementation of a more general argument to recover the energy estimates in the Galerkin method.

The second main novelty is given by the generality of the confinement condition under consideration, which comprises at once all of the three components of the displacement vector field. Such a confinement conditions were first considered in the papers [START_REF] Ciarlet | An obstacle problem for elliptic membrane shells[END_REF], [START_REF] Ciarlet | Un problème de confinement pour une coque membranaire linéairement élastique de type elliptique[END_REF], [START_REF] Ciarlet | An obstacle problem for Koiter's shells[END_REF], and [START_REF] Ciarlet | A confinement problem for a linearly elastic Koiter's shell[END_REF]. Other types of confinement conditions which are more amenable in the context of numerical simulations are discussed in the paper [START_REF] Piersanti | Numerical methods for static shallow shells lying over an obstacle[END_REF].

Finally, the method we propose here for recovering the initial condition for the first derivative in time of the displacement slightly differs from the one used in [START_REF] Bock | On hyperbolic contact problems[END_REF], [START_REF] Bock | Dynamic contact problem for viscoelastic von Kármán-Donnell shells[END_REF], [START_REF] Bock | A vibrating thermoelastic plate in a contact with an obstacle[END_REF], and [START_REF] Bock | On the solutions of a dynamic contact problem for a thermoelastic von Kármán plate, Nonlinear Anal[END_REF].

The paper is organised as follows. First, some notations and background are provided. Secondly, the main existence theorem for a dynamic linearly elastic body confined in a half space is established. Thirdly, and finally, some final comments about the uniqueness of the solution are made.

Geometrical preliminaries

For details about the classical notions of differential geometry recalled in this section, see, e.g. [START_REF] Ciarlet | Mathematical Elasticity[END_REF] or [START_REF] Ciarlet | An Introduction to Differential Geometry with Applications to Elasticity[END_REF].

Latin indices, except when they are used for indexing sequences, take their values in the set {1, 2, 3}, and the summation convention with respect to repeated indices is systematically used in conjunction with this rule.

Given an open subset Ω ⊂ R 3 , notations such as L 2 (Ω) and H 1 (Ω) denote the standard Lebesgue and Sobolev spaces. The notation D(Ω) designates the space of functions that are infinitely differentiable over Ω and have a compact support in Ω. The notation • X designates the norm of a vector space X.

Spaces of vector-valued functions are denoted by boldface letters. Lebesgue-Bochner spaces (see, e.g., [START_REF] Leoni | A First Course in Sobolev Spaces[END_REF]) are designated by the notation L p (0, T ; X), where 1 ≤ p ≤ ∞, T > 0, and X is a Banach space satisfying the Radon-Nikodym property. The notation M([0, T ]; X) designates the space of X-valued measures defined over the compact interval [0, T ] (see, e.g., [START_REF] Diestel | Vector measures[END_REF] and [START_REF] Dinculeanu | Vector measures[END_REF]). The notation X * designates the dual space of a vector space X and the notation X * •, • X denotes the duality pair between X * and X. The notation C 0 ([0, T ]; X) denotes the space of X-valued continuous functions defined over the compact interval [0, T ] and the special notation •, • X denotes the duality pair between (C 0 ([0, T ]; X)) * and C 0 ([0, T ]; X). The notations η and η denote the first weak derivative with respect to t ∈ (0, T ) and the second weak derivative with respect to t ∈ (0, T ) of a scalar function η defined over the interval (0, T ). The notations η and η denote the first weak derivative with respect to t ∈ (0, T ) and the second weak derivative with respect to t ∈ (0, T ) of a vector field η defined over the interval (0, T ). A domain Ω ⊂ R 3 is a nonepmpty, open, bounded and connected subset with Lipschitz continuous boundary Γ, the set Ω being locally on the same side of Γ. The notation dx designates the volume element in Ω, the symbol dΓ designates the area element along Γ. Finally, let Γ 0 and Γ 1 be a dΓ-measurable portion of the boundary such that Γ 0 ∩ Γ 1 = ∅, Γ = Γ 0 ∪ Γ 1 , and area Γ 0 > 0.

As a model of the three-dimensional "physical" space R 3 , we take a real three-dimensional affine Euclidean space, i.e., a set in which a point O has been chosen as the origin and with which a real three-dimensional Euclidean space, denoted E 3 , is associated. We equip E 3 with an orthonormal basis consisting of three vectors e i . The Euclidean inner product of two elements a and b of E 3 is denoted by a • b; the Euclidean norm of any a ∈ E 3 is denoted by |a|; the Kronecker symbol is denoted by δ ij .

The definition of R 3 as an affine Euclidean space means that with any point x ∈ R 3 is associated an uniquely defined vector Ox ∈ E 3 . The origin O ∈ R 3 and the orthonormal vectors e i ∈ E 3 together constitute a Cartesian frame in R 3 and the three components x i of the vector Ox over the basis formed by e i are called Cartesian coordinates of x ∈ R 3 , or the Cartesian components of Ox ∈ E 3 . Once a Cartesian frame has been chosen, any point x ∈ R 3 may be thus identified with the vector Ox = x i e i ∈ E 3 . We then denote

∂ i = ∂/∂x i .
The set Ω is the reference configuration occupied by a linearly elastic elastic body in absence of applied body forces. We assume that Ω is a natural state, i.e., that the body is stress-free in this configuration. We also assume, following [START_REF] Ciarlet | Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity[END_REF], that the constituting material is isotropic, homogeneous, and linearly elastic. Under these assumptions, the behaviour of the linearly elastic material is governed by its two Lamé constants λ ≥ 0 and µ > 0. The positive constant ρ designates the mass density of the linearly elastic body per unit volume.

We also assume that the linearly elastic body to be subjected to applied body forces in its interior, whose density per unit volume is defined by means of its

contravariant components f i ∈ L ∞ (0, T ; L 2 (Ω)) over the vectors e i .
In what follows, "a.e." stands for "almost everywhere". Define the space

V (Ω) := {v = (v i ) ∈ H 1 (Ω); v = 0 on Γ 0 },
and equip it with the norm

v V (Ω) := i v i 2 H 1 (Ω) 1/2 .
Next, we define the three-dimensional elasticity tensor in Cartesian coordinates and we denote its components by A ijkl . We recall that the contravariant components of this tensor are defined by (see, e.g., [START_REF] Ciarlet | Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity[END_REF])

A ijkl := λδ ij δ kl + µ(δ ik δ jl + δ il δ jk ),
and that

A ijkl = A jikl = A klij ∈ C 1 (Ω).
For each v ∈ H 1 (Ω) we consider the linearised change of metric tensor e(v),

whose components e i j (v) are defined by

e i j (v) := 1 2 (∂ j v i + ∂ i v j ) ∈ L 2 (Ω).
This tensor is symmetric, i.e., e i j (v) = e j i (v), for all v ∈ H 1 (Ω). Likewise, we can define the time-dependent version of the linearised change of metric tensor by considering the operator

ẽi j : L 2 (0, T ; H 1 (Ω)) → L 2 (0, T ; L 2 (Ω)) defined by ẽi j (v)(t) = e i j (v(t)), for all v ∈ L 2 (0, T ; H 1 (Ω)),
for almost all ("a.a." in what follows) t ∈ (0, T ). It is easy to see that such an operator is well-defined, linear and continuous. It can be easily verified (cf., e.g., [START_REF] Piersanti | An existence and uniqueness theorem for the dynamics of flexural shells[END_REF]) that the continuity constant is independent of t ∈ (0, T ).

95

To begin with, we state Korn's inequality in Cartesian coordinates (see, e.g., Theorem 6.3-6 of [START_REF] Ciarlet | Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity[END_REF]).

Theorem 1.

Let Ω be a domain in R 3 and let Γ 0 be a nonzero area subset of the whole boundary Γ. Then, there exists a constant C > 0 such that

C -1 v H 1 (Ω) ≤ e(v) L 2 (Ω) ≤ C v H 1 (Ω) ,
for all v ∈ V (Ω).

Various proofs have been given of this delicate inequality; see in particular [START_REF] Friedrichs | On the boundary-value problems of the theory of elasticity and Korn's inequality[END_REF], [START_REF] Gobert | Une inégalité fondamentale de la théorie de l'élasticité[END_REF], [START_REF] Hlaváček | On inequalities of Korn's type. I. Boundary-value problems for elliptic system of partial differential equations[END_REF], [START_REF] Hlaváček | On inequalities of Korn's type. II. Applications to linear elasticity[END_REF], page 110 of [START_REF] Duvaut | Inequalities in Mechanics and Physics[END_REF], Sect. 6.3 of [START_REF] Nečas | Mathematical theory of elastic and elasto-plastic bodies: an introduction[END_REF]; in [START_REF] Temam | Problèmes mathématiques en plasticité[END_REF], Korn's inequal-100 ity is proved in the space W 1,p (Ω), 1 ≤ p ≤ ∞; an elementary proof is given in [START_REF] Nitsche | On Korn's second inequality[END_REF] (see also Appendix (A) in [START_REF] Miyoshi | Foundations of the numerical analysis of plasticity[END_REF]).

A natural formulation of the time-dependent obstacle problem for a linearly elastic body

In this paper, we consider a specific obstacle problem for a linearly elastic body subjected to a confinement condition, expressing that any admissible displacement vector field v i e i , must be such that all the points of the corresponding deformed configuration remain in a half-space of the form

H := {x ∈ R 3 ; Ox • q ≥ 0},
where q is a nonzero vector given once and for all. Let us denote by I the identity mapping I : Ω → E 3 and let us assume that the undeformed reference configuration satisfies

I(x) • q > 0, for all x ∈ Ω,
or, in other words, there is no contact between the obstacle and the reference 105 configuration when no applied body forces are acting on the reference configuration. Let us observe that this condition is assumed only for physical reasons and that it is not exploited in the forthcoming proofs.

The general confinement condition can be thus formulated as follows: any admissible displacement vector field must satisfy

(I(x) + v i (x)e i ) • q ≥ 0,
for all x ∈ Ω or, possibly, only for a.a. x ∈ Ω when the covariant components v i are required to belong to the Sobolev space H 1 (Ω). The subset U (Ω) of admissible displacements thus takes the form

U (Ω) := {v =(v i ) ∈ H 1 (Ω); v = 0 on Γ 0 and (I + v i e i ) • q ≥ 0 a.e. in Ω}.
We emphasise that the vectorial confinement condition above, which was originally considered in [START_REF] Ciarlet | An obstacle problem for elliptic membrane shells[END_REF], [START_REF] Ciarlet | Un problème de confinement pour une coque membranaire linéairement élastique de type elliptique[END_REF], [START_REF] Ciarlet | An obstacle problem for Koiter's shells[END_REF] and [START_REF] Ciarlet | A confinement problem for a linearly elastic Koiter's shell[END_REF], considerably departs from the scalar conditions favoured by many authors (see, e.g., [START_REF] Bock | On hyperbolic contact problems[END_REF], [START_REF] Bock | Dynamic contact problem for viscoelastic von Kármán-Donnell shells[END_REF] and [START_REF] Bock | On the solutions of a dynamic contact problem for a thermoelastic von Kármán plate, Nonlinear Anal[END_REF]). Such a confinement condition renders the analysis substantially more difficult, as the constraint now bears on a vector field, the displacement vector field of the reference configuration, instead of on only a single scalar-valued function.

A natural formulation of the corresponding time-dependent obstacle problem takes the form of a set of hyperbolic three-dimensional variational inequalities ("three-dimensional", in the sense that they are posed over the threedimensional subset Ω), which can be derived by slightly modifying the model proposed by Xiao in the papers [START_REF] Xiao | Asymptotic analysis of dynamic problems for linearly elastic shellsjustification of equations for dynamic flexural shells[END_REF], [START_REF] Xiao | Asymptotic analysis of dynamic problems for linearly elastic shellsjustification of equations for dynamic membrane shells[END_REF] and [START_REF] Xiao | Asymptotic analysis of dynamic problems for linearly elastic shellsjustification of equations for dynamic Koiter shells[END_REF].

Let us introduce the problem P(Ω), which constitutes the point of departure of our analysis.

Problem P(Ω). Find u = (u i ) : (0, T ) → V (Ω) such that u ∈ L ∞ (0, T ; U (Ω)), u ∈ L ∞ (0, T ; L 2 (Ω)), ü ∈ M([0, T ]; L 2 (Ω)),
that satisfies the following variational inequalities

2ρ ü(t), v(t) -u(t) L 2 (Ω) + T 0 Ω A ijkl e k l (u(t))e i j (v(t) -u(t)) dx dt ≥ T 0 Ω f i (t)(v i (t) -u i (t)) dx dt, for all v ∈ D(0, T ; V (Ω)) such that v(t) ∈ U (Ω)
for a.a. t ∈ (0, T ), and that satisfies the following initial conditions

     u(0) = u 0 , u(0) = u 1 , (1)
where u 0 = (u i,0 ) ∈ U (Ω), and u 1 = (u i,1 ) ∈ L 2 (Ω) are prescribed.

Observe that the "acceleration term" in Problem P(Ω) is described in terms of a vector-valued measure. Note in passing that the concept of solution of Problem P(Ω) is inspired by the one given on page 403 of [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]. The concept of solution of Problem P(Ω) will be thoroughly explained in the proof of Theorem 6, which constitutes the main result of this paper.

We recall a very important inequality which is used to study evolutionary problems: Gronwall's inequality (see, e.g., the seminal paper [START_REF] Gronwall | Note on the derivatives with respect to a parameter of the solutions of a system of differential equations[END_REF] and Theorem 1.1 in Chapter III of [START_REF] Hartman | Ordinary Differential Equations[END_REF]).

Proof of existence of solutions to Problem P(Ω)

Let us recall a compactness result proved by Simons (see, e.g., Corollary 4

of [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]), which will be used in what follows to recover the initial conditions. In what follows, the symbol " →" denotes a continuous embedding, whereas the symbol " → →" denotes a compact embedding.

Theorem 3. Let T > 0 and let X, Y and Z be three Banach spaces such that

X → → Y → Z.
Let (f n ) ∞ n=1 be a bounded sequence in L ∞ (0, T ; X) and assume that the sequence ( ḟn ) ∞ n=1 is bounded in L ∞ (0, T ; Z). Then, there exists a subsequence, still denoted

(f n ) ∞ n=1 , that converges in the space C 0 ([0, T ]; Y ).
In what follows we identify the spaces L 2 (Ω) and L 2 (Ω) with their respective dual spaces, and we equip them with the following inner products

(v, w) ∈ L 2 (Ω) × L 2 (Ω) → Ω vw dx, (v, w) ∈ L 2 (Ω) × L 2 (Ω) → Ω v i w i dx.
We observe that the following chain of immersions holds

V (Ω) → → L 2 (Ω) → → V * (Ω),
viz., following the notation of [START_REF] Kyritsi-Yiallourou | Handbook of Applied Analysis[END_REF],

V (Ω), L 2 (Ω), V * (Ω) is an evolution triple (or Gelfand triple).
Let us also recall a result on vector-valued measures proved by Zinger in the paper [START_REF] Zinger | Linear functionals on the space of continuous mappings of a compact Hausdorff space into a Banach space[END_REF] (see also, e.g., page 182 of [START_REF] Diestel | Vector measures[END_REF], and page 380 of [START_REF] Dinculeanu | Vector measures[END_REF]). There exists an isomorphism between (C 0 (ω; X)) * and the space of the regular Borel measures with finite variation taking values in X * . In particular, for each F ∈ (C 0 (ω; X)) * , there exists a unique regular Borel measure µ : F → X * in M(ω; X * ) with finite variation such that

α, F X = ω X * dµ, α X ,
for all α ∈ C 0 (ω; X).

Let us denote the penalty parameter by κ and let us introduce the corresponding penalised problem P(κ; Ω).

Problem P(κ; Ω). Find u κ = (u i,κ ) : (0, T ) → V (Ω) such that u κ ∈ L ∞ (0, T ; V (Ω)), uκ ∈ L ∞ (0, T ; L 2 (Ω)), üκ ∈ L ∞ (0, T ; V * (Ω)),
that satisifes the following nonlinear variational equations

2ρ V * (Ω) üi,κ (t)e i , v j e j V (Ω) + Ω A ijkl e k l (u κ (t))e i j (v) dx - 1 κ Ω {[I + u i,κ (t)e i ] • q} -(v i e i • q) dx = Ω f i (t)v i dx,
for all v ∈ V (Ω), in the sense of distributions in (0, T ), and that satisfies the initial conditions (1).

We first prove, by Galerkin method, that Problem P(κ; Ω) admits a solution.

Theorem 5. For each κ > 0, Problem P(κ; Ω) admits a solution.

Proof. The proof is carried out via a Galerkin argument and is subdivided into three steps. To begin with, we fix κ > 0.

(a) Construction of a Galerkin approximation. In order to construct such a scheme, we rely on the fact that V (Ω) is an infinite dimensional separable Hilbert space which is also dense in L 2 (Ω), in order to infer the existence of an orthogonal basis (w k ) ∞ k=1 of the space V (Ω), whose elements also constitute a Hilbert basis of the space L 2 (Ω).

The existence of such a basis is assured by the spectral theorem (Theorem 6.2-1 of [START_REF] Raviart | Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles[END_REF]). For each positive integer m ≥ 1, we denote by E m the following m-dimensional linear hull

E m := Span (w k ) m k=1 ⊂ V (Ω) ⊂ L 2 (Ω).
Since each element of this Hilbert basis is independent of the variable t, we have that

w k ∈ L ∞ (0, T ; V (Ω)) for each integer 1 ≤ k ≤ m.
We now discretise Problem P(κ; Ω) and, in order to keep the notation simple, we drop the dependence of the vector fields entering the variational equations on the penalty parameter κ. Observe that the duality pair between E m and its dual coincides with the inner product of L 2 (Ω) introduced beforehand.

Problem P m (κ; Ω). Find functions c k : [0, T ] → R, 1 ≤ k ≤ m, such that u m (t) := m k=1 c k (t)w k , for a.a. t ∈ (0, T ),
and satisfying the following penalised variational equations a.e. in (0, T )

2ρ

Ω üm i (t)w p i dx + Ω A ijkl e k l (u m (t))e i j (w p ) dx - 1 κ Ω {[I + u m i (t)e i ] • q} -(w p i e i • q) dx = Ω f i (t)w p i dx, for each integer 1 ≤ p ≤ m.
Such a function u m must satisfy, in addition, the following initial conditions

u m (0) = u m 0 , um (0) = u m 1 , u m 0 ∈ V (Ω) and u m 0 → u 0 in V (Ω) as m → ∞, u m 1 ∈ L 2 (Ω) and u m 1 → u 1 in L 2 (Ω) as m → ∞, (2)
where the initial data u m 0 and u m 1 are, respectively, the projections of u 0 and u 1 onto the finite dimensional space E m .

170

We immediately observe that the projections of u 0 = (u i,0 ) and u 1 = (u i,1 ) onto E m can be expanded as follows (cf., e.g., Theorem 4.9-1 of [START_REF] Ciarlet | Linear and Nonlinear Functional Analysis with Applications[END_REF])

u m 0 = m k=1 Ω u i,0 w k i dx + Ω ∂ j u i,0 ∂ j w k i dx w k , u m 1 = m k=1 Ω u i,1 w k i dx w k .
Since the elements of the Hilbert basis do not depend on the time variable we can take the coefficients c k as well as their derivatives outside the integral sign, getting a m × m nonlinear system of second order ordinary differential equations with respect to the variable t. Such a system can be rewritten in the form 2ρ C(t) = -

Ω

A ijkl e k l (w r )e i j (w p ) dx m p,r=1

C(t) + 1 κ Ω {[I + (C(t) • (w 1 i . . . w m i ))e i ] • q} -(w p i e i • q) dx m p=1 + Ω f i (t)w p i dx m p=1 (3)
where C(t) := (c 1 (t) . . . c m (t)), and satisfies the following initial conditions

c k (0) = Ω u i,0 w k i dx + Ω ∂ j u i,0 ∂ j w k i dx, ċk (0) = Ω u i,1 w k i dx.
Observe that the negative part operator is a Lipschitz continuous function, i.e.,

|b --a -| ≤ |b -a|, for all a, b ∈ R. (4)
By the Cauchy-Lipschitz theorem (cf., e.g., Theorem 3.8-1 of [START_REF] Ciarlet | Linear and Nonlinear Functional Analysis with Applications[END_REF]), we deduce that for each integer m ≥ 1 there exists a unique global solution u m to Problem P m (κ; Ω), defined a.e. over the interval (0, T ), such that

u m ∈ L ∞ (0, T ; E m ), um ∈ L ∞ (0, T ; E m ), üm ∈ L ∞ (0, T ; E m ). (5)
(b) Energy estimates for the Galerkin scheme. Let us multiply the variational equations in Problem P m (κ; Ω) by ċk (t), with 0 < t < T , and sum with respect to k varying in the discrete set {1, . . . , m}. As a result, we obtain that the penalised variational equations in Problem P m (κ; Ω) take the form

ρ d dt Ω um i (t) um i (t) dx + 1 2 d dt Ω A ijkl e k l (u m (t))e i j (u m (t)) dx + 1 2κ d dt Ω [I + u m i (t)e i ] • q -2 dx = Ω f i (t) um i (t) dx, (6)
for a.a. t ∈ (0, T ).

Observe that the differentiation of the negative part is obtained as a result of the same computational steps as in Stampacchia's theorem (cf., e.g., [START_REF] Stampacchia | Èquations elliptiques du second ordre à coefficients discontinus[END_REF]), together with an application of Theorem 8.28 of [START_REF] Leoni | A First Course in Sobolev Spaces[END_REF]. The change in sign of the penalty term is due to the properties of the Heavyside function.

Carrying out an integration over the interval (0, t), where 0 < t ≤ T , changes

(6) into ρ Ω um i (t) um i (t) dx + 1 2 Ω A ijkl e k l (u m (t))e i j (u m (t)) dx + 1 2κ Ω [I + u m i (t)e i ] • q -2 dx = ρ Ω u m i,1 u m i,1 dx + 1 2 Ω A ijkl e k l (u m 0 )e i j (u m 0 ) dx + 1 2κ Ω [I + u m i (0)e i ] • q -2 dx + t 0 Ω f i (τ ) um i (τ ) dx dτ. (7)
By Cauchy-Schwarz inequality, there exists a constant C > 0 such that

t 0 Ω f i (τ ) um i (τ ) dx dτ ≤ T 0 f (t) 2 L 2 (Ω) dt 1/2 t 0 um (τ ) 2 L 2 (Ω) dτ 1/2 ≤ 1 2 T 0 f (t) 2 L 2 (Ω) dt + t 0 um (τ ) 2 L 2 (Ω) dτ . (8)
Since u 0 ∈ U (Ω), we have that

Ω [I + u m i (0)e i ] • q -2 dx → 0, as m → ∞.
As a result, there exists a positive integer m(κ) such that

Ω [I + u m i (0)e i ] • q -2 dx ≤ κ, for all m ≥ m(κ). (9)
By the uniform positive-definiteness of the elasticity tensor (A ijkl ), Korn's inequality (Theorem 1), ( 7), [START_REF] Fundos | Existence theorems of Hartmann-Stampacchia type for hemivariational inequalities and applications[END_REF], and (9), we obtain that there exists a real constant C > 0 independent of u m (and so independent of t, m and κ) for which the following estimate holds for all m ≥ m(κ)

1 C um (t) 2 L 2 (Ω) + u m (t) 2 V (Ω) + 1 κ C [I + u m i (t)e i ] • q -2 L 2 (Ω) ≤ u 1 2 L 2 (Ω) + u 0 2 V (Ω) + f 2 L ∞ (0,T ;L 2 (Ω)) + t 0 um (τ ) 2 L 2 (Ω) + u m (τ ) 2 V (Ω) dτ + 1 κ t 0 [I + u m i (τ )e i ] • q -2 L 2 (Ω)
dτ.

(

) 10
An application of the Gronwall's inequality (Theorem 2) with a ≡ C > 0 and

b ≡ C u 1 2 L 2 (Ω) + u 0 2 V (Ω) + f 2 L ∞ (0,T ;L 2 (Ω)) ≥ 0
gives the following upper bound

t 0 um (τ ) 2 L 2 (Ω) + u m (τ ) 2 V (Ω) dτ + 1 κ t 0 [I + u m i (τ )e i ] • q -2 L 2 (Ω) dτ ≤ CT e CT u 1 2 L 2 (Ω) + u 0 2 V (Ω) + f 2 L ∞ (0,T ;L 2 (Ω)) , (11)
for all t ∈ [0, T ].

Therefore, we obtain that

(u m ) ∞ m=1 is uniformly bounded with respect to m in L ∞ (0, T ; V (Ω)), ( um ) ∞ m=1 is uniformly bounded with respect to m in L ∞ (0, T ; L 2 (Ω)), (12)
and, moreover, by [START_REF] Papageorgiou | Nonlinear elliptic inclusions with unilateral constraint and dependence on the gradient[END_REF], there exists a positive uniform constant L such that

0 ≤ [I + u m i e i ] • q -2 L 2 (0,T ;L 2 (Ω)) ≤ Lκ. ( 13
)
Since the following direct sum decomposition holds true

V (Ω) = E m ⊕ (E m ) ⊥ ,
we get that for any v ∈ V (Ω), with v V (Ω) ≤ 1 and a.a. t ∈ (0, T ), the variational equations in Problem P m (κ; Ω) give

| V * (Ω) üm i (t)e i , v j e j V (Ω) | ≤ f L ∞ (0,T ;L 2 (Ω)) + C u m L ∞ (0,T ;V (Ω)) + 1 κ [I + u m i e i ] • q - L 2 (0,T ;L 2 (Ω))
, and, by ( 12) and ( 13), we thus infer that there exists a constant

C κ > 0, independent of m, such that üm L ∞ (0,T ;V * (Ω)) ≤ C κ . (14)
(c) Passage to the limit and retrieval of Problem P(κ; Ω). By ( 12), ( 13) and ( 14) we can infer that there exist subsequences, still denoted (u m ) ∞ m=1 , ( um ) ∞ m=1 and ( üm ) ∞ m=1 such that the following convergences take place

u m * u κ , in L ∞ (0, T ; V (Ω)) as m → ∞, um * uκ , in L ∞ (0, T ; L 2 (Ω)) as m → ∞, üm * üκ , in L ∞ (0, T ; V * (Ω)) as m → ∞, κ -1 [I + u m i e i ] • q - χ κ , in L 2 (0, T ; L 2 (Ω)) as m → ∞. (15)
By the Sobolev embedding theorem (Theorem 10.1.20 of [START_REF] Kyritsi-Yiallourou | Handbook of Applied Analysis[END_REF]), we obtain

u m u κ , in C 0 ([0, T ]; L 2 (Ω)) as m → ∞, um uκ , in C 0 ([0, T ]; V * (Ω)) as m → ∞, (16)
An application of Theorem 8.28 of [START_REF] Leoni | A First Course in Sobolev Spaces[END_REF] to the fourth convergence of the process [START_REF] Mugnai | Maximum principles for inhomogeneous elliptic inequalities on complete Riemannian manifolds[END_REF] gives

κ -1 [I + u m i e i ] • q - χ κ , in L 2 ((0, T ) × Ω) as m → ∞. (17)
By ( 4), the first convergence of ( 16) and the weak convergence ( 17), Theorem 8.28 of [START_REF] Leoni | A First Course in Sobolev Spaces[END_REF] and Theorem 8.62 of [START_REF] Leoni | A First Course in Sobolev Spaces[END_REF], we are in a position to apply Theorem 9.13-2 of [START_REF] Ciarlet | Linear and Nonlinear Functional Analysis with Applications[END_REF] (where the involved monotone operator is nothing but the negative part operator ) and, so, to obtain

χ κ = κ -1 [I + u i,κ e i ] • q -∈ L 2 ((0, T ) × Ω). ( 18
)
We now verify that u κ is a solution to the penalised variational equations in Problem (P(κ; Ω)). Let ψ ∈ D(0, T ) and let µ ≥ 1 be any integer. For each m ≥ µ, we have 2ρ

T 0 Ω üm i (t)v i dxψ(t) dt + T 0 Ω A ijkl e k l (u m (t))e i j (v) dxψ(t) dt - 1 κ T 0 Ω [I + u m i (t)e i ] • q -(v i e i • q) dxψ(t) dt = T 0 Ω f i (t)v i dxψ(t) dt, ( 19
)
for all v ∈ E µ .
Keeping in mind (4), the convergence process ( 15), ( 18), the arbitrariness of ψ ∈ D(0, T ), as well as the fact that

µ≥1 E µ • V (Ω) = V (Ω),
a passage to the limit as m → ∞ in [START_REF] Ciarlet | An obstacle problem for Koiter's shells[END_REF] shows that u κ is a solution to the penalised variational equations in Problem P(κ; Ω).

The last thing that we have to check is the validity of the initial conditions for u κ . Let us introduce the operator

L 0 : C 0 ([0, T ]; L 2 (Ω)) → L 2 (Ω) defined
in a way such that L 0 (v) := v(0). Such an operator L 0 turns out to be linear and continuous and, therefore, by the first convergence of ( 16), we get that

u m 0 u κ (0), in L 2 (Ω). Since u m 0 → u 0 in V (Ω), we deduce that u κ (0) = u 0 . 180
Similarly, let us introduce the operator

L 1 : C 0 ([0, T ]; V * (Ω)) → V * (Ω)
defined in a way such that L 1 (v) := v(0). Such an operator L 1 turns out to be linear and continuous and, therefore, by the second convergence of ( 16), we get that

u m 1 uκ (0), in V * (Ω). Since u m 1 → u 1 in L 2 (Ω), we deduce that uκ (0) = u 1 .
We have thus shown that u κ is a solution of Problem P(κ; Ω). This completes the proof.

We are now in a position to prove the existence of solutions of Problem P(Ω), which constitutes the main result of this paper.

Theorem 6. For each κ > 0, let u κ denote a solution to Problem P(Ω).

Assume also that the following "uniformity property" holds: There exists a number t0 > 0, independent of κ, such that

[I + u i,κ (t)e i ] • q ≥ 0 a.e. in Ω,
for a.a. 0 < t < t0 , for all κ > 0.

Then, Problem P(Ω) admits a solution.

Proof. By the energy estimate [START_REF] Rȃdulescu | Perturbations of hemivariational inequalities with constraints[END_REF] in Theorem 5, it can be easily observed that there exists a positive constant

c = c(u 0 , u 1 , f ) such that 1 C uκ 2 L ∞ (0,T ;L 2 (Ω)) + u κ 2 L ∞ (0,T ;V (Ω)) ≤ c.
As a result, the sequences (u κ ) κ>0 and ( uκ ) κ>0 are uniformly bounded in L ∞ (0, T ; V (Ω)) and L ∞ (0, T ; L 2 (Ω)), respectively.

190

Let us consider, for a.a. 0 < t < T , the partial differential equation associated with Problem P(κ; Ω)

2ρ üκ (t) + Au κ (t) - 1 κ N u κ (t) = f (t), in V * (Ω), (20)
where the operator A : V (Ω) → V * (Ω) defined by

V * (Ω) Au, v V (Ω) := Ω A ijkl e k l (u)e i j (v) dx, for all u, v ∈ V (Ω),
is linear and continuous.

Similarly, we define the nonlinear operator N : V (Ω) → V * (Ω) as

V * (Ω) N u, v V (Ω) := Ω [I + u i e i ] • q - v i e i • q dx, for all u, v ∈ V (Ω).
Let us prove the uniform boundedness of the sequence (N u κ ) κ>0 by observing that

1 κ T 0    sup v∈V (Ω) v V (Ω) ≤1 V * (Ω) N u κ (t), v V (Ω)    dt ≤ f L ∞ (0,T ;L 2 (Ω)) + u κ L ∞ (0,T ;V (Ω)) + sup v∈V (Ω) v V (Ω) ≤1 V * (Ω) uκ (T ) -uκ (0), v V (Ω) ,
where the last term in the right hand side derives from an application of Corollary 10.1.26 of [START_REF] Kyritsi-Yiallourou | Handbook of Applied Analysis[END_REF]. We make use of this strategy to gain insight into a uniform bound for the nonlinear term, since nothing is known about the boundedness of the sequence ( üκ ) κ>0 yet.

195

Observe that, by Theorem 4, the following chain of embeddings holds

L 1 (0, T ; V * (Ω)) → C 0 ([0, T ]; V * (Ω)) * → C 0 ([0, T ]; L 2 (Ω)) * ∼ = M([0, T ]; L 2 (Ω)).
An application of ( 15) and ( 16) thus gives that the sequence (N u κ ) κ>0 is bounded in L 1 (0, T ; V (Ω)). Therefore, a fortiori, we have

( üκ ) κ>0 is bounded in C 0 ([0, T ]; L 2 (Ω)) * .
Hence, up to passing to a subsequence, we get that the following convergence process takes place

u κ * u, in L ∞ (0, T ; V (Ω)) as κ → 0, uκ * u, in L ∞ (0, T ; L 2 (Ω)) as κ → 0, üκ * ũ, in (C 0 ([0, T ]; L 2 (Ω))) * as κ → 0. ( 21
)
We immediately deduce, by Theorem 4, that there exists a unique vector-valued

measure µ ∈ M([0, T ]; L 2 (Ω)) such that ũ(t), σ(t) L 2 (Ω) = T 0 Ω dµ i (t)σ i (t) dx dt,
for all σ ∈ C 0 ([0, T ]; L 2 (Ω)). Clearly, the vector-valued measure µ is regular (cf., e.g., [START_REF] Diestel | Vector measures[END_REF]).

By Theorem 3, the following convergence holds, up to passing to a subsequence

u κ → u, in C 0 ([0, T ]; L 2 (Ω)). (22)
Besides, by [START_REF] Rȃdulescu | Global well-posedness for a class of fourth order nonlinear strongly damped wave equations[END_REF] we have

0 ≤ [I + u i,κ e i ] • q - L 2 (0,T ;L 2 (Ω)) ≤ √ Lκ. (23)
Consequently, by ( 4), ( 22) and ( 23), we get

[I + u i (t)e i ] • q -= 0, a.e. in Ω, for a.a. t ∈ (0, T ), (24)
i.e., u(t) ∈ U (Ω), for a.a. t ∈ (0, T ).

Given any v ∈ D(0, T ; V (Ω)) such that v(t) ∈ U (Ω) for a.a. t ∈ (0, T ), we use (v -u κ ) as a test function in the variational equations of Problem P(κ; Ω), getting 2ρ üi,κ (t)e i , (v j (t) -u j,κ (t))e j L 2 (Ω)

+ T 0 Ω A ijkl e k l (u κ (t))e i j (v(t) -u κ (t)) dx dt - 1 κ T 0 Ω {[I + u i,κ (t)e i ] • q} -((v j (t) -u j,κ (t))e j • q) dx dt = T 0 Ω f i (t)(v i (t) -u i,κ (t)) dx dt. (25)
Besides, we observe that the third integral term of ( 25) is such that

1 κ T 0 Ω [I + u i,κ (t)e i ] • q -((v j (t) -u j,κ (t))e j • q) dx dt ≥ 0, (26)
for all κ > 0, since v(t) ∈ U (Ω) for a.a. t ∈ (0, T ).

By virtue of the continuity of the mappings ẽi j and the convergence process [START_REF] Piersanti | Numerical methods for static shallow shells lying over an obstacle[END_REF] we get that, for all v ∈ L 2 (0, T ; V (Ω)), the mapping

u ∈ L 2 (0, T ; V (Ω)) → T 0 Ω A ijkl ẽi j (u)(t)ẽ k l (v)(t) dx dt
is linear and continuous. An application of the convergence process (21) thus gives

T 0 Ω A ijkl ẽi j (u κ )(t)ẽ k l (v)(t) dx dt → T 0 Ω A ijkl ẽi j (u)(t)ẽ k l (v)(t) dx dt, (27)
as κ → 0.

200 By the convergence process [START_REF] Piersanti | Numerical methods for static shallow shells lying over an obstacle[END_REF], and [START_REF] Ciarlet | An Introduction to Differential Geometry with Applications to Elasticity[END_REF] we infer the following convergence üi,κ (t)e i , (v j (t) -u j,κ (t))e j → ũi (t)e i , (v j (t) -u j (t))e j ,

as κ → 0.

By the convergence process [START_REF] Piersanti | Numerical methods for static shallow shells lying over an obstacle[END_REF], and the continuity of the bilinear form

(u, v) ∈ L 2 (0, T ; V (Ω))×L 2 (0, T ; V (Ω)) → T 0 Ω A ijkl ẽi j (u)(t)ẽ k l (v)(t) dx dt,
we obtain, in particular, that

T 0 Ω A ijkl ẽi j (u)(t)ẽ k l (u)(t) dx dt ≤ lim inf κ→0 T 0 Ω A ijkl ẽi j (u κ )(t)ẽ k l (u κ )(t) dx dt. (29)
Combining ( 26), ( 27), [START_REF] Friedrichs | On the boundary-value problems of the theory of elasticity and Korn's inequality[END_REF], and (29), we immediately deduce that the limit u is a solution to the variational inequalities in Problem (P(Ω)).

We can observe that, by the convergence process (21), the vector-valued measure µ ∈ M([0, T ]; L 2 (Ω)) can be interpreted as the acceleration of the limit displacement u. Indeed, by the classical definition of weak derivative, we have that, for each i,

T 0 ui,κ (t)ϕ (t) dt = - T 0 üi,κ (t)ϕ(t) dt, for all ϕ ∈ D(0, T ).
By the properties of Lebesgue-Bochner integrals we have that, for all v ∈ L 2 (Ω) and all ϕ ∈ D(0, T ), it results

Ω T 0 ui,κ (t)(ϕ (t)v i ) dt dx = - Ω T 0 üi,κ (t)(ϕ(t)v i ) dt dx, so that, letting κ → 0 (see Comment 3 of Chapter 4 of [49]) gives Ω T 0 ui (t)ϕ (t) dt v i dx = T 0 Ω ui (t)v i dx ϕ (t) dt = -ũ(t), ϕ(t)v L 2 (Ω) = - T 0 Ω dµ i (t)(v i ϕ(t)) dt,
derivative, and, finally, the last equality holds true by Theorem 4.

To sum up, we have obtained that

T 0 Ω ui (t)v i dx ϕ (t) dt = - T 0 Ω dµ i (t)(v i ϕ(t)) dt,
for all ϕ ∈ D(0, T ) and all v ∈ L 2 (Ω).

We can thus regard the vector-valued measure µ as the second weak derivative with respect to t ∈ (0, T ) of the limit displacement u obtained via the process [START_REF] Piersanti | Numerical methods for static shallow shells lying over an obstacle[END_REF]. This justifies the following change in the notation µ = ü, and the symbol ü is now an element of M([0, T ]; L 2 (Ω)).

In conclusion, we have shown that u is in the set U (Ω) and that satisfies the variational inequalities in Problem P(Ω), namely, 2ρ ü(t), v(t) -u(t) L 2 (Ω)

+ T 0 Ω
A ijkl e k l (u(t))e i j (v(t) -u(t)) dx dt

≥ T 0 Ω f i (t)(v i (t) -u i (t)) dx dt, for all v ∈ D(0, T ; V (Ω)) such that v(t) ∈ U (Ω) for a.a. t ∈ (0, T ).
The last thing to check is the validity of the initial conditions for u. Let us introduce the operator L 0 : C 0 ([0, T ]; L 2 (Ω)) → L 2 (Ω) defined in a way such that L 0 (v) := v(0). Such an operator turns out to be linear and continuous and, by the convergence [START_REF] Ciarlet | An Introduction to Differential Geometry with Applications to Elasticity[END_REF], we get that

u κ (0) → u(0) = u 0 , in L 2 (Ω).
For what concerns the initial condition for the first derivative of u with respect to t, we present an argument, making use of the assumed "uniformity property", that slightly differs from the ones used in [START_REF] Bock | On hyperbolic contact problems[END_REF], [START_REF] Bock | Dynamic contact problem for viscoelastic von Kármán-Donnell shells[END_REF], and [START_REF] Bock | On the solutions of a dynamic contact problem for a thermoelastic von Kármán plate, Nonlinear Anal[END_REF]. For sake of clarity, we present all the computations in detail. Observe that, by virtue of the "uniformity property", we have [I + u i,κ (t)e i ] • q ≥ 0 a.e. in Ω, for a.a. 0 < t < t0 , and for all κ > 0.

210

As a result, for a.a. 0 < t < t0 , equation ( 20) takes the simpler form

2ρ üκ (t) + Au κ (t) = f (t), in V * (Ω), (30)
since we have N u κ (t) = 0 in V * (Ω), for a.a. 0 < t < t 0 , for all κ > 0.

Since f = (f i ) ∈ L ∞ (0, T ; L 2 (Ω)), we deduce that ( üκ ) κ>0 is bounded in L ∞ (0, t0 ; V * (Ω)) and, up to extracting a subsequence, we get that the following convergence takes place as κ → 0 üκ ü, in L 2 (0, t0 ; V * (Ω)).

Hence, by the convergence process [START_REF] Hlaváček | On inequalities of Korn's type. II. Applications to linear elasticity[END_REF] and the Sobolev embedding theorem (Theorem 10.1.20 of [START_REF] Kyritsi-Yiallourou | Handbook of Applied Analysis[END_REF]), the following convergence holds uκ u, in C 0 ([0, t0 ]; V * (Ω)). [START_REF] Duvaut | Inequalities in Mechanics and Physics[END_REF] Let us thus introduce the operator L1 : C 0 ([0, t0 ]; V * (Ω)) → V * (Ω) defined in a way such that L1 (v) := v(0), for all v ∈ C 0 ([0, t0 ]; V * (Ω)). Such an operator L1 is linear and continuous and, by the convergence [START_REF] Duvaut | Inequalities in Mechanics and Physics[END_REF] and the reflexivity of the space V * (Ω), we are in a position to recover the initial condition u(0) = u 1 .

215

In conclusion, we have shown that u is a solution of Problem P(Ω) and the proof is thus complete.

About the uniqueness of the solution

To conclude the investigation, we observe that the following phenomenon that occurs in the early stage. We can indeed show that, 2ρ ü(t) + Au(t) = f (t), in V * (Ω), [START_REF] Nečas | Mathematical theory of elastic and elasto-plastic bodies: an introduction[END_REF] admits a unique solution, for a.a. 0 < t < t0 . In this direction, we follow [START_REF] Evans | Partial Differential Equations, Second Edition[END_REF] (Theorem 4, Section 7.2).

220

To see this, let us show that the only solution to the initial value problem 2ρ ü(t) + Au(t) = 0, in V * (Ω), for a.a. 0 < t < t0 , u(0) = 0, A ijkl e k l (v(0))e i j (v(0)) dx = 0.

u(0) = 0, (34)
We thus infer, u(s) L 2 (Ω) = 0, for all 0 ≤ s ≤ t0 . By the arbitrariness of s, we conclude that the solution u is uniquely defined in the interval [0, t0 ].

In conclusion, all the solutions to Problem P(Ω) coincide in the interval [0, t0 ].

A sufficient condition ensuring the "uniformity property"

Let us recall the "uniformity property" that we used to prove Theorem 6:

There exists a number t0 > 0, independent of κ, such that [I + u i,κ (t)e i ] • q ≥ 0 a.e. in Ω, for a.a. 0 < t < t0 , for all κ > 0.

In this section we identify a simple sufficient condition that insures the validity of the "uniformity property". Let us consider applied body forces f such that each one of their components f i satisfies f i (t) = 0 a.e. in Ω, for a.a. 0 < t < τ 0 , for some τ 0 > 0. As a result, almost all numbers t0 between 0 and τ 0 ensure the validity of the "uniformity property".

Theorem 2 .

 2 Let T > 0 and suppose that the function y : [0, T ] → R is absolutely continuous and such that dy dt (t) ≤ a(t)y(t) + b(t), a.e. in (0, T ), where a, b ∈ L 1 (0, T ) and a, b ≥ 0 a.e. in (0, T ). Then, it results y(t) ≤ y(0) + t 0 b(s) ds e t 0 a(s) ds , for all t ∈ [0, T ].

Theorem 4 .

 4 Let ω be a compact Hausdorff space and let X be a Banach space satisfying the Radon-Nikodym property. Let F be the collection of Borel sets of 145 ω.

is u ≡ 0 . 0 -

 00 To this aim, for any fixed 0 ≤ s ≤ t0 , let us define the functionv(t) := τ ) dτ , 0 ≤ t ≤ s, 0 , s < t ≤ t0 ,a.e. in Ω, with v ∈ C 0 ([0, t0 ]; V (Ω)). Since u(0) = 0 = v(s), an application of the integration by parts formula (Corollary 10.1.26 of[START_REF] Kyritsi-Yiallourou | Handbook of Applied Analysis[END_REF]) givess 2ρ V * (Ω) ui (t)e i , vj (t)e j V (Ω) + Ω A ijkl e k l (u(t))e i j (v(t)) dx dt = 0.Since v(t) = -u(t), for all 0 ≤ t ≤ s, the latter formula becomes s 0 2ρ V * (Ω) ui (t)e i , u j (t)e j V (Ω) +ΩA ijkl e k l ( v(t))e i j (v(t)) dx dt = 0.Again, by integration by parts formula (Corollary 10.1.26 of[START_REF] Kyritsi-Yiallourou | Handbook of Applied Analysis[END_REF]), we get ijkl e k l (v(t))e i j (v(t)) dx dt = ρ u(s) 2 L 2 (Ω) + 1 2 Ω

Acknowledgements

The author is greatly indebted to Professor Igor Bock, Professor Philippe G.

Ciarlet, and Professor Cristinel Mardare for their valuable advice.

The author acknowledges partial support by the ERC advanced grant 668998 (OCLOC) under the EU's H2020 research program.

where the first equality holds by Fubini's theorem, the second equality holds by Theorem 4, the third convergence of the process [START_REF] Piersanti | Numerical methods for static shallow shells lying over an obstacle[END_REF] and the definition of weak


domain: phys sdv

text:
Danahe Mohammed

Guillaume Charras

Eléonore Vercruysse

Marie Versaevel

Joséphine Lantoine

Laura Alaimo

Céline Bruyère

Marine Luciano

Karine Glinel

Geoffrey Delhaye

Olivier Théodoly

Sylvain Gabriele
email: sylvain.gabriele@umons.ac.be

Substrate area confinement is a key determinant of cell velocity in collective migration

Collective cell migration is fundamental throughout development, wound healing and in many diseases. Although much effort has focused on cell-cell junctions, a role for physical confinement in collective cell migration remains unclear. Here we used adhesive microstripes of varying widths to mimic the spatial confinement experienced by follower cells within epithelial tissues. Our results reveal that the substrate area confinement is sufficient to modulate the three-dimensional (3D) cellular morphology without the need for intercellular adhesive cues. Our findings show a direct correlation between the migration velocity of confined cells and their cell-substrate adhesive area. Closer examination revealed that adhesive area confinement reduces lamellipodial protrusive forces, decreases the number of focal complexes at the leading edge and prevents the maturation of focal adhesions at the trailing edge, leading together to less effective

forward propelling forces. The release of follower confinement required for the emergence of leader cells is associated with a three-fold increase in contractile stress and a ten-fold increase in protrusive forces, providing together a sufficient stress to generate highly motile mesenchymal cells. These findings demonstrate that epithelial confinement alone can induce follower-like behaviours and identify substrate adhesive area confinement as a key determinant of cell velocity in collective migration.

Cell migration is a fundamental biological process in which individual cells orchestrate their movement by interacting with each other and with their environment. In two-dimensional (2D) environments, cells migrate as large epithelial sheets during many biological processes, such as tissue morphogenesis [START_REF] Keller | Shaping the Vertebrate Body Plan by Polarized Embryonic Cell Movements[END_REF] , wound closure 2 , embryonic development [START_REF] Farooqui | Multiple rows of cells behind an epithelial wound edge extend cryptic lamellipodia to collectively drive cell-sheet movement[END_REF] or tumor metastasis [START_REF] Ilina | Mechanisms of collective cell migration a glance[END_REF] . Migration of epithelium-derived cells consists of basic sequential events occurring in a cyclic fashion [START_REF] Ridley | Cell Migration: Integrating Signals from Front to Back[END_REF] . This migration mode depends on the formation of adhesion sites, actin-driven protrusion generation, and actomyosin contractility that together create stress within the cell [START_REF] Murrell | Forcing cells into shape: the mechanics of actomyosin contractility[END_REF] . Migration of 2D epithelial sheets occurs under a broad range of external constraints that induce the appearance of highly motile mesenchymal cells at the leading front that guide follower cells [START_REF] Mayor | The front and rear of collective cell migration[END_REF] . Among the different environmental constraints, recent evidence suggests that a 2D confinement may modulate the transition between collective and single cell migration [START_REF] Friedl | Cancer Invasion and the Microenvironment: Plasticity and Reciprocity[END_REF] by inducing a restriction of cell-cell contacts, leading to the dispersion of cells from a collective sheet [START_REF] Wong | Collective and individual migration following the epithelialmesenchymal transition[END_REF] . Eventually, leader cells tend to escape and migrate separately leading to the progressive dissociation of the cell cluster. Behind the leader cells, cell compaction impairs the formation of a classical leading edge, suggesting that physical confinement affects not only cell morphology but also the core biological mechanisms driving migration of follower cells.

Cadherin-mediated interactions between neighbouring cells contribute to the polarization of migrating cell groups and it has been shown that impairing cadherin function dramatically alters collective migration dynamics [START_REF] Mayor | The front and rear of collective cell migration[END_REF] . Although much effort has focused on the importance of cell-cell junctions [START_REF] Vedula | Emerging modes of collective cell migration induced by geometrical constraints[END_REF] , the effect of physical 2D confinement on the migration of single cells or epithelial cell sheets has been recognized but not yet elucidated. This is surprising because cells in epithelial sheets are often naturally compacted and have access to smaller adhesion area with the substrate. Recent reports suggest that density mediated contact inhibition of locomotion [START_REF] Abercrombie | Observations on the social behaviour of cells in tissue culture. I. Speed of movement of chick heart fibroblasts in relation to their mutual contacts[END_REF][START_REF] Stoker | Density Dependent Inhibition of Cell Growth in Culture[END_REF] can be described using an analogy between a cell monolayer and a bidimensional "jammed" colloidal system [START_REF] Angelini | Glass-like dynamics of collective cell migration[END_REF][START_REF] Garcia | Physics of active jamming during collective cellular motion in a monolayer[END_REF] , emphasizing the role of spatial confinement as a control parameter for wound healing and epithelial cell migration. However, our current understanding of even the most basic aspects of epithelial cell migration under physical confinement remains poor.

Here we study how the restriction of the cell-substrate contact area can modulate epithelial migration. We used adhesive micropatterned stripes to examine the motion of confined fish epithelial keratocytes, a classic model system to investigate single cell motility [START_REF] Keren | Mechanism of shape determination in motile cells[END_REF][START_REF] Lacayo | Emergence of Large-Scale Cell Morphology and Movement from Local Actin Filament Growth Dynamics[END_REF][START_REF] Fournier | Force transmission in migrating cells[END_REF] . A wide range of physical 2D confinements, similar to those encountered physiologically by cells when migrating collectively, was imposed by varying the width of adhesive microstripes. Furthermore, we use keratocyte explants as an efficient primary model of collective cell migration. The modulation of the available adhesive area in confined environments emerges as a key physical parameter that regulates cell morphology, driving forces and migrating velocity in collective cell migration.

Followers are elongated, laterally confined and slower

When scales are removed from a fish and deposited onto a substrate homogeneously coated with fibronectin (FN), this leads to the formation of a 2D migrating epithelial tissue that spreads onto the substrate replicating wound healing (Fig. 1A and Supplementary Movie 1).

As shown in Fig. 1A, there are clear differences between follower and leader cells after 14 h in explant culture. Leader cells were polarized and elongated along the direction perpendicular to the direction of migration. They possessed a broad actin-rich lamellipodium, whereas their microtubules were wrapped around the nucleus in the cell body. In contrast, followers were mostly elongated along the migration direction, exhibited a prominent cortical actin network and a microtubule network that extended throughout the cell. The mean aspect ratio, as denoted by the ratio of the cell axis perpendicular to the migration direction to the cell axis parallel to the migration direction, was 0.440.11 for confined cells (n=42) and 1.580.24 for leaders (n=47), suggesting important symmetry breaking dependent on cell position (Fig. 1B). Furthermore, followers have a smaller mean width (11.12.3 µm, n=52, Supplementary Fig. 1A) than leaders (22.52.6 µm, n=49, Supplementary Fig. 1B). As shown in Figs. 1C-D, followers had a smaller projected area (19049 µm 2 , n=42) and were slower (0.030.05 µm/s, n=40) than leaders (47520 µm 2 , n=47 and 0.050.08 µm/s, n=44).

Together, these results raise the possibility that spatial confinement experienced by epithelial cells might control their morphology and migration velocity.

Confinement regulates cell morphologies and velocities

To address this question, we examined the motile behaviour of individual keratocytes migrating on FN microstripes of well-defined widths ranging from 5 to 20 µm (Fig 1E ) to reproduce the range of lateral 2D confinements encountered within epithelial tissues (Supplementary Fig. 1). As shown in Fig. 1F and Supplementary Movie 2, individual keratocytes migrating on narrow microstripes modified their typical fan-shaped morphology and increased their long axis to elongate in the direction of migration. We found that the cell aspect ratio decreased linearly with the microstripe width (Fig. 2A, R 2 = 0.948). Interestingly, spatial confinement also modulated the fraction of stationary cells that decreased from 0.980.01 on the narrowest microstripe to 0.070.01 on the largest one (n=48, Fig. 2B). The linear adaptation of the cell projected area to the lateral confinement imposed by the microstripe width was mainly driven by a reorganization of the lamellipodium (Fig. 2C).

Time-lapse recording of individual keratocytes migrating on microstripes indicated

that cells moved with a constant velocity that can be modulated by the microstripe width, as observed by slope changes (Fig. 2D). Our results showed that the evolution of migration velocity with the micropattern width was described by two distinct domains with a sharp transition around 10 µm (Fig. 2E), corresponding to a spreading area of ~190 µm 2 (Fig. 2F).

For widths below 10 µm, keratocytes migrated very slowly (~0.02 µm/s), whereas above 10 µm the migration velocity increased linearly with microstripe width to reach ~0.35 µm/s on 20 µm wide microstripes, similar to the velocity observed in unconfined cells [START_REF] Riaz | Persistence of fanshaped keratocytes is a matrix-rigidity-dependent mechanism that requires α5β1 integrin engagement[END_REF] . To confirm these observations, we next designed a pattern composed of five interconnected microstripes of increasing widths: 5, 9, 13, 17 and 21 µm (Figs. 2G andH). The migration of individual confined cells (n=13) on interconnected microstripes was tracked by time-lapse microscopy during several forward and backward travels (Supplementary Movie 3). The small difference in width between two successive segments (i.e. 4 µm) ensured a smooth transition in velocity.

As shown in Fig. 2I, the velocity increased from narrow to large microstripes (forth trajectory), then cells turned back at the end of the larger microstripe and the migrating distance over time decreased from large to narrow microstripes (back trajectory). The overlay of two successive forth trajectories of the same cell indicated a robust and reproducible adaptation of migrating behaviour to physical confinement (Fig. 2J). The instantaneous velocity on forth and back travels (Fig. 2K) was obtained by differentiating the sigmoidal fit of the migrating distance versus time (Fig. 2I). Next, we grouped follower cells in bins centered on the widths of the microstripes. The average area of followers within each bin was calculated and plotted as a function of the area of cells confined on microstripes with a width equal to the bin of interest (Supplementary Fig. 3). We found a linear correlation (slope = 0.99  0.04 and R 2 = 0.9015, n=93) between the projected area of followers and the projected area of confined cells for different microstripe widths. Confined cells cannot maintain their migrating phenotype on microstripes narrower than 11 µm, which is just below the mean width of followers (w= 12.1±2.2 µm, n= 52, Supplementary Fig. 1).

Together, these results suggest that lateral confinement, which is the dominant form of confinement experienced by follower cells, regulates morphologies and velocities of migrating cells.

2D physical confinement modulates 3D cell morphology

We next investigated the three-dimensional (3D) cytoskeletal organization (F-actin, microtubules and DNA) of confined keratocytes by confocal microscopy. The leading edge of fan-shaped cells was composed of a flat lamellipodium enriched with F-actin [START_REF] Verkhovsky | Orientational Order of the Lamellipodial Actin Network as Demonstrated in Living Motile Cells[END_REF] , whereas the trailing edge was characterized by a spherical cell body containing a dense network of microtubules wrapped around the nucleus (Fig. 3A). In contrast, confined keratocytes displayed an elongated morphology with a thicker leading edge and a large accumulation of F-actin on both cell sides. Surprisingly, our results revealed that microtubules extended to the leading edge of confined cells (Fig. 3B), as observed in followers during collective migration (Fig. 3C). These findings suggest that physical confinement alone is sufficient to reorganize actin and microtubule networks without the need for intercellular adhesive cues. Our results showed that isolated cells exhibited a random orientation of microtubules (Supplementary Fig. 4A-C), whereas confined cells (Supplementary Fig. 4D-F) and followers (Supplementary Fig. 4G-J) were characterized by a network of microtubules oriented parallel to the axis of migration. In addition, cells confined on microstripes showed similar values of coverage area of microtubules to overall cell area (45.5±5.9%) than followers (40.6±8.5%), whereas unconfined cells exhibited lower values (18.3±1.2%). Together, these results strengthen the morphological similarities between cells confined on narrow adhesive microstripes and follower cells compacted within epithelial tissues (Supplementary Fig. 4K).

We acquired cell height profiles along the migration axis of glutaraldehyde fixed isolated and confined keratocytes using atomic force microscopy (Fig. 3D and Supplementary Fig. 5). The rounded cell body of unconfined cells formed an angle of ~78±4° (n=11) with the flat leading edge (Fig. 3E), indicating an abrupt transition between leading and trailing edges.

In contrast, confined keratocytes exhibited a more elongated morphology and a lower transition angle (~42±7°, n=13), suggesting a smooth transition between both subcompartments. We then quantified the mean height of both cell microcompartments by acquiring profiles perpendicular to the cell migration axis for keratocytes on 20, 10 and 5 µm wide microstripes (Fig. 3G). Surprisingly, our results showed that the cell body flattened significantly on narrow stripes (5 µm wide) down to 1.30.2 µm (Fig. 3H,n=13), corresponding to a decrease of ~59% in height for confined cells. Conversely, we observed a significant thickening of the leading edge that reached 58050 nm on 5 µm wide microstripes (Fig. 3I,n=11), corresponding to a ~285% increase of the thickness. On 20 µm wide microstripes, keratocytes adopted a fan-shaped morphology characterized by a mean volume of the cell body about 8 times higher than the leading edge volume. Despite large morphological modifications of cell body (Fig. 3J) and leading edge (Fig. 3K) in confined cells, the total cell volume remained constant (Fig. 3L).

Together, these results show that 2D confinement induces significant 3D modifications of cell morphology with a redistribution of cytoskeletal components and large variations of volume between leading and trailing edges, while keeping the total cellular volume constant.

Thicker lamellipodia exert less protrusive forces

Based on the central role of F-actin assembly and disassembly in keratocyte motility [START_REF] Ofer | Actin disassembly clock determines shape and speed of lamellipodial fragments[END_REF] , we next hypothesized that the slower velocity of keratocytes confined on microstripes might be a consequence of the leading edge thickening, which may reduce the efficiency of polymerization driven membrane protrusion. To test this experimentally, we measured the lamellipodial protrusive forces by placing an AFM cantilever of calibrated spring constant perpendicular to the surface in front of the leading edge, as introduced previously [START_REF] Prass | Direct measurement of the lamellipodial protrusive force in a migrating cell[END_REF] . The vertical position of the cantilever was measured optically over time (Figs. 4A-B and Supplementary Movie 4). When the lamellipodium was stalled at higher forces, the velocity of the lamellipodium decreased eventually to zero. The first derivative of the deflection-time curves was used to obtain the cantilever speed versus time (Fig. 4C) and the lamellipodial protrusive force was determined from the mean stall force, when the cantilever movement stopped. Our results showed that protrusive forces exerted by the leading edge increased with cell velocity (Fig. 4D), which was itself controlled by the microstripe width. As a consequence, protrusive forces increased linearly with microstripe width (Fig. 4E). Assuming a mean force of ~8 pN generated per actin filament [START_REF] Mogilner | Cell motility driven by actin polymerization[END_REF] and knowing the height of the lamellipodium (Fig. 2I), we estimated that only ~25 actin filaments pushed on a contact area of 3.5 µm 2 in highly confined cells. This corresponded to ~7 filaments/µm 2 . In contrast, ~300 actin filaments pushed on a contact area of 2.5 µm 2 in unconfined cells, corresponding to ~120 filaments/µm 2 . The pressure exerted by actin filaments was therefore significantly modulated by the width of the microstripe (Fig. 4F).

Together, these results revealed that physical confinement imposed by narrow microstripes leads to less effective protrusive force generation in thicker leading edges that reduce the migration velocity.

Escaping tissues requires the actin cytoskeleton remodeling

Next we sought to understand the mechanisms allowing leader cells to escape from the tissue.

The escape process from primary epithelial sheets can be described as a succession of three states: (i) confined, (ii) transition and (iii) unconfined (Fig. 5A and Supplementary Movie 5).

We reproduced this process using a new micropattern geometry composed of a linear microstripe of 12 µm wide connected to a disk of 50 µm in diameter (Fig. 5B and Supplementary Movie 6). As soon as the cell extremity reached the circular zone, the leading edge tended to adopt a fan shape, whereas the cell body was still confined on the narrow microstripe.

We found that the cell projected area increased for cells escaping a tissue (~75%, Fig. 5D) and a microstripe (~71%, Fig. 5E). Spatiotemporal kymograph analysis of the escape process indicated that the three successive steps were associated with unsynchronized velocity changes in the leading and trailing edges (Fig. 5F). Indeed, confined cells entering the circular disk stopped the progression of their leading edge (Fig. 5G) until their trailing edge had exited the microstripe. At this transition stage, keratocytes reorganized their actin cytoskeleton to extend their leading edge on the circular disk. Using Sir-actin to label F-actin in live cells (Fig 5I and Supplementary Movie 7), we showed the presence of two intense fluorescent signals in confined cells located on their lateral sides and oriented along to the migration axis, as observed in immunostainings (Fig. 3B). Interestingly, the F-actin signal disappeared at the transition between the microstripe and the circular disk (Fig. 5J) and unconfined cells formed a large lamellipodium enriched with F-actin. To get insight into actin reorganization, we performed additional time-lapse experiments with GFP-UTR keratocytes (Supplementary Fig. 6A and Movie 8). We confirmed an increase of the spreading area during the transition (Supplementary Fig. 6B). During the confined stage (Supplementary Fig. 6C), cells generated large and dynamic protrusive extensions composed of densely packed bundles of actin filaments that extended parallel to the microstripe axis (Supplementary Movie 8). Lateral membrane ruffles were highly dynamic (Supplementary Fig. 6D-F) and symmetric with a mean angle of ~31°, corresponding to a velocity of ~0.15 μm/sec. Previous reports have shown that membrane ruffles, which form in conditions of low cell-substrate adhesion, lead to a significantly reduced lamellipodium persistence and an inefficient cell migration [START_REF] Borm | Membrane ruffles in cell migration: indicators of inefficient lamellipodia adhesion and compartments of actin filament reorganization[END_REF] . Our observations indicated that membrane ruffles were associated with confined migration modes, which were characterized by a low number of focal complexes in the leading edge.

Interestingly, membrane ruffles disappeared during the relaxation stage when a network of actin filament formed at the cell edge, leading to an efficient migration mode characterized by a dynamic actin retrograde flow and higher protrusive forces (Supplementary Fig. 6G-I).

These results reveal that the transition from followers to leaders required a profound remodeling of the actin network to adopt larger spreading areas and faster migration speeds.

Migration velocity is regulated by adhesive area

One hypothesis to explain how physical confinement affects cell migration is a modification of the spatial distribution of cell-substrate adhesions that may, in turn, affect traction forces exerted on the substrate. To address this issue, we determined the distribution of vinculincontaining adhesions during the escape process with confocal microscopy (Fig. 6A). We observed numerous focal complexes (≤1 μm 2 ) at the leading edge of confined and unconfined cells, whereas mature elongated adhesions were mainly found at the trailing edges of unconfined cells. The number of focal complexes in the leading edge increased with the spreading area leading to a constant density of focal complexes (Fig. 6B). The mean area of focal complexes remained constant, despite large modifications of the leading edge morphology during the escape process (Fig. 6C). We next sought to determine the influence of physical confinement on mature focal adhesions (≥3 μm 2 ) at the cell rear. To compare the individual role of both adhesion types, we normalized the area of focal complexes at the front and mature focal adhesions at the rear by the total cell spreading area. We found that adhesion area in the leading edge increased slightly, whereas focal adhesions at the rear matured significantly (Fig. 6D). Interestingly, the normalization of the speed of confined cells migrating on adhesive microstripes of various widths to their total adhesive area demonstrated that cell velocity was directly correlated to adhesive area (Fig. 6E), suggesting that physical confinement regulates cell velocity through cell-substrate adhesions. Taken together, these results disclose a distinctive pattern of adhesions, which is directly correlated with physical confinement and cell migration speed.

We used traction force microscopy (TFM) to quantify the amount of contractile forces exerted on the substrate by keratocytes escaping a confined zone (Fig. 6F). Confined cells exerted two-times less contractile stress than isolated cells, which reached a plateau of total stress around 9 kPa (Fig. 6G). Intense contractile forces that were mostly distributed in the leading edge region of confined cells shifted to the trailing edge during the escape process.

Our findings show that the amount of traction forces increased in leading edges as confinement was released, whereas traction forces in trailing edges were significantly higher in unconfined cells (Fig. 6H).

We next investigated the role of cell-cell adhesions by forming "trains" of migrating cells on FN-coated microstripes of 15 µm wide (Fig. 6I andJ). During its spreading, a nontrypsinized tissue forms multicellular protrusions, known as "fingers", that extend on the microstripes and lead to trains of cells of different lengths. Such collectively migrating trains arise from the native epithelial monolayer which was not subjected to a trypsin treatment to preserve endogenous cell-cell contacts (Fig. 6K and Supplementary Fig. 7). We quantified the projected area and the migrating distance of trains of cells composed of n=2, 3 and 4 cells during time-lapse experiments (Figs. 6L andM). All trains of cells show a persistent migration (Supplementary Movie 9) and behave as a cohesive unit during the whole experiment (Fig. 6N), independently of the cohort size. Moreover, the spreading area increased linearly with the number of cells in the cohort (Fig. 6O). By normalizing the train velocity by the velocity of single cells confined on the same microstripe (Fig. 6L), we found that the velocity of cell trains (n=2, 3 or 4 cells) with mature cell-cell adhesions was not statistically different than isolated cells (Fig. 6P). As shown in Supplementary Fig. 8, followers (0.119  0.023) and confined cells (0.115  0.018) exhibited statistically the same amount of normalized vinculin area, whereas leader cells showed larger values (0.253  0.029), which were close to those observed for individual cells on fibronectin (0.239  0.032).

Taken together, these results confirm that a physical confinement modulates the adhesiveness of keratocytes and suggest that the establishment of head-to-tail adhesive interactions does not modulate the migrating velocity of confined epithelial cells. This result was confirmed for very large trains of cells (n=13) that exhibited a similar migrating velocity than isolated cells (Fig. 6P).

Cohesiveness was usually associated with a reduced migration speed [START_REF] Rolli | Switchable adhesive substrates: Revealing geometry dependence in collective cell behavior[END_REF] and whenever rapid migration occurs physiologically, cells tend to downregulate E-cadherin and dissociate through a complete epithelial-to-mesenchymal transition (EMT). However, in many physiological situations, cells undergo an incomplete EMT, in which E-cadherin adhesion is weakened to enable dynamic flexibility for each individual cell within the group, while keeping a certain degree of cohesiveness [START_REF] Trepat | Cell migration[END_REF] . As summarized in Supplementary Fig. 9, our results establish for the first time that the substrate area confinement resulting from a lateral cell compaction is a key mechanism in the modulation of cell velocity in collective migration.

Methods

Stamp fabrication and microcontact printing. Microstripes with different widths (from 5 to 20 µm) and a relaxation pattern (a microstripe of 12 µm wide connected to a disk of 50 µm in diameter) are created using a silicon master fabricated by deep reactive-ion etching from a chromium photomask (Toppan Photomask, Corbeil Essonnes, France). The silicon surface was passivated with a fluorosilane (tridecafluoro-1,1,2,2-tetrahydrooctyl-1-trichlorosilane, Gelest) for 30 min, and microstamps were obtained by molding the silanized silicon master with polydimethylsiloxane (PDMS) (Sylgard 184 Silicone Elastomer Kit; Dow Corning, Midland, MI) cured for 4 h at 60°C. PDMS stamps were oxidized in an ultraviolet/O3 oven for 7 min and inked with a FN solution (18 mg/ml) from human plasma (Sigma, St Louis, MO) for 1 h at room temperature. Then PDMS stamps were dried under filtered air and gently pressed onto a flat PDMS-coated glass coverslip. Uncoated regions were passivated by incubating the glass substrates for 5 min in a 1% Pluronic F-127 solution (BASF, Mount Olive, NJ) and washed three times in PBS [START_REF] Lantoine | Matrix stiffness modulates formation and activity of neuronal networks of controlled architectures[END_REF][START_REF] Versaevel | Cell confinement: putting the squeeze on the nucleus[END_REF] . The error bars on the microstripe width correspond to the experimental variability of the microstripe width. The experimental variability on the microstripe width mainly depends on the quality of the microstructured PDMS stamp and also on the applied pressure during the microprinting process. The mean value of each microstripe width and its associated standard deviation was obtained by measuring the width of each FN microstripe in epifluorescent mode (12 ≤ n ≤ 16 per width).

The Primo method based on a Light Induced Molecular Adsorption of Proteins (LIMAP) technology (Alveole Company, France) was used to create a pattern of FN composed of five interconnected microstripes of increasing widths (from 5 to 21 µm, with an increment of 4 µm). The 5, 9, 13 and 17 µm wide segments each had a length of 60 µm, whereas the 21 µm wide segment was 80 µm long.

Preparation of polydimethylsiloxane elastomers. Polydimethylsiloxane (PDMS) substrates of 3 MPa were prepared from the commercially available Sylgard 184 silicone elastomer kit (Dow Corning, Midland, MI) by mixing the base and the curing agent as 10:1 w/w ratio [START_REF] Versaevel | Micropatterning Hydroxy-PAAm Hydrogels and Sylgard 184 Silicone elastomers with tunable elastic moduli[END_REF] .

Pre-polymer solutions were mixed thoroughly for at least 5 min, degassed, and spin-coated at 5000 rpm on 25 mm glass coverslips [START_REF] Coppée | Influence of chain interdiffusion between immiscible polymers on dewetting dynamics[END_REF] . PDMS was then cured for 3 hours at 60°C. Samples were stored at room temperature in a vacuum desiccator.

Cell culture. Keratocytes are harvested from scale of Central American cichlid Hypsophrys

Nicaraguensis. A Fish scale was putted onto a glass coverslip of 25 mm previously washed in a 30% ethanol solution and dried. The scale was sandwiched between two 25 mm diameter glass coverslips and cultured in Leibovitz's Media (L-15) supplemented with 10% FBS, 1% antibiotic-antimycotic, 14.2 mM HEPES and 30% deionized water at room temperature for 12 hours [START_REF] Riaz | Persistence of fanshaped keratocytes is a matrix-rigidity-dependent mechanism that requires α5β1 integrin engagement[END_REF] . Individual keratocytes were then dissociated by incubating the tissue in a trypsin solution (1 ml per glass slide) for 5 min and resuspended in 4 ml of L-15 Leibovitz complete medium. Suspended cells were then deposited on microprinted coverslips to investigate their migration behaviour. All experiments were made between 2 and 8 hours after cell seeding. To ensure that stationary cells were alive, only cells that actively probed their microenvironment by generating dynamic lamellipodial protrusions over the course of the experiment (~1800 sec) were used.

Immunocytochemistry. Intracellular components were made visible using fluorescent staining techniques. Fish keratocytes were rinsed 3 times with PBS (pH ~7.4), fixed and permeabilized with a first 1 min incubation in 0.05% glutaraldehyde and 0.1% triton X-100 followed by a first rinse in PBS then a second 10 min incubation in a 0.2% solution of glutaraldehyde. For vinculin immunostaining, cells were permeabilized and fixed with a solution of 4% paraformaldehyde (Electron Microscopy Sciences, PA) and 0.05% Triton X-100 (Sigma) in PBS (PAA Laboratories) for 15 min and washed three times in PBS. After and analyzed with Origin 8.5 (OriginLab, Northampton, MA). Selected cells were highlighted and their centroids at each time point were recorded [START_REF] Riaz | Persistence of fanshaped keratocytes is a matrix-rigidity-dependent mechanism that requires α5β1 integrin engagement[END_REF] . A cell was considered as "stationary" when its center of mass did not move over a period of 1800 sec. Stationary cells possessed two lamellipodia at opposite sides of their cell body, which was localized at the center of the cell. For each microstripe width, the migration velocity was determined from the evolution of the distance as a function of time. Distance versus time curves were fitted with a linear regression to determine the slope that corresponds to the mean migration velocity. Then the average of slopes (from 12 to 16 cells, pooled from 3 different experiments) was taken to determine the migration velocity on a specific microstripe width.

Traction force microscopy data measurement and analysis. Hydroxy-polyacrylamide (hydroxy-PAAm) hydrogels 30 containing 1:50 volume of carboxylate-modified fluorescence latex beads (0.2 µm Fluospheres, Molecular Probes, Eugene, OR) were fabricated on 25-mm diameter glass coverslips according to a protocol previously described [START_REF] Grevesse | Preparation of hydroxy-PAAm hydrogels for decoupling the effects of mechanotransduction cues[END_REF] . The Young's modulus of the gel was estimated to be 12 kPa using Dynamic Mechanical Analysis (Mettler Toledo, Switzerland). FN-coated PDMS stamps were allowed to react with the activated gel for 1 h at room temperature to create FN-coated adhesive islands on the gel surface. After removal of the excess of FN by washing with PBS, unprinted areas of the gel were passivated with a bovine serum albumin (BSA) solution overnight. Fluorescence images of the gel containing fluorescent beads immediately beneath the patterned cells were taken during the cell migration and the bead displacements were measured in Matlab between consecutive frames. The traction field was then calculated from the displacement field by solving the inverse of the Boussinesq solution from the displacement field on the surface of an elastic halfspace to obtain the traction field when the mechanical properties of the gel are known [START_REF] Versaevel | Spatial coordination between cell and nuclear shape within micropatterned endothelial cells[END_REF] .

The interior of the cell was subdivided into 64 µm 2 squares to approximate the discretized localization of contractile forces.

Determination of the lamellipodial protrusive forces. An atomic force microscope (AFM) cantilever (Bruker, SNL-10) was glued in the vertical position to a glass micropipette, which was controlled by a motorized 3-axis micromanipulator. The AFM cantilever of calibrated spring constant was placed vertically perpendicular to the substrate surface, in front of the leading edge of an individual keratocyte migrating on a FN-coated microstripe. The load force applied by a migrating cell on the advancing leading edge exerts in response an equal and opposite protrusive force to the cantilever. The spring constant of the cantilever was measured according to the thermal noise method [START_REF] Butt | Calculation of thermal noise in atomic force microscopy[END_REF] and was found to be ~0.06 N/m. The load force applied by a migrating cell on the advancing leading edge exerts in response an equal and opposite protrusive force to the cantilever. The vertical position of the cantilever was measured optically using a high magnification objective, from initial contact with the leading edge at t=0 to the deflection of the cantilever at t+∂t. The cantilever deflection increased with time until a stall force was reached. Protrusive forces were then estimated using the Hook's law F=k.x, where F is the force (N), k the spring constant of the AFM cantilever (N/m) and x the deflection of the cantilever (µm).

AFM profile. Cells confined on FN microstripes of 5, 10 and 20 µm wide were permeabilized in a solution of 0.05% glutaraldehyde and 0.1% Triton X-100 in PBS for 1 min then fixed in a solution of 0.2% glutaraldehyde in PBS for 10 minutes. Fixed cells were finally dehydrated with water/alcohol solutions of increasing ethanol ratios (10%, 20%, 50%, 70% and 100% of ethanol). Dehydrated samples were mounted on an ICON Dimension Atomic Force Microscope (Bruker Company) working in Peak Force Quantitative Nanomechanical Mapping (QNM) mode. AFM profiles were acquired on dried cells using calibrated SNL-10 (Bruker) cantilevers. We used a scan rate of 0.2 Hz and a scan size ranging from 55 µm 2 to 4040 µm 2 to acquire AFM profiles perpendicular to the long cell axis. We impose a typical thickness interval for each AFM profile of 2 µm and 8 to 12 AFM profiles were acquired per cell (n=8) to cover the whole cell length, depending on the confinement condition (see Rebuttal Figure 5). Finally, each AFM profile was integrated and the mean cellular volume was estimated by summing all profile areas multiplied by profile interval for cells confined on FN microstripes of 5, 10 and 20 µm wide (n=8 for each conditions, Supplementary Fig. 5).

F-actin live imaging. Fish keratocytes were incubated in a solution of SirActin at 80 nM for 1h in medium Leibovitz-15. Then, the medium was changed with a solution of 20 µM of Verapamil in medium. SirActin-labeled cells were visualized with a Cy5 cube. The concentration of the probe in the media was kept constant at 80 nM during the whole experiment to get a constant signal. The image correction tool introduced by T. Peng and coworkers [START_REF] Peng | A BaSiC tool for background and shading correction of optical microscopy images[END_REF] has been used to improve the intensity quantification of Sir-actin time-lapse experiments. This method corrects the intensity profile of each frame and removes both spatial shading effects and the temporal drift.

Statistical analysis. Differences in means between groups were evaluated by two-tailed

Student's t-tests performed in Origin 8.5 (OriginLab, Northampton, MA). For multiple comparisons the differences were determined by using an analysis of variance (ANOVA) followed by Tukey post-hoc test. *p < 0.05, **p < 0.01, ***p < 0.001 and n.s. is not significant. Unless otherwise stated, all data are presented as mean ± standard deviation (s.d.).

  permeabilization and fixation, cells were blocked with a solution of 5% FBS. Coverslips were rinsed extensively with PBS and then incubated for 45 min at 37°C with Alexa Fluor 488 phalloidin (Invitrogen, 1:200) for staining filamentous actin, DAPI (Invitrogen, 1:200) to visualize the nuclei. To visualize vinculin, the samples were first incubated a primary antibody (anti-vinculin antibody produced in mouse, Sigma-Aldrich, HVIN-1 clone, 1:200) or anti-tubulin antibody produced in mouse (1:200) then tetramethylrhodamine-labelled secondary antibody (goat anti-mouse 1:200, Sigma-Aldrich) was used for 45 min at 37°C. Slides were mounted in Slow Fade Gold Antifade (Molecular Probes, Invitrogen). Epifluorescence and confocal microscopy imaging. Fixed and immunofluorescence-stained preparations were observed in epifluorescence and confocal mode with a Nikon Eclipse Ti-E motorized inverted microscope (Nikon C1 scanhead; Nikon, Japan) equipped with ×40 DIC, ×40 Plan Apo (NA 1.45, oil immersion), ×60 Plan Apo (NA 1.45, oil immersion) and ×100 Plan Apo (NA 1.45, oil immersion) objectives, two lasers (Ar-ion 488 nm; HeNe, 543 nm) and a modulable diode (408 nm). Epifluorescence images were recorded with a Roper QuantEM:512SC EMCCD camera (Photometrics, Tucson, AZ) using NIS Elements Advanced Research 4.0 software (Nikon) (Photometrics, Tucson, AZ) using Capture-Pro so ware (Photometrics). Confocal images were acquired with NIS Elements Advanced Research 4.0 software (Nikon) by using small Z-depth increments between focal sections (0.15 µm). Cell tracking. Time-lapse microscopy experiments were done to follow the keratocyte migration. These measurements were performed in DIC mode with a Nikon Eclipse Ti-E motorized inverted microscope at ×40 magnification. Tracking of the keratocytes and kymographs were done with NIS Elements Advanced Research 4.0 software (Nikon, Japan)

Figure 1 -

 1 Figure 1-Epithelial cells are confined within epithelial tissues. (A) Growing epithelial tissue is composed of confined (left) and leader (right) cells. Confined (in red) and leader (in green) epithelial cells are characterized by different (B) aspect ratios, (C) projected areas and (D) velocities. (E) A microcontact printing technique was used to create fibronectin-coated

Figure 2 -

 2 Figure 2-Cell morphologies and migrating velocities are regulated by the lateral confinement. (A) The cell aspect ratio changes linearly with the width of the micropatterned adhesive microstripe. Errors bars indicate s.d. with 12≤ n ≤ 16 for each point. Black line is a linear fit with R 2 =0.948. (B) The fraction of stationary cells is modulated by the microstripe width. Errors bars indicate s.d. with n=48 for each bar. (C) Evolution of the projected area of the leading edge (in black), the whole cell (in light grey), and the cell body (in dark grey) as a function of the microstripe width. Errors bars indicate s.d. with 12≤ n ≤ 16 for each point, pooled from three different experiments. (D) Temporal evolution of the migration distance on

Figure 3 -

 3 Figure 3-2D confinement modulates the 3D cellular morphology. Orthogonal views acquired by confocal microscopy of individual keratocytes migrating on (A) a homogeneous FN-coated substrate and (B) a FN microstripe of 15 µm wide. (C) Orthogonal confocal view of an epithelial tissue. (D) AFM was used to acquire cell profiles along a line running from the cell rear to the cell front at the cell midline (dotted line) for isolated and confined cells. (E) and (F) shows representative height profiles measured for an isolated cell and a confined cell, respectively. (G) Schematic representation showing the position of the lateral AFM

Figure 4 -

 4 Figure 4-Thicker lamellipodia exert less protrusive forces. The protrusive force exerted by the leading edge decreases with confinement. (A) Schematic representation (side view) of a migrating cell deflecting an AFM cantilever. (B) Top view phase micrographs of the cantilever deflection in response to the pushing force exerted by the leading edge of the oncoming cell. The dashed red line represents the initial position of the cantilever and the dashed blue line shows the cantilever deflection. Contact between the cell and the cantilever

Figure 5 -

 5 Figure 5-Escaping a tissue requires the reorganization of the actin cytoskeleton. (A) Time series showing the three main stages allowing a leader cell to escape from an epithelial tissue. (B) The escape process was reproduced by using micropatterns composed of adhesive microstripes of 12 µm wide connected to a circular disk of 50 µm in diameter. Scale bars are 30 µm and 10 µm in (A) and (B), respectively. (C) Representation of the outline of a cell escaping from a micropatterned stripe for different time points. Evolution of the cell area during the 3 different stages (confined, transition and unconfined) for cells escaping from (D) a tissue (n=8) and (E) a 12 µm wide micropatterned stripe (n=12). Errors bars indicate s.d, **p < 0.01 and ***p < 0.001. The dashed line represents the axis used for kymographs. (F) A typical kymograph shows three successive stages at the front and at the rear of the cell

Figure 6 -

 6 Figure 6-Migration velocity is correlated to cell-substrate adhesive area. (A) Typical fluorescence images of a confined and unconfined keratocytes stained for vinculin. Zooms indicate the presence of focal complexes in the leading edge and mature focal adhesion in the trailing edges. Scale bars are 12 µm. (B) Evolution of the number of focal complexes as a function of the cell area for the confined state (in red), the transition (in blue) and the unconfined state (in green). (C) Mean area per focal adhesions for the three successive stages (n=16 for confined, n=17 for transition, n=16 for unconfined and n.s. is not significant). (D) Normalized adhesion area for the leading (plain bars) and trailing (dashed bars) edges for the three successive stages (n=16 for confined, n=17 for transition and n=16 for unconfined). Error bars indicate s.d. with *p<0.05 and ***p<0.001. (E) Normalization of the cell velocity by the adhesive area as a function of the cell spreading area. (F) Sequence of traction force maps during the three main stages (confined, transition and unconfined) of the escape process. The scale bar is 15 µm. (G) Evolution of the total stress exerted by migrating keratocytes

Acknowledgements

The authors gratefully acknowledge Carl-Philip Heisenberg for kindly providing the GFP-UTR zebrafish line, Andrew Kennard for his technical advices for culture of zebrafish embryos, Mathieu Surin for his technical support with AFM measurements. The authors thank Martial Balland, Beth L. Pruitt, Michael Sixt, Daniel Fletcher, Julie Theriot and Jiandong Ding for insightful discussions. This work was conducted with the financial support from the Belgian National Fund for Scientific Research (F.R.S.-FNRS, Crédit de Recherches -J009916F) and FEDER Prostem. D.M, J.L., L.A., C.B., E.V. and M.L. are financially supported by FRIA (F.R.S.-FNRS). G.C. is supported by an ERC CoG grant (MolCellTissMech, agreement 647186).

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon request.