Dorin Bucur

Antonin Chambolle

AND MICKA Alessandro Giacomini

Ë L Nahon

A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS

Keywords: 1991 Mathematics Subject Classification. 49Q10, 76D07, 76D55, 35R35 Free discontinuity problems, Stokes flow, Navier boundary conditions, drag. A

In this paper we study obstacles immersed in a Stokes flow with Navier boundary conditions. We prove the existence and regularity of an obstacle with minimal drag, among all shapes of prescribed volume and controlled surface area, taking into account that these shapes may naturally develop geometric features of codimension 1. The existence is carried out in the framework of free discontinuity problems and leads to a relaxed solution in the space of special functions of bounded deformation (SBD). In dimension 2, we prove that the solution is classical.

and that the associated velocity field u is equal to a constant vector V ∞ on the walls of the channel.

The obstacle E experiences a force, whose component in direction of V ∞ will be denoted by Drag(E) and is usually called the drag force. If we further assume that the velocity of the fluid satisfies the Stokes equation in Ω \ E and obeys to Navier boundary conditions on ∂E, the expression of the drag force turns out to be given (up to a multiplicative constant) by (1.1) Drag(E) = 2µ

Ω\E |e(u)| 2 dx + β ∂E |u| 2 dH d-1 ,
where e(u) := 1 2 (Du+(Du) * ) denotes the symmetrized gradient of u and β > 0 is the friction coefficient (we refer to Subsection 3.2 for details).

We are interested in minimizing the drag force among all obstacles E with a prescribed volume and controlled surface area. Precisely we look for the existence of such an optimal obstacle and for its qualitative properties. The existence question is not very relevant as soon as one imposes strong geometric constraints on the admissible obstacles (e.g. convexity, uniform cone conditions, etc.) since this may hide some specific features which would naturally occur. Indeed, letting the geometry of the obstacle to be completely free, some qualitative behavior (blocked by rigid geometric constraints) can be observed. This is the case of our problem, where the optimal obstacle (that we prove to exist without imposing any geometric or topological constraint) may be composed, roughly speaking as a union of a body with the prescribed volume and pieces of surfaces of dimension d-1. Those surfaces do not have volume, but count for the total surface area H d-1 (∂E) and of course have a strong influence on the flow.

Penalizing the surface area and the volume, the model problem we are interested in can be written as min

E Drag(E) + cH d-1 (∂E) + f (|E|) ,
where c > 0 and f : (0, |Ω|) → R ∪ {+∞} is a lower semicontinuous function. Roughly speaking, the terms involving perimeter and volume can be thought as a price to pay in order to build the obstacle E, and we can give the two relevant choices of function f : f (m) = +∞1 {m =m 0 } for some m 0 ∈ (0, |Ω|), or f (m) = -λm for some λ > 0.

Many similar optimisation problems have been considered under the "no-slip" boundary condition, meaning flows for which u = 0 at ∂E. Under volume constraint and an a priori symmetry hypothesis around an axis parallel to the flow, the minimal drag question has been studied in [START_REF] Watson | Towards the minimum drag on a body of given volume in slow viscous flow[END_REF] on smooth surfaces. In [START_REF] Pironneau | On optimum profiles in Stokes flow[END_REF], still under symmetry hypotheses, it was conjectured that the optimal profile in three dimensions is a prolate spheroid with sharp ends of angle of 120 degrees. In the same symmetry context, let us also mention the slender body approximation of [START_REF] Tuck | Toward the calculation and minimization of stokes drag on bodies of arbitrary shape[END_REF]. We also refer the reader to the paper by Sverák [START_REF] Šverák | On optimal shape design[END_REF] who, in two dimensions, proves the existence of an optimal obstacle under topological hypotheses, namely that the obstacle has at most a given number of connected components (in particular this number can be equal to 1). The proof is genuinely two dimensional and can not be extended to higher dimensions.

The Navier boundary condition gives many new challenges, namely the possible apparition of lower dimensional structures in the obstacle that minimize the drag, something which was absent under the no-slip condition. The Navier boundary condition may be seen as a partial adherence to the boundary of the obstacle, and it may be asymptotically obtained as a limit of flows with perfect slip on an obstacle with rough boundary. More precisely, a periodic microstructure with the right scaling on the boundary is modelled at the limit by a Navier boundary condition, as was proved in [START_REF] Casado-Díaz | Asymptotic behavior of a viscous fluid with slip boundary conditions on a slightly rough wall[END_REF]. In dimension higher than two it is also necessary to take into account more complex geometries for the microstructure, which at the limit produce an anisotropic factor that favors certain directions of the flow. Moreover, infinitesimal boundary perturbations can dramatically modify the solution of the Stokes equation with Navier boundary conditions, while in presence of no-slip boundary conditions the solution remains stable. We refer the reader to [START_REF] Bucur | Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions[END_REF] for an analysis of those phenomena and for a discussion on the pertinence of the Navier boundary conditions in physical models.

For a fixed obstacle E, the minimization of the drag with respect to the friction parameter β of the Navier conditions (meaning, from a physical point of view, with respect to the microstructure on the boundary) has been studied in [START_REF] Bonnivard | Microshape control, riblets, and drag minimization[END_REF], for both Stokes and Navier-Stokes flows. While for Stokes flows the drag is increasing with the friction parameter, an important observation which occurs for the Navier-Stokes equation is that the monotonicity of the drag with respect to the parameter β does not hold. This is a reason for which the results we give in this paper for the Stokes flows are not expected to hold, as such, for the Navier-Stokes equation.

Since the stationary velocity field associated to a Lipschitz obstacle E turns out to be characterized variationally as the minimizer of the right hand side of (1.1) in the class of admissible velocities

V reg E,V∞ (Ω) = u ∈ H 1 (Ω \ E; R d ) : divu = 0, u |∂E • ν E = 0, u |∂Ω = V ∞
(see (3.4) in Subsection 3.1 for more details), we can conveniently rephrase the minimization problem by letting also the velocity fields intervene explicitely in the form

(1.2) min E,u∈V reg E,V∞ (Ω) 2µ Ω\E |e(u)| 2 dx + β ∂E |u| 2 dH d-1 + cH d-1 (∂E) + f (|E|) .
The first main goal of the paper is to find suitable relaxations of problem (1.2) for which we can prove the existence of minimizers without any a priori constraint on the regularity or the topology of the sets E.

In order to avoid unnatural geometric restrictions on the obstacle E, it is natural in view of the third term appearing in (1.2) to let it vary within the class of sets of finite perimeter (see Subsection 2.2), and replace the topological boundary with reduced one ∂ * E.

In order to describe properly obstacles with very narrow spikes which in the limit degenerate to (d-1)-surfaces and that cannot be taken into account through the reduced boundary, it is convenient to consider admissible velocity fields which can be discontinuous outside E (see Subsection 3.3). Since the symmetrized gradient e(u) is involved explicitly in (1.2), a natural family for the admissible velocities is given by the space of functions of bounded deformation SBD. The natural relaxation of the energy takes the form (see Remark 4.11 for further comments)

J (E, u) :=2µ Ω\E |e(u)| 2 dx + β ∂ * E |u + | 2 dH d-1 + β Ju\∂ * E [|u + | 2 + |u -| 2 ] dH d-1 + cH d-1 (∂ * E) + 2cH d-1 (J u \ ∂ * E) + f (|E|), (1.3)
where u is set equal to zero a.e. in E, while J u denotes the discontinuity set of u and u ± are the traces of u on ∂ * E and J u (the trace u -vanishes on ∂ * E by the choice of orientation, while u + is on the outward side).

Within this framework the global obstacle is given by E∪J u , so that it contains also lower dimensional parts, namely J u \ ∂ * E: roughly speaking, for the optimal velocity these discontinuous regions generate (d -1)-surfaces which correspond to volumeless, lower dimensional subsets of the optimal obstacle.

Admissible velocities must be tangent to the obstacles, meaning that not only u is tangent to ∂ * E, but also the two traces u ± are orthogonal to the normal ν u along the jump set J u . The contribution of the Navier surface term takes naturally into account the contribution from both sides given by u ± . Concerning the perimeter term, we count twice the lower dimensional parts because we see the relaxed obstacle as a limit of regular obstacles, such that points of J u \ ∂ * E correspond to thin parts of the regular obstacle that collapse on a lower-dimensional structure. We could also see the perimeter term as a price to pay in order to construct the obstacle and just keep H d-1 (∂ * E ∪ J u ) instead, and the main results of the paper would not be affected.

The relaxed optimization problem can be seen as a minimization problem on the pairs (E, u) which has the features of classical geometrical problems for E coupled with a free discontinuity problem for u, with a surface term depending on the traces which are subject to suitable tangency constraints and boundary conditions.

The first main results of the paper (Theorem 4.8) concerns the existence of minimizers for the relaxed functional J in (1.3) among the class of admissible configurations (see Definition 4.1 for the precise definition).

The main difficulties we have to face in order to prove that the problem is well posed are the following:

(a) the closure of the non-penetration constraint for the velocity on ∂ * E ∪ J u under the natural weak convergence of the problem; (b) the lower semicontinuity of energies of the form

(1.4) Ju [|u + | 2 + |u -| 2 ] dH d-1
associated to the Navier conditions. Point (a) is a consequence of a lower semicontinuity result for the energy

Ju |u + • ν u | + |u -• ν u | dH d-1
which is proved in Theorem 5.2, by resorting to recent lower semicontinuity results for functionals on SBD by Friedrich, Perugini and Solombrino [START_REF] Friedrich | Lower semicontinuity for functionals defined on piecewise rigid functions and on GSBD[END_REF].

The energy of point (b) naturally appears in a scalar setting when dealing with shape optimization problems involving Robin boundary conditions (see e.g. [START_REF] Bucur | A variational approach to the isoperimetric inequality for the Robin eigenvalue problem[END_REF][START_REF] Bucur | Shape optimization problems with Robin conditions on the free boundary[END_REF][START_REF] Bucur | Faber-Krahn inequalities for the Robin-Laplacian: a free discontinuity approach[END_REF][START_REF] Caffarelli | A free boundary problem related to thermal insulation[END_REF]), and it is easily seen to enjoy lower semicontinuity properties by working with sections. The lower semicontinuity result in the vectorial SBD setting is given by Theorem 5.4 and cannot rely on an easy argument by sections, which instead would yield the lower semicontinuity of an energy of the form

Ju |u + • ξ| 2 + |u -• ξ| 2 |ξ • ν u | dH d-1
with ξ ∈ R d with |ξ| = 1: the optimization in ξ in order to recover (1.4) does not seem feasible in dimension d ≥ 3. We thus follow a different strategy based on a blow up argument in which we reconstruct the vector quantities u ± by controlling them along a sufficiently high number of directions (see Subsection 5.3 for details): in this way we can deal with more general energy densities of the form φ(u + ) + φ(u -), where φ is a lower semicontinuous function.

The second main result of the paper (see Theorem 4.10) concerns the regularity of the relaxed minimizers of (1.3). Provided that the volume penalization function f is Lipschitz and that we are in two dimensions, we prove that for a minimizer (E, u) of J , the optimal obstacle E ∪ J u is a closed set, while the optimal velocity u is a smooth Sobolev function outside the obstacle, recovering somehow the classical setting of the problem. More precisely we show that

(1.5) H 1 (Ω ∩ ∂ * E ∪ J u \ (∂ * E ∪ J u )) = 0,
so that the optimal obstacle can be described as the closed set obtained by the complement of the connected components of Ω \ ∂ * E ∪ J u on which u does not vanish identically.

The technical ideas to prove (1.5) stem from the pioneering result of De Giorgi, Carriero and Leaci on the Mumford-Shah problem [START_REF] Mumford | Optimal approximations by piecewise smooth functions and associated variational problems[END_REF][START_REF] De Giorgi | Existence theorem for a minimum problem with free discontinuity set[END_REF], where the key of the proof is a decay estimate obtained by a contradiction/compactness argument. For vectorial problems, a similar strategy, but definitely more involved, was used for the Griffith fracture problem in [START_REF] Conti | Existence of strong minimizers for the Griffith static fracture model in dimension two[END_REF] (for the two-dimensional case) and in [START_REF] Chambolle | Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy[END_REF] (for higher dimension). In the fracture problem, the key compactness result relies on the possibility to approximate a field u ∈ SBD([-1, 1] d ) with a small jump set by a Sobolev function which is locally controlled in H 1 (via the classical Korn inequality).

In our case, we follow a similar approximation procedure, but we have to handle two additional constraints: incompressibility and non-penetration at the jumps. From a technical point of view, this is problematic since the bound in [START_REF] Conti | Existence of strong minimizers for the Griffith static fracture model in dimension two[END_REF] in not strong enough to stay in divergence-free vector fields and the method in [START_REF] Chambolle | Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy[END_REF] creates new jumps on which the non-penetration constraint is not a priori verified. However, when restricted to two dimensions, the method of [START_REF] Chambolle | Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy[END_REF] leads to a stronger result, so that both constraints can be handled.

The paper is organized as follows. In Section 2 we fix the notation and recall some basic facts concerning sets of finite perimeter, functions of bounded deformation and Hausdorff convergence of compact sets. Section 3 is devoted to the precise exposition of the drag optimization problem. In Section 4 we detail the relaxation of the problem in the family of obstacle of finite perimeter and with velocities of bounded deformation, and formulate the main results of the paper concerning the existence of minimizers (in any dimension) and their regularity in dimension two. The proof of the existence of minimizers is given in Section 6, and it is based on some technical results for SBD functions collected in Section 5, while the regularity result is proved in Section 7.

Notations and Preliminaries

2.1. Basic notation. If E ⊆ R d , we will denote with |E| its d-dimensional Lebesgue measure, and by H d-1 (E) its (d -1)-dimensional Hausdorff measure: we refer to [START_REF] Evans | Measure theory and fine properties of functions[END_REF]Chapter 2] for a precise definition, recalling that for sufficiently regular sets H d-1 coincides with the usual area measure. Moreover, we denote by E c the complementary set of E, and by 1 E its characteristic function, i.e., 1 E (x) = 1 if

x ∈ E, 1 E (x) = 0 otherwise. In addition we will say that E 1 E 2 if E 1 ⊂ E 2 . Finally we will denote with Q x,r ⊆ R d the cube of center x and side r: when x = 0, we will simply write Q r .

If A ⊆ R d is open and 1 ≤ p ≤ +∞, we denote by L p (A) the usual space of p-summable functions on A with norm indicated by • p . W 1,p (A) will stand for the Sobolev space of functions in L p (A) whose gradient in the sense of distributions belongs to L p (A; R d ). Finally, given a finite dimensional unitary space Y , we will denote by M b (A; Y ) the space of Y -valued Radon measures on A, which can be identified with the dual of Y -valued continuous functions on A vanishing at the boundary.

We will denote by M d×m the set of d × m matrices with values in R: when d = m we will denote by M d×d sym the subspace of d × d symmetric matrices. For a ∈ R d and b ∈ R m we will denote with a ⊗ b the element of M d×m such that

(a ⊗ b) ij = a i b j , while if a, b ∈ R d we denote with a b the matrix in M d×d sym such that (a b) ij = a i b j + a j b i 2 .
Given ξ ∈ R d with |ξ| = 1, we denote with ξ ⊥ the hyperplane through the origin orthogonal to ξ.

If E ⊆ R d , we set (2.1) E ξ := π ξ ⊥ (E),
where π denotes the orthogonal projection, and for y ∈ ξ ⊥ we set

(2.2) E ξ y := {t ∈ R : y + tξ ∈ E}.
2.2. Functions of bounded variation and sets of finite perimeter.

If Ω ⊆ R d is open, we say that u ∈ BV (Ω; R m ) if u ∈ L 1 (Ω; R m
) and its derivative in the sense of distributions is a finite Radon measure on Ω, i.e., Du ∈ M b (Ω; M d×m ). BV (Ω; R m ) is called the space of functions of bounded variation on Ω with values in R m and it is a Banach space under the norm u BV (Ω;R m ) := u L 1 (Ω;R m ) + Du M b (Ω;M d×m ) . We call |Du|(Ω) := Du M b (Ω;M d×m ) the total variation of u. We refer the reader to [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] for an exhaustive treatment of the space BV .

We say that u ∈ SBV (Ω; R m ) if u ∈ BV (Ω; R m ) and its distributional derivative can be written in the form

Du = ∇u dx + (u + -u -) ⊗ ν u H d-1 J u ,
where ∇u ∈ L 1 (Ω; M d×m ) denotes the approximate gradient of u, J u denotes the set of approximate jumps of u, u + and u -are the traces of u on J u , and ν u (x) is the normal to J u at x. Note that if u ∈ SBV (Ω; R m ), then the singular part of Du is concentrated on J u which is a countably H d-1 -rectifiable set: there exists a set E with H d-1 (E) = 0 and a sequence (

M i ) i∈N of C 1 -submanifolds of R d such that J u ⊆ E ∪ i∈N M i .
We will say that

E ⊆ R d with |E| < +∞ has finite perimeter if 1 E ∈ BV (R d ).
The perimeter of E is defined as

P er(E) = |D1 E |(R d ).
It turns out that

D1 E = ν E H d-1 ∂ * E, P er(E) = H d-1 (∂ * E),
where in the sense of distributions is a finite Radon measure on Ω, i.e., Eu ∈ M b (Ω; M d×d sym ). BD(Ω) is called the space of functions of bounded deformation on Ω. We refer the reader to [START_REF] Temam | Functions of bounded deformation[END_REF][START_REF] Temam | Variational problems of plasticity[END_REF] for the main properties of the space BD.

∂
We will make use of a subspace of BD(Ω) called the space of special functions of bounded deformation introduced in [START_REF] Ambrosio | Fine properties of functions with bounded deformation[END_REF]. We say that u ∈ SBD(Ω) if u ∈ BD(Ω) and its symmetrized distributional derivative can be written in the form

Eu = e(u) dx + (u + -u -) ν u H d-1 J u ,
where e(u) ∈ L 1 (Ω; M d×d sym ) denotes the approximate symmetrized gradient of u, J u denotes the set of approximate jumps of u, u + and u -are the traces of u on J u , and ν u (x) is the normal to J u at x. As in the case of functions of bounded variation, J u is a H d-1 -countably rectifiable set.

We will use the following compactness and lower semicontinuity result proved in [START_REF] Bellettini | Compactness and lower semicontinuity properties in SBD(Ω)[END_REF].

J ξ u := {x ∈ J u : (u + (x) -u -(x)) • ξ = 0}
The following result holds true (see [START_REF] Ambrosio | Fine properties of functions with bounded deformation[END_REF]).

Obstacles in Stokes fluids and drag minimization

In this section we explain the drag problem for an obstacle immersed in a stationary flow. where µ > 0 is a viscosity parameter, e(u) the symmetrized gradient of u (also denoted by D(u)) and p is the pressure, we require

(3.2) div σ = 0 in Ω \ E.
(d) Navier conditions on the obstacle: we have

(σν) τ = βu on ∂E,
where β > 0 is a friction parameter, and (σν) τ denotes the tangential component of force σν.

The stationary flow has the following variational characterization: u is the minimizer of the energy

(3.3) E(u) := 2µ Ω\E |e(u)| 2 dx + β ∂E |u| 2 dH d-1
among the class of (sufficiently regular) admissible fields

(3.4) V reg E,V (Ω) := {v ∈ H 1 (Ω \ E; R d ) :
v satisfies points (a) and (b)}, where H d-1 stands for the (d -1)-dimensional Hausdorff measures, which reduces to the area measure on sufficiently regular sets. Indeed if u is a minimizer, and ϕ is an admissible variation, so that ϕ = 0 on ∂Ω, we get

0 = 2µ Ω\E e(u) : e(ϕ) dx + β ∂E u • ϕ dH d-1 = 2µ Ω\E e(u) : ∇ϕ dx + β ∂E u • ϕ dH d-1 = -2µ Ω\E div e(u) • ϕ dx + ∂E [-2µe(u)ν + βu] • ϕ dH d-1
In particular, choosing ϕ with compact support in Ω \ E we have 2µdiv e(u) = ∇p for some pressure field p: as a consequence σ := -pI d + 2µe(u) satisfies (3.2) of condition (c).

Since the admissible functions ϕ are tangent to ∂E, the optimality condition reduces to

(3.5) 0 = ∂E [-2µe(u)ν + βu] • ϕ dH d-1 = ∂E [-σν + βu] • ϕ dH d-1 .
Notice that every tangential vector field η on ∂E can be extended to a divergence free vector field on Ω \ E which vanishes on ∂Ω, hence it is the trace of an admissible variation ϕ: indeed any extension W which vanishes on ∂Ω has a divergence with zero mean, so that considering W 1 with div W 1 = div W with W 1 = 0 on ∂Ω and on ∂E (whose existence is guaranteed, for example by [6, Theorem IV.3.1])), the required extension is given by W -W 1 . We conclude that the optimality condition (3.5) yields the Navier condition of point (b).

3.2. The drag force. Assume now that the external vector field V is equal to a constant V ∞ ∈ R d \{0}, i.e. the obstacle E is immersed in a uniform flow. The flow is perturbed near E, assuming the value u, and the obstacle experiences a force which has a component in the direction V ∞ which is given by

Drag(E) := ∂E σν • V ∞ |V ∞ | dH d-1 ,
which is called the drag force on E in the direction of the flow. We claim that

(3.6) Drag(E) = 1 |V ∞ | E(u),
where E(u) is the energy defined in (3.3). Using the facts that σ is symmetric and with zero divergence (so that also the vector field σV ∞ is divergence free), and that u = V ∞ on ∂Ω, we may write

∂E σν • V ∞ dH d-1 = ∂E σV ∞ • ν dH d-1 = ∂Ω σV ∞ • ν dH d-1 = ∂Ω σu • ν dH d-1 = Ω\E div (σu) dx + ∂E σu • ν dH d-1 = Ω\E σ : ∇u dx + ∂E σν • u dH d-1 . (3.7)
Using again that σ is symmetric and that u is divergence free, together with the constitutive equation (3.1), we have

Ω\E σ : ∇u dx = Ω\E σ : e(u) dx = Ω\E (-p I d + 2µe(u)) : e(u) dx = Ω\E (-p div u + 2µ|e(u| 2 ) dx = 2µ Ω\E |e(u)| 2 dx,
while in view of the Navier conditions on ∂E and the fact that u is tangent to the obstacle

∂E σν • u dH d-1 = ∂E (σν) τ • u dH d-1 = β ∂E |u| 2 dH d-1 .
Inserting into (3.7), we get that (3.6) follows.

3.3. The optimization problem. Let c > 0 and let f : (0, |Ω|) → R ∪ {+∞} be a lower semicontinuous functions that is not identically equal to +∞. We are concerned with the following optimization problem:

min E Drag(E) + cH d-1 (∂E) + f (|E|) .
We are thus interested in finding the optimal shape of an obstacle which minimizes the drag force, under a penalization involving its perimeter and its volume.

In view of the energetic characterization of the drag force established in Subsection 3.2, we can formulate the problem as a minimization problem among the pairs (E, u), where u is a velocity field belonging to the family V reg E,V∞ (Ω) defined in (3.4):

min E,u∈V reg E,V∞ (Ω) 2µ |V ∞ | Ω\E |e(u)| 2 dx + β |V ∞ | ∂E |u| 2 dH d-1 + cH d-1 (∂E) + f (|E|) .
Setting all the constants equal to 1, and replacing V ∞ by a given divergence free velocity vector field V as in Subsection 3.1, the drag minimization problem above is a particular case of the following shape optimization problem (3.8) min

E,u∈V reg E,V (Ω) Ω\E |e(u)| 2 dx + ∂E |u| 2 dH d-1 + H d-1 (∂E) + f (|E|) .
If we want to apply the direct method of the calculus of variations to the problem, i.e., if we want to recover a minimizer by looking at minimizing sequences (E n , u n ) n∈N , the following considerations are quite natural.

(a) Since the problem involves the perimeter of E, the sequence (E n ) n∈N is relatively compact in the family of sets of finite perimeter (see Section 2). (b) Concerning the velocities, it turns out naturally that it is convenient to consider also discontinuous vector fields. Indeed if u n → u in some sense, and ∂E n collapses in some parts generating a surface Γ outside the limit set E, the limit velocity field u can present, in general, discontinuities across Γ.

E n E

Γ

We thus expect an extra term in the surface integral related to the Navier conditions, which amounts at least to

Γ\∂E [|u + | 2 + |u -| 2 ] dH d-1 ,
where u ± are the two traces from both sides of Γ. The previous considerations yield to formulate a relaxed version of problem (3.8) in which E varies among the family of sets of finite perimeter contained in Ω, while the family of associated admissible velocity fields u is naturally contained in the space of special functions of bounded deformation SBD(Ω) (see Section 2).

In Section 4, we will give a precise formulation of problem in this weak setting, which guarantees existence of optimal solutions, describing in particular how the boundary conditions on ∂Ω and on the obstacle have to be rephrased in this context.

A relaxed formulation of the shape optimization problem and statements of the main results

Let Ω ⊆ R d be open, bounded and with a Lipschitz boundary, and let V ∈ C 1 (R d ; R d ) be a divergence free vector field. In order to deal conveniently with the boundary conditions, let us consider Ω ⊆ R d open and bounded such that Ω Ω .

The following definition deals with the family of admissible configurations in the relaxed setting.

Definition 4.1 (The class A(V ) of admissible obstacle-velocity configurations). We say that (E, u) is an admissible configuration for the external velocity V , and we will write

(E, u) ∈ A(V ), if E ⊆ Ω is a set of finite perimeter, while u ∈ SBD(Ω ) ∩ L 2 (Ω ; R d )
is such that u = 0 a.e. on E and the following conditions are satisfied.

u ± • ν = 0 on ∂ * E ∪ J u ,
where ν denotes the normal to the rectifiable set ∂ * E ∪ J u .

Remark 4.2. The crucial difference between admissible velocities in the present framework and those of the family V reg E,V (Ω) introduced before (see (3.4)) is that they may have discontinuities outside of E. Within the new setting, the global obstacle is given by E ∪ J u i.e. it may contain (d -1) dimensional parts.

Given (E, u) ∈ A(V ), concerning the traces of u on ∂ * E, we will denote with u + the trace in the direction of the external normal ν E , so that u -= 0 H d-1 -a.e. on ∂ * E.

Concerning the non-penetration constraint, notice that it suffices to require it only on J u , since it is then automatically verified also on ∂ * E. Indeed for H d-1 -a.e. x ∈ ∂ * E \J u , we have u -(x) = u + (x) = 0 and the constraint is verified, while for H d-1 -a.e. x ∈ J u ∩ ∂ * E the two rectifiable sets J u and ∂ * E share the same normal vector.

Remark 4.3. The space SBD(Ω ) is naturally a subspace of L 1 (Ω ; R d ): we require for admissibility that u ∈ L 2 (Ω ; R d ) to ensure that the velocity field has finite kinetic energy. It will turn out that velocities in SBD(Ω ) which are interesting for our problem (i.e., with finite energy) are automatically elements of L 2 (Ω ; R d ) (see Theorem 5.1).

Remark 4.4 (On the boundary condition).

If (E, u) ∈ A(V ), then u ∈ SBD(Ω ) with u = V a.e. on Ω \ Ω, so that J u ∩ ∂Ω = {x ∈ ∂Ω : γ(u)(x) = V (x)}
, where γ(u) is the trace of u on ∂Ω coming from Ω (i.e., the usual trace of u seen as an element of SBD(Ω)). We conclude that within the present framework, the boundary condition is somehow relaxed: a possible mismatch between u and V on ∂Ω is admitted, but then the zone is counted as a jump part of the velocity field, and consequently as a part of the obstacle ∂ * E ∪ J u , and will carry a contribution for the energy (see (4.2) below). Such a relaxation of the boundary condition is a feature which is common to several applications of functions of bounded variation to problems in continuum mechanics (see for example [START_REF] Francfort | Existence and convergence for quasi-static evolution in brittle fracture[END_REF][START_REF] Dal Maso | Quasistatic crack growth in nonlinear elasticity[END_REF] in connection to fracture mechanics or [START_REF] Dal Maso | Quasistatic evolution problems for linearly elasticperfectly plastic materials[END_REF] for problems in plasticity).

Remark 4.5. Given (E, u) ∈ A(V ), the obstacle E ∪ J u may touch ∂Ω only on those part where V is tangent to Ω: this is due to the fact that on (∂ * E ∪ J u ) ∩ ∂Ω, the two sets share H d-1 -a.e. the same normal, and u + = V (if the orientation is suitably chosen).

W ∈ H 1 (Ω\E; R d ) such that W = V on ∂Ω, W = 0 on ∂E and div W = 0. Indeed if ϕ ∈ C ∞ (R d
) is such that ϕ = 1 on a neighborhood of R d \ Ω and ϕ = 0 on a neighborhood of E, we can consider the vector field V 1 := ϕV , whose divergence has zero mean on Ω \ E (by Gauss theorem). Then we can find

V 2 ∈ H 1 0 (Ω \ E; R d ) such that div V = div V 1 (see [6, Theorem IV.3.1]), so that the field W := V 1 -V 2 is an admissible choice.
In particular we get that (E, W ) ∈ A(V ), so that the class of admissible configurations is not empty.

Let (4.1)

f : [0, |Ω|] → R ∪ {+∞} be lower semicontinuous, not identically equal to +∞.

For every (E, u) ∈ A(V ), let us set (normalizing to 1 the constants involved in the drag force problem)

J (E, u) := Ω |e(u)| 2 dx + ∂ * E |u + | 2 dH d-1 + Ju\∂ * E [|u + | 2 + |u -| 2 ] dH d-1 + H d-1 (∂ * E) + 2H d-1 (J u \ ∂ * E) + f (|E|). (4.2)
Remark 4.7. Concerning the volume integral in J (E, u), the density e(u) is equal to e(V ) a.e. on Ω \ Ω and equal to 0 a.e. on E: as a consequence we could replace it with an integral on Ω \ E without affecting the minimization of J . Concerning the Navier energy and the surface penalization for ∂ * E ∪J u , notice that it counts also for the possible mismatch at the boundary between u and V as pointed out in Remark 4.4: the mismatch is thus "penalized" by the energy of the problem.

The previous observations show that the larger domain Ω plays only an instrumental role for the problem, as it can be replaced by any open domain strictly containing Ω.

The first main result of the paper is the following Theorem 4.8 (Existence of optimal obstacles). Let Ω ⊆ R d be a bounded open set with Lipschitz boundary, V ∈ C 1 (R d ; R d ) a divergence-free vector field, and f a function satisfying (4.1). Let the family of admissible configurations A(V ) be given by Definition 4.1 and let J be the functional defined in (4.2). Then the problem

(4.3) min (E,u)∈A(V )
J (E, u) admits a solution.

Remark 4.9. We recover the original drag minimization problem when V is a constant nonzero vector V ∞ , and we restore properly in the functional the physical constants µ and β, together with the perimeter penalization constant c.

The second main result of the paper concerns the regularity of minimizers in the two dimensional setting. Let (E, u) ∈ A(V ) be a solution to (4.3) according to Theorem 4.8. Then

H 1 Ω ∩ (J u ∪ ∂ * E \ (J u ∪ ∂ * E)) = 0, and u ∈ C ∞ (Ω \ J u ∪ ∂ * E; R 2 ).
Theorem 4.8 will be proved in Section 6, on the basis of some technical results established in 5. The proof of Theorem 4.10 will be addressed in Section 7.

Remark 4.11. In order to prove that the functional J (E, u) is the relaxation of the energy appearing in the original problem 3.8 in the sense of the lower semicontinuous envelope of the Calculus of Variations, we need to approximate in energy any (E, u) ∈ A(V ) through "regular" configurations (E n , u n ) ∈ A(V ), where E n has Lipschitz boundary and u n ∈ H 1 (Ω \ E n ; R d ). This reminds the situation studied in [START_REF] Braides | A relaxation result for energies defined on pairs setfunction and applications[END_REF], which can be extended to the case of energies involving only the symmetrized gradient like in the study of material voids in linearly elastic materials (in this direction, see for example [START_REF] Crismale | Equilibrium configurations for epitaxially strained films and material voids in three-dimensional linear elasticity[END_REF]). However the constraints of the our problem make the analysis very hard to be carried out: more specifically, admissibility requires the divergence free condition div u n = 0 and the tangency constraint u n ⊥ ∂E n , and it is not clear how to enforce them within the by now available approximation procedures.

Some technical results in SBD

In this section we collect some technical properties concerning the space SBD that will be fundamental in the proof of Theorem 4.8. In particular in Theorem 5.1 we will prove that admissible velocity vector fields enjoy higher summability properties (indeed they belong to L 2d d-1 ). In Theorem 5.3 we will prove that velocity fields u with u ± tangent to the discontinuity set J u form a closed set under the natural convergence of minimizing sequences for the main optimization problem. Finally in Theorem 5.4 we will prove a lower semicontinuity result for surface energies depending on the traces, which entails in particular the lower semicontinuity of the term associated to the Navier conditions.

E(u) := Ω |e(u)| 2 dx + Ju |u + | 2 + |u -| 2 dH d-1 < +∞. Then u ∈ L 2d d-1 (Ω) with u 2d d-1 ≤ C E(u),
where C depends on d and diam(Ω) only.

Proof. It suffices to follow the strategy of the proof of the classical embedding of BD into L d/d-1 explained in [START_REF] Temam | Variational problems of plasticity[END_REF], but concentrating on the square of the components. Let us consider the unit vector

ξ := 1 √ d (1, 1, . . . , 1) ∈ R d .
Employing the characterization by sections recalled in Section 2, for H d-1 -a.e. y ∈ ξ ⊥ we have

ûξ y ∈ SBV (Ω ξ y ) with Ω ξ y |(û ξ y ) | 2 dt + t∈J ûξ y |(û ξ y ) + (t)| 2 + |(û ξ y ) -(t)| 2 < +∞.
Then we can write for a.e.

t ∈ R ûξ y 2 L ∞ (Ω ξ y ) ≤ D(û ξ y ) 2 (Ω ξ y ) = Ω ξ y 2|û ξ y (û ξ y ) |dt + t∈J ûξ y |(û ξ y ) + (t)| 2 -|(û ξ y ) -(t)| 2 ≤ 1 2 ûξ y 2 L ∞ (Ω ξ y ) + 2|Ω ξ y | Ω ξ y (û ξ y ) 2 dt + t∈J ûξ y (û ξ y ) + (t) 2 + (û ξ y ) -(t) 2 , (5.1)
Let us set

g ξ (x) := Ω ξ y |(û ξ y ) | 2 dt + t∈J ûξ y |(û ξ y ) + (t)| 2 + |(û ξ y ) -(t)| 2 ,
where y := π ξ ⊥ (x), i.e., the projection of x on the hyperplane ξ ⊥ . g ξ (x) only depends on the projection of x on ξ ⊥ and

ξ ⊥ g ξ dH d-1 = Ω |e(u)ξ • ξ| 2 dx + Ju |u + | 2 + |u -| 2 |ξ • ν| dH d-1 ≤ C Ω |e(u)| 2 dx + Ju |u + | 2 + |u -| 2 dH d-1
where C depends only on d. Thanks to (5.1) we have

(5.2) |ξ • u| 2 ≤ Cg ξ a.e. on Ω,
where C depends on the diameter of Ω, and from now on all the constants C that appear depend on n, diam(Ω). For every k = 1, . . . , d -1, we can write

ξ = 1 √ d e k + d -1 d h k ,
where e k is the k-th vector of the canonical base, and h k is the unit vector in the direction √ dξ -e k . Reasoning as above on the decomposition

ξ • u = d -1 d h k • u + 1 √ d e k • u
we obtain a similar estimate

(5.3) |ξ • u| 2 ≤ C (g h k + g e k ) ,
Multiplying inequality (5.2) with inequalities (5.3) for k = 1, . . . , d -1, we obtain reasoning as in [32, Chapter II, Theorem 1.2]

(ξ • u) 2 d d-1 ≤ C Ω |e(u)| 2 dx + Ju |u + | 2 + |u -| 2 dH d-1 .
Since this estimate does not depend on the particular choice of the basis and hence holds for any ξ with norm one, the theorem is proved. 5.2. Closure of the non-penetration constraint. In the context of equi-Lipschitz boundaries, the preservation of the non-penetration property for a sequence of Sobolev functions converging weakly, comes rather directly via the divergence theorem (we refer the reader, for instance, to [START_REF] Bucur | Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions[END_REF]). However, in the case of collapsing boundaries, so that the limit function lives on both sides of a surface and in absence of any smoothness of the limit set, this technique does not work. The proof of the nonpenetration preservation requires different technical arguments that we handle in the SBD context.

Let us start with the following lower semicontinuity result.

Theorem 5.2. Let Ω ⊆ R d be a bounded open set, and let (u n ) n∈N be a sequence in SBD(Ω) such that

sup n Ω |e(u n )| 2 dx + H d-1 (J un ) < +∞ with u n → u in measure
for some u ∈ SBD(Ω). Then

Ju |u + • ν u | + |u -• ν u | dH d-1 ≤ lim inf n→+∞ Ju n |u + n • ν un | + |u -• ν un | dH d-1 .
Proof. Let us consider a countable set of functions {ϕ h : h ∈ N} which is dense with respect to • ∞ inside the set

f ∈ C 0 c (]0, +∞[) : +∞ 0 f dt = 0 and f ∞ ≤ 1 .
Given ε > 0, let us consider

g h,k (x) := 1 2 |x-x k | 2 0 ϕ h (t) dt,
where

{x k : k ∈ N} is a countable and dense set in B ε (0) ⊂ R d with x 0 = 0. Clearly g h,k ∈ C 1 c (R d ) with G h,k (x) := ∇g h,k (x) = ϕ h 1 2 |x -x k | 2 (x -x k ).
We have that G h,k is a continuous conservative vector field with compact support on R d . Let us set for (i,

j) ∈ R d × R d and ν ∈ R d with |ν| = 1 f ε (i, j, ν) := sup h,k (G h,k (i) -G h,k (j)) • ν.
By construction f ε is a symmetric jointly convex function according to [28, Definition 3.1]. We claim that for i = j

(5.4) |i • ν| + |j • ν| ≤ f ε (i, j, ν) ≤ |i • ν| + |j • ν| + 2ε.
In view of the lower semicontinuity result [28, Theorem 5.1] we have lim inf

n→+∞ Ju n f ε (u + n , u - n , ν un ) dH d-1 ≥ Ju f ε (u + , u -, ν u ) dH d-1 .
We can thus write lim inf

n→+∞ Ju n |u + n • ν un | + |u - n • ν un | dH d-1 + 2εH d-1 (J un ) ≥ lim inf n→+∞ Ju n f ε (u + n , u - n , ν un ) dH d-1 ≥ Ju f ε (u + , u -, ν u ) dH d-1 ≥ Ju |u + • ν u | + |u -• ν u | dH d-1 ,
so that the result follows taking into account the bound on H d-1 (J un ) and letting ε → 0.

In order to complete the proof, we need to show claim (5.4). The estimate from above follows from

[G h,k (i) -G h,k (j)] • ν ≤ |(i -x k ) • ν| + |(j -x k ) • ν| ≤ |i • ν| + |j • ν| + 2ε since ϕ h ∞ ≤ 1 and |x k | < ε.
Let us prove the estimate from below. We select x kn → 0 such that |i -x kn | = |j -x kn | (which is always possibile in view of the density of {x k : k ∈ N} inside B ε (0) and since i = j) and then ϕ hn such that for n → +∞

ϕ hn 1 2 |i -x kn | 2 → i • ν |i • ν| + η and ϕ hn 1 2 |j -x kn | 2 → - j • ν |j • ν| + η ,
where η > 0. By definition of f ε we infer that

f ε (i, j, ν) ≥ |i • ν| + |j • ν| -2η,
so that the estimate from below follows by sending η → 0.

We are now in a position to prove the main result of the section.

Theorem 5.3 (Closure of the non-penetration constraint on the jump set). Let Ω ⊆ R d be a bounded open set, and let (u n ) n∈N be a sequence in SBD(Ω) such that

sup n Ω |e(u n )| 2 dx + H d-1 (J un ) < +∞ and u n → u in measure for some u ∈ SBD(Ω). If u ± n • ν un = 0 H d-1 -a.e. on J un , then u ± • ν u = 0 H d-1 -a.e. on J u .
Proof. By Theorem 5.2 we may write

Ju |u + • ν u | + |u -• ν u | dH d-1 ≤ lim inf n→+∞ Ju n |u + n • ν un | + |u -• ν un | dH d-1 = 0,
so that the result follows.

5.3.

A lower semicontinuity result for surface energies in SBD. In this section we deal with the lower semicontinuity of the surface term of the functional J in (4.2) connected with the Navier conditions on the obstacle. The following lower semicontinuity result holds true.

Theorem 5.4. Let Ω ⊆ R d be an open set, u n , u ∈ SBD(Ω) such that

u n → u strongly in L 1 (Ω; R d ) and sup n Ω |e(u n )| 2 dx + H d-1 (J un ) < +∞.
Then if φ : R d → [0, +∞] is a lower semicontinuous function, we have

Ju [φ(u + ) + φ(u -)] dH d-1 ≤ lim inf n→+∞ Ju n [φ(u + n ) + φ(u - n )] dH d-1 .
This applies in particular to φ(u) = |u| 2 and φ(u) = 1 {u =0} , which will be of interest to us.

Proof. Notice first that φ may be supposed to be continuous. Indeed for any lower-semicontinuous nonnegative φ, by considering a sequence of continuous nonnegative functions φ k φ we get

Ju [φ(u + ) + φ(u -)] dH d-1 = lim inf k→∞ Ju [φ k (u + ) + φ k (u -)] dH d-1 ≤ lim inf k→∞ lim inf n→+∞ Ju n [φ k (u + n ) + φ k (u - n )] dH d-1 ≤ lim inf n→+∞ Ju n [φ(u + n ) + φ(u - n )] dH d-1
Through a by now standard blow-up argument ( see Remark 5.6), we can reduce the problem to the following lower semicontinuity result. Let Q 1 ⊆ R d be the unit square centred at 0, and let us set

H := Q 1 ∩ {x d = 0} and Q ± 1 := Q 1 ∩ {x d ≷ 0}. Given u ± ∈ R d with u + = u -and u n ∈ SBD(Q 1 ) with (5.5) u n → u := u + 1 Q + 1 + u -1 Q - 1 strongly in L 1 (Q 1 ; R d ), (5.6) sup n H d-1 (J un ) < +∞ and (5.7) e(u n ) → 0 strongly in L 1 (Q 1 ; M d×d sym ), then (5.8) φ(u + ) + φ(u -) ≤ lim inf n→+∞ Ju n [φ(u + n ) + φ(u - n )] dH d-1 .
We now divide the proof in several steps, and we will employ the characterization by sections of SBD functions explained in Section 2.

Step 1. Let ε > 0 be given. We fix δ > 0 and N ∈ N with N > d: these numbers will be subject to several constraints that will appear during the proof. Let us fix N unit vectors {ξ i } 1≤i≤N such that (5.9)

|e d • ξ i -1| < δ
and such that any subset of d of them forms a basis of R d . Moreover, we may assume in addition that (5.10)

(u + -u -) • ξ i = 0
for every i = 1, . . . , N . Thanks to (5.5) and (5.6), we can fix a > 0 small such that setting

H ± := H × {±a} = H ± ae d , we have (u n ) |H ± → u ± strongly in L 1 (H ± ; R d ) and (5.11) ∀n ∈ N : H d-1 (J un ∩ H ± ) = 0.
Step 2. We claim that, up to a subsequence, we can find H - ε ⊂ H -with (5.12)

H d-1 (H -\ H - ε )
< ε such that for every i = 1, . . . , N , for every y ∈ H - ε and for every n ∈ N (5.13)

H - ε ∩ J un = ∅, and (5.17) (u n ) |H -→ u - uniformly on H - ε . Indeed, if the number δ appearing in (5.9) is small enough, we can find A - ε ⊆ H -with (5.18)

(5.14) H 0 ((J un ) ξ i y ) < +∞, H 0 ((J un ) ξ i y ∩R + ) ≥ 1 . Moreover setting (u n ) ξ i y := u n (y + tξ i ) • ξ i , for every y ∈ H - ε we have (u n ) ξ i y ∈ SBV ((Q 1 ) ξ i y ), (5.15) J (un)
H d-1 (H -\ A - ε ) <
ε 2 and such that for every y ∈ A - ε the lines {y + tξ i : t ∈ R} intersect H + for every i = 1, . . . , N . In view of (5.5), (5.6) and (5.7), and since pointwise convergence implies almost uniform convergence, we can find

N ε ⊂ A - ε with (5.19) H d-1 (N ε ) < ε 2 and such that, up to a subsequence (5.20) (u n ) ξ i y -u ξ i y L 1 → 0 uniformly for y ∈ A - ε \ N ε (5.21) [ (u n ) ξ i y ] L 1 → 0 uniformly for y ∈ A - ε \ N ε (5.22) (u n ) |H -→ u - uniformly on A - ε \ N ε , and for every y ∈ A - ε \ N ε (5.23) H 0 ((J un ) ξ i y ) < +∞.
Notice that for n large enough and for every y ∈ A - ε \ N ε we have

(5.24) (J un ) ξ i y = ∅.
Indeed otherwise, we would get for n k → +∞ the existence of

y k ∈ A - ε \N ε with (u n k ) ξ i y k ∈ W 1,1 ((Q 1 ) ξ i y k
), and (5.22) together with (5.21) would yield

(u n k ) ξ i y k -u - 1 → 0
against (5.20) (recall that by the choice (5.10) of the ξ i , the functions u ξ i y have a jump). The claim follows by setting

H - ε := A ε \ N ε ∪ n (J un ∩ H -) .
Indeed (5.12) follows from (5. Step 3. For every i = 1, . . . , N , let us consider the set J i,- n given by the first point of intersection (with t > 0) of the line {y + tξ i : t ∈ R} with the jump set J un as y varies in the set H - ε defined in Step 2 (recall (5.14) and (5.15)). In view of (5.16) and (5.17), we can find η n → 0 such that for every

x ∈ J i,- n with ν un • ξ i > 0 (5.25) |u - n (x) • ξ i -u -• ξ i | < η n .
Step 4. We claim that, for δ small enough and N large enough, up to a subsequence, we can find Jn ⊆ J un with (5.26)

H d-1 ( J- n ) ≥ 1 -c ε ,
where c ε → 0 as ε → 0, and such that for every x ∈ J-

n (5.27) x ∈ J i,- n for d different indices i ∈ {1, . . . , N },
where J i,- n is defined in Step 3. Moreover, we can orient ν un on Jn in such a way that (5.28) e d • ν un > 0 and ξ i • ν un > 0 for every i = 1, . . . , N .

Intuitively speaking, the points in Jn are seen from H - ε under d different directions: moreover the associated lines cut the jump transversaly, from the "lower" to the "upper" part. Indeed, in view of the definition of ξ i (which form a very small angle with e d as δ → 0) and of the area formula (cf for instance [26, Sec. 3.2]), we can assume that δ is so small that for every i = 1, . . . , N (5.29)

H d-1 (J i,- n ) ≥ J i,- n |ν un • ξ i | dH d-1 = H d-1 ((H - ε ) ξ i ) = 1 1 + ĉδ H d-1 (H - ε ),
where the notation (H - ε ) ξ i is defined in (2.1) and where ĉδ → 0, so that, taking into account (5.12), for small δ we have (5.30)

H d-1 (J i,- n ) ≥ 1 -2ε.
By Lemma 5.5 below (with X = J un , µ = H d-1 , and M given by the family of Borel sets) if N is large enough we can find an index ī such that (5.31)

H d-1     J ī,- n \ i 1 <i 2 <•••<i d i h =1,...,N J i 1 ,- n ∩ J i 2 ,- n ∩ • • • ∩ J i d ,- n     < ε.
Intuitively speaking, most of the points in J ī,- n are seen from H - ε at least under d different directions: we call this set Jn , i.e.,

(5.32) J-

n := J ī,- n ∩ i 1 <i 2 <•••<i d i h =1,...,N J i 1 ,- n ∩ J i 2 ,- n ∩ • • • ∩ J i d ,- n .
In view of (5.30) and (5.31) we get (5.33)

H d-1 ( J- n ) ≥ 1 -3ε. Finally, if we set G n,ε := {x ∈ J- n : |ν un • ξī| > ε} and B n,ε := J- n \ G n,ε , coming back to (5.29) we have H d-1 (G n,ε ) + ε 2 H d-1 (B n,ε ) > 1 -3ε, so that H d-1 (G n,ε ) > 1 -3ε -ε 2 C,
where C := sup n H d-1 (J un ) < +∞. Finally we orient the normal ν un on G n,ε in such a way that

ν un • ξī > ε.
The inequalities (5.28) then also hold true on G n,ε if δ is small enough thanks to (5.9). Reducing Jn to G n,ε if necessary, the full claim follows taking into account (5.32) and (5.33).

Step 5. Let Jn ⊆ J un be the set given by Step 4. Since the points of this set are seen from H - ε under d different directions, in view of (5.25) we infer that there exists ηn → 0 such that for every x ∈ J-

n |u - n (x) -u -| < ηn .
Reasoning in a similar way starting from the upper part H + ε , and employing the opposite directions {-ξ i : i = 1, . . . , N }, we can construct J+ n ⊆ J un with ν un oriented such that again e d • ν un > 0 and ξ i • ν un > 0 for every i = 1, . . . , N , such that (5.34)

H d-1 ( J+ n ) ≥ 1 -c ε
with c ε → 0 as ε → 0, and such that for every x ∈ J+

n |u + n (x) -u + | < ηn .
Notice that for x ∈ Jn ∩ J+ n , the orientation chosen is compatible with that of (5.28), so that indeed u - n (x) and u + n (x) are the two traces of u n at x. Lemma 5.5. Let (X, M, µ) be a finite measure space. Let ε > 0 and d ≥ 2. Then there exists N ∈ N that only depends on µ(X), ε, d such that if {E i } i=1,...,N is a family of sets in M, we can find ī such that

µ   Eī \ j 1 <j 2 <•••<j d (E j 1 ∩ E j 2 ∩ • • • ∩ E j d )   < ε.
Proof. Up to dividing ε by µ(X) we suppose without loss of generality that µ(X) = 1. It is enough to prove that for any d ≥ 2, ε > 0, there is some N (d, ε) ≥ 1 such that any family of N ≥ N (d, ε) of sets (E i ) 1≤i≤N there is some i that verifies

µ   E i \ J⊂[1,N ]\{i},|J|=d-1 j∈J E j   < ε,
meaning that there is some i such that every point of E i outside a set of measure less than ε is in (at least) d -1 other sets E j (for j = i).

We prove it by recursion. If d = 2, let N := 1 ε , where [•] denotes the integer part. Given (E i ) 1≤i≤N , let us consider the sets E i \ 1≤j≤N,j =i E j 1≤i≤N . These are disjoint and µ(X) = 1, so there is some For every k ∈ [1, N ], the sets E k,i \ 1≤j≤M,j =i E k,j 1≤i≤M are disjoints so there is some i k such that

i such that µ   E i \ 1≤j≤N,j =i E j   ≤ 1 N ≤ ε,
µ   E k,i k \ 1≤i≤M,i =i k E k,i   ≤ 1 M ≤ ε 2 . Considering the sets (E k,i k ) 1≤k≤N , since N = N d, ε 2 we find some k such that µ   E k,i k \ K⊂[1,N ]\{k},|K|=d-1 k∈K E k,i k   ≤ ε 2 .
This means that outside a set of measure at most ε 2 , every point of E k,i k is in d -1 sets of the form E k,i k for k = k, and similarly every point outside a set of measure at most ε 2 is also in one set of the form E k,i for some i = i k . We conclude that outside of measure at most ε, every point of E k,i k belongs to d other sets, meaning N (d + 1, ε) is well-defined and

N (d + 1, ε) ≤ N d, ε 2
µ n := [φ(u + n ) + φ(u - n )]H d-1
J un and assume that (up to a subsequence)

µ n * µ weakly* in M b (Ω)
for some Radon measure µ on Ω, the conclusion follows if we show that

µ ≥ [φ(u + ) + φ(u -)]H d-1 J u as measures on Ω.
With this aim is sufficient to show that (5.35)

dµ dH d-1 (x) ≥ [φ(u + (x)) + φ(u -(x))] for H d-1 -a.e. x ∈ J u ,
where dµ dH d-1 denotes the Radon-Nykodim derivative of µ with respect to H d-1 (restricted to J u ). Let us assume (up to subsequences) that

λ n := H d-1 J un * λ weakly* in M b (Ω),
and that

|e(u n )| dx * f dx weakly* in M b (Ω),
where f ∈ L 1 (Ω) (this is possible since (e(u n )) n∈N is bounded in L 2 ). Let x ∈ J u be such that

dµ dH d-1 (x) = lim r→0 µ(Q x,r ) r d-1 , lim r→0 λ(Q x,r ) r d-1 < +∞, lim r→0 1 r d-1 Qr(x)
|f | dx = 0, and (having choosen the axis so that ν u (x) = e d ), for r → 0

+ u(x + r•) → u + (x)1 Q + 1 + u -(x)1 Q - 1 strongly in L 1 (Q 1 ; R d ).
Since H d-1 -a.e. x ∈ J u satisfies these properties, it suffices to concentrate on such points to prove inequality (5.35).

Let r k → 0 be such that µ(∂Q x,r k ) = λ(∂Q x,r k ) = 0.

Since by weak convergence and the relation above we have µ n (Q x,r k ) → µ(Q x,r k ), and similarly for λ, we can choose n k +∞ such that

µ(Q x,r k ) ≥ µ n k (Q x,r k ) - r d-1 k k , λ(Q x,r k ) ≥ λ n k (Q x,r k ) - r d-1 k k ,
and

Qx,r k |f | dx ≥ Qx,r k |e(u n k )| dx - r d-1 k k .
Moreover, setting v k (y) := u n k (x + r k y) we can assume also

v k → u + (x)1 Q + 1 + u -(x)1 Q - 1 strongly in L 1 (Q 1 ; R d ).
We get

Q 1 |e(v k )| dx = 1 r d-1 k Qx,r k |e(u n k )| dx ≤ 1 r d-1 k Qx,r k |f | dx + 1 k → 0 and H d-1 (J v k ) = 1 r d-1 k H d-1 (J un k ∩ Q x,r k ) = λ n k (Q x,r k ) r d-1 k ≤ λ(Q x,r k ) r d-1 k + 1 k → c < +∞,
so that, using the lower semicontinuity (5.8) concerning functions on the unit square (and to which the proof of the Theorem has been reduced)

dµ dH d-1 (x) = lim k→+∞ µ(Q x,r k ) r d-1 k ≥ lim inf k→+∞ µ n k (Q x,r k ) r d-1 k = lim inf k→+∞ Jv k [φ(v + k ) + φ(v - k )] dH d-1 ≥ φ(u + (x)) + φ(u -(x))
and (5.35) follows.

6. Existence of minimizers: proof of Theorem 4.8

We are now in a position to prove the first main result of the paper.

Proof of Theorem 4.8. Let (E n , u n ) n∈N be a minimizing sequence: since the function f is not identically equal to +∞, and in view of Remark 4.6, there exists C > 0 such that

J (E n , u n ) ≤ C.
Since u n = 0 a.e. on E n we may write

∂ * En |u + n | 2 dH d-1 + Ju n \∂ * En [|u + n | 2 + |u - n | 2 ] dH d-1 = Ju n [|u + n | 2 + |u - n | 2 ] dH d-1
so that we infer

H d-1 (∂ * E n ) ≤ C and Ω |e(u n )| 2 dx + H d-1 (J un ) + Ju n [|u + n | 2 + |u - n | 2 ] dH d-1 ≤ C.
Notice that

|E(u n )|(Ω ) = Ω |e(u n )| dx + Ju n |u + n -u - n | dH d-1 ≤ Ω \Ω |e(V )| dx + Ω |e(u n )| dx + Ju n [|u + n | + |u - n |] dH d-1 ≤ Ω \Ω |e(V )| dx + 1 2 |Ω| + Ω |e(u n )| 2 dx + 2H d-1 (J un ) + Ju n [|u + n | 2 + |u - n | 2 ] dH d-1 ≤ C,
for some C > 0. Moreover, thanks to Theorem 5.1 applied to u -V we may assume also that (6.1)

u n L 2d d-1 (Ω ) ≤ C.
By the compactness result in SBD (see Theorem 2.1), there exist a subsequence (u n k ) k∈N and u ∈ SBD(Ω ) with u = V on Ω \ Ω and such that (6.2)

u n k → u strongly in L 1 (Ω ; R d ), (6.3) e(u n k ) e(u)
weakly in L 2 (Ω ; M d×d sym ), and

H d-1 (J u ) ≤ lim inf k→+∞ H d-1 (J un k ).
Concerning the sets E n k , we may assume, up to a further subsequence if necessary, that there exists a set of fine perimeter E ⊆ Ω such that (6.4)

1 En k → 1 E strongly in L 1 (R d ) with H d-1 (∂ * E) ≤ lim inf k→+∞ H d-1 (∂ * E n k ).
In particular we get (6.5)

f (|E|) ≤ lim inf n→+∞ f (|E n |).
Let us prove that

(6.6) (E, u) ∈ A(V ).
In view of (6.1) we infer that u ∈ L 2d d-1 (Ω ; R d ) so that in particular u ∈ L 2 (Ω ; R d ). Moreover u = V on Ω \ Ω, while u = 0 a.e. on E thanks to (6.2) and (6.4).

Since the divergence constraint is intended in the sense of distributions on Ω, this passes easily to the limit thanks to (6.2). Moreover, in view of Theorem 5.3 we deduce

u ± ⊥ ν u on J u .
In particular this entails

u + ⊥ ν E on ∂ * E ∩ Ω,
since for x ∈ ∂ * E we have either x ∈ J u or u + (x) = 0. We conclude that the non-penetration constraint for the velocity field holds on ∂ * E and on J u \ ∂ * E, so that (6.6) holds true.

Let us prove the pair (E, u) is a minimizer for the problem. Thanks to (6.3) we get

Ω |e(u)| 2 dx ≤ lim inf k→+∞ Ω |e(u n k )| 2 dx,
while in view of Theorem 5.4 we have that

Ju [|u + | 2 + |u -| 2 ] dH d-1 ≤ lim inf k→+∞ Ju n k [|u + n k | 2 + |u - n k | 2 ] dH d-1 ,
which entails

∂ * E |u + | 2 dH d-1 + Ju\∂ * E [|u + | 2 + |u -| 2 ] dH d-1 ≤ lim inf k→+∞ ∂ * En k [u + n k | 2 dH d-1 + Ju n k \∂ * En k [|u + n k | 2 + |u - n k | 2 ] dH d-1 (6.7)
since u = 0 a.e. on E and u n k = 0 a.e. on E n k . Let us prove that

(6.8) 2H d-1 (J u \ ∂ * E) + H d-1 (∂ * E) ≤ lim inf k→+∞ 2H d-1 (J un k \ ∂ * E n k ) + H d-1 (∂ * E n k ) .
Let us choose h ∈ R d such that

H d-1 ({x ∈ ∂ * E ∪ J u : u + (x) = h}) = H d-1 ({x ∈ ∂ * E ∪ J u : u -(x) = h}) = H d-1 ({x ∈ ∂ * E n k ∪ J un k : u + n k (x) = h}) = H d-1 ({x ∈ ∂ * E n k ∪ J un k : u - n k (x) = h}) = 0.
This is possible because for example the sets {x ∈ ∂ * E ∪ J u : u + (x) = h} are disjoint as h varies, and similarly for the other sets. In particular, setting

v h := u + h1 E and v h n k := u n k + h1 En k we have J v h = J u ∪ J 1 E = ∂ * E ∪ J u and J v h n k = J un k ∪ J 1 En k = ∂ * E n k ∪ J un k
up to H d-1 -negligible sets. If we apply Theorem 5.4 with the choice φ h (s) = 1 {s =h} to the sequence (v h n k ) k∈N we get

H d-1 (∂ * E) + 2H d-1 (J u \ ∂ * E) = J v h [φ h ((v h ) + ) + φ h ((v h ) -)]dH d-1 ≤ lim inf k→+∞ J v h n k [φ h ((v h n k ) + ) + φ h ((v h n k ) -)]dH d-1 = lim inf k→+∞ H d-1 (∂ * E n k ) + 2H d-1 (J un k \ ∂ * E n k ) (6.9)
so that (6.8) holds true. Gathering (6.3), (6.7), (6.5) and (6.8), we deduce

J (E, u) ≤ lim inf k→+∞ J (E n k , u n k )
so that, taking into account (6.6), the pair (E, u) is a minimizer of the main problem (4.3), and the proof is concluded.

Regularity of two-dimensional minimizers: proof of Theorem 4.10

This section is devoted to the proof of Theorem 4.10 concerning the regularity of minimizers in dimension two.

As mentioned in the Introduction, the general strategy used by De Giorgi, Carriero and Leaci for the Mumford-Shah problem in [START_REF] De Giorgi | Existence theorem for a minimum problem with free discontinuity set[END_REF] faces the new difficulties given by the vectorial context, considered in [START_REF] Conti | Existence of strong minimizers for the Griffith static fracture model in dimension two[END_REF][START_REF] Chambolle | Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy[END_REF] in connection to the Griffith fracture problem, and also by extra conditions proper to our problem, that is incompressibility and non-penetration for the velocity fields. We follow the main lines of [START_REF] Conti | Existence of strong minimizers for the Griffith static fracture model in dimension two[END_REF][START_REF] Chambolle | Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy[END_REF]: however technical difficulties allow us to deal only with dimension 2 (see point (a) below).

Since our drag problem involves pairs (E, u) as admissible configurations, and some points of ∂ * E may not be jump points of u, it will be useful to deal with pairs (J, u), where J is a rectifiable set and u is a function whose jumps are contained (up to H 1 -negligible sets) in J and satisfy the constraints of zero divergence and non-penetration. More precisely we formulate the following definition.

Definition 7.1 (The class V). Let Ω ⊆ R 2 be an open set. We say that (J, u) ∈ V(Ω) if J ⊆ Ω is a rectifiable set, and u ∈ SBD(Ω) is such that div u = 0 in the sense of distributions in Ω, H 1 (J u \J) = 0 and u ± |J • ν J = 0 H 1 -a.e. on J.
The structure of the section is the following.

(a) In Section 7.1 we prove a fundamental approximation lemma (Smoothing Lemma 7.2), which allows us to approximate every (J, u) ∈ V(Q 1 ) with H 1 (J) small by a configuration (J \Q r , v) ∈ V(Q 1 ), where v is a Sobolev function in the slightly smaller square Q r with a control on the energy. The idea is that the jumps of u in Q r are "smoothed out", giving rise to the function v which preserves the divergence free constraint together with the non-penetration condition. This result is inspired by [START_REF] Chambolle | Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy[END_REF], and it is here that the dimension two is fundamental.

(b) In Section 7.2 we prove regularity for local minimizers of a Griffith functional

G(J, u) := Ω |e(u)| 2 dx + H 1 (J),
defined on pairs (J, u) ∈ V(Ω). The kind of local minimality considered is very weak, and inspired by the kind of competitors that can be constructed thanks to the Smoothing Lemma 7.2. The key result to get regularity is given by the decay estimate contained in Proposition 7.7.

Regularity for minimizers of the Griffith energy is then used in Section 7.3 to prove Theorem 4.10, that is to show the regularity of minimizers of the drag problem. (c) Finally, motivated by the regularity result of Theorem 4.10, in Section 7.4 we describe a different relaxation of the drag problem which involves topologically closed obstacles and Sobolev velocities: the regularity result can be used to prove that such a formulation is well posed in dimension two.

7.1. The smoothing lemma. We fix a standard radial, smooth, nonnegative mollifier ρ with support in a disc of radius 1/8 and denote

ρ δ (x) := δ -2 ρ x δ .
The main result of the section is the following smoothing lemma which is in the spirit of [START_REF] Chambolle | Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy[END_REF].

Lemma 7.2 (Smoothing Lemma

). There exist C, η > 0 such that for any (J, u) ∈ V(Q 1 ) with H 1 (J) < η, then letting δ := H 1 (J)

1 2 there exist r ∈]1 -δ 1 2 , 1[ and v ∈ SBD(Q 1 ) ∩ H 1 (Q r )
such that the following items hold true.

(a) H 0 (J ∩ ∂Q r ) = 0 and for every 0 < s < r

H 1 (J ∩ (Q r \ Q r-s )) ≤ Cδ 3 2 s. (b) {v = u} ⊆ Q r and (J \ Q r , v) ∈ V(Q 1 ). (c) It holds e(v) L 2 (Q 1 ) ≤ (1 + Cδ 1 6 ) e(u) L 2 (Q 1 ) . (d) There exists a cut-off function ϕ ∈ C ∞ (Q r , [0, 1]) with ϕ = 0 on Q r \ Q r-δ , ϕ = 1 on Q r-4δ ,
and such that e(v) -ϕρ δ * e(u) L 2 (Qr) ≤ Cδ

1 6 e(u) L 2 (Q 1 ) .
Proof. The proof follows the strategy introduced in [START_REF] Chambolle | Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy[END_REF], and some parts will be referred directly to that paper. However, since our conclusion is slightly different, we prefer to develop some computations in detail. We will use the notation a b when a ≤ Cb for some dimensional constant C. We divide the proof in several steps.

Step 1: Subdivision in small squares. Let us set

N := 1 + H 1 (J) -1 2 ,
where [•] denotes the integer part. In the following we will assume that H 1 (J) is arbitrary small, so that N is arbitrarily large. For convenience in the construction, we will set δ = 1/N ≤ H 1 (J)

1 2 , which (mildly) differs from the choice of the statement: yet since δ is asymptotically equivalent to H 1 (J)

δ k := δr 2 k and r k = N - 1 2 k δ .
Then we consider a partition (up to a negligible set) of Q r into cubes obtained by filling Q r 0 with cubes of side δ 0 and denoted by (q 0,j ) j , and then each Q r k \ Q r k-1 with cubes of side δ k and denoted (q k,j ) j (note that there is only one way to do this).

For any square q = z + [-t, t] 2 , we write

q := z + - 8 7 t, 8 7 t 2
and q := (q ) .

We will set q k,j := (q k,j ) . We may notice that with our choices

(7.1) ∀k ≥ 1 : q k,j Q r k+1 \ Q r k-2 ,
and {q k,j } k,j is a covering of Q r with a fixed finite number of overlapping: indeed each q k,j meets at most 8 neighbours q p,i , and they all verify |k -p| ≤ 1, meaning δ k /δ p ∈ 1 2 , 1, 2 . This is because the factor 8 7 above is chosen such that 8

7 3 < 3 2 .
Step 2: Choice of the square Q r . We now make a convenient choice of r such that the density of J near ∂Q r is small, following an approach similar to [18, Theorem 2.1]. We claim that there exist C, η > 0 such that for δ < η we can choose

r ∈]1- √ δ, 1[ with H 0 (J ∩∂Q r ) = 0, (7.2) ∀ s ∈]0, r[ : H 1 (J ∩ (Q r \ Q r-s )) ≤ Cδ
Qr\Qr -2 |e(u)| 2 dx < Cδ 1 2 Q 1 |e(u)| 2 dx.
Consider indeed the measure µ on [0, 1] defined as

µ(E) := H 1 (J ∩ Q E ) H 1 (J) + Q E |e(u)| 2 dx Q 1 |e(u)| 2 dx ,
where

Q E := ∪ r∈E ∂Q r is the cubic shell associated to E ⊂ [0, 1]. It suffices to prove that we can find r ∈]1 -δ 1 2 , 1[ such that (7.4) H 0 (J ∩ ∂Q r ) = 0,
and, denoting I s r := [r -s, r[ for 0 < s < r, (7.5)

µ(I s r ) ≤ Ĉδ -1 2 s,
where Ĉ > 0 is a suitable constant which we fix below. Indeed, if δ is small enough this implies that (recall that H 1 (J) behaves like δ 2 )

H 1 (J ∩ (Q r \ Q r-s )) ≤ H 1 (J)µ(I s r ) ≤ Ĉδ 3 2 s and Qr\Q r-4δr |e(u)| 2 dx ≤ Ĉδ -1 2 (4δr) Q 1 |e(u)| 2 dx ≤ 4 Ĉδ 1 2 Q 1 |e(u)| 2 dx,
so that (7.2) and (7.3) follow by choosing C := 4 Ĉ. Let I 1 be the union of all intervals that do not satisfy (7.5). If (

I s i r i ) is a Vitali covering of I, then 2 = µ([0, 1]) ≥ i µ(I s i r i ) > Ĉδ -1 2 i |I s i r i | = Ĉδ -1 2 5 i |5I s i r i | ≥ Ĉδ -1 2 5 |I 1 | hence |I 1 | < 10 Ĉ δ 1 2
. Let I 2 := π x (J) ∪ π y (J), where π x , π y denote the projection on the coordinate axis: we have asymptotically

|I 2 | ≤ 2δ 2 . If C > 10, this implies that for δ small enough ]1 - √ δ, 1[\(I 1 ∪ I 2 ) = ∅,
which yields the existence of r which verifies claims (7.4) and (7.5).

Step 3: A first approximation. In view of (7.2) and of (7.1), for every k ≥ 1 we have

H 1 (J u ∩ q k,j ) δ 3 2 δ k , while if δ is small enough (recall that H 1 (J) behaves like δ 2 and r ∈]1 -δ 1 2 , 1[) H 1 (J u ∩ q 0,j ) ≤ H 1 (J u ) δδ 0 .
This means that the jump set of u in every cube of the constructed subdivision is arbitrarily small compared to its sides.

Thanks to [15, Proposition 3], and taking into account the preceding inequalities , for every (k, j) there is a set ω k,j ⊂ q k,j and an affine function a k,j with e(a k,j ) = 0, such that

(7.6) |ω k,j | δ k H 1 (J u ∩ q k,j ) δδ 2 k (7.7) q k,j \ω k,j |u -a k,j | 4 dx δ k q k,j |e(u)| 2 dx 2 ,
and the function v k,j := u + (a k,j -u)1 ω k,j verifies

q k,j |e(ρ δ k * v k,j ) -ρ δ k * e(u)| 2 dx H 1 (J u ∩ q k,j ) δ k 1 3 q k,j |e(u)| 2 dx δ 1 3 q j,k |e(u)| 2 dx, (7.8)
(see [15, p. 1389]) where ρ is the mollifier defined at the beginning of the section. Notice that in view of our construction (namely the choice of r), we have

(7.9) |ω k,j | |q k,j |,
and this is where we most use the fact that we are in two dimensions. We now let (ϕ k,j ) be a partition of unity associated to the covering (q k,j ) of Q r and such that

|∇ϕ k,j | 1 δ k . Let us set w := 1 Q 1 \Qr u + 1 Qr k,j ϕ k,j w k,j where w k,j := ρ δ k * v k,j .
We claim that (7.10)

w ∈ SBD(Q 1 ) ∩ H 1 (Q r ), {w = u} ⊂ Q r , H 1 (J w \ J) = 0, (7.11) e(w) - k,j ϕ k,j ρ δ k * e(u) L 2 (Qr) δ 1 6 e(u) L 2 (Q 1 )
, and (7.12) the trace of w and u on ∂Q r coincide.

We postpone the proof of these claims to Step 5. Let us set ϕ := (0,j)∈K ϕ 0,j , where K denotes the set of indices such that q 0,j has a distance greater than 2δr from ∂Q r . Since r ∈]1 -δ 1 2 , 1[, in view of the definition of the set of indices K, we get that the function ϕ vanishes on Q \ Q r-δ and it is equal to 1 on Q r-4δ .

We can write

e(w) - k,j ϕ k,j ρ δ k * e(u) = e(w) -ϕρ δ * e(u) - (k,j) ∈K ϕ k,j ρ δ k * e(u).
Thanks to (7.3) we have

(k,j) ∈K ϕ k,j ρ δ k * e(u) 2 L 2 (Qr) = (k,j) ∈K ϕ k,j ρ δ k * e(u) 2 L 2 (Qr\Q r-2δr ) (k,j) ∈K ϕ j,k ρ δ k * e(u) 2 L 2 (Q 1 \Q r-2δr ) e(u) 2 L 2 (Q 1 \Q r-3δr ) δ 1 2 e(u) 2 L 2 (Q 1 ) ,
so that in view of (7.11) we conclude (7.13) e(w) -ϕρ δ * e(u) L 2 (Qr) δ

1 6 e(u) L 2 (Q 1 ) .
Moreover we may write

e(w) L 2 (Qr) ≤ ϕρ δ * e(u) L 2 (Qr) + e(w) -ϕρ δ * e(u) L 2 (Qr) = ϕρ δ * e(u) L 2 (Q r-δ ) + e(w) -ϕρ δ * e(u) L 2 (Qr) ≤ e(u) L 2 (Q 1 ) + e(w) -ϕρ δ * e(u) L 2 (Qr) ,
so that taking into account (7.13) we deduce

(7.14) e(w) L 2 (Q 1 ) ≤ (1 + Cδ 1 6 ) e(u) L 2 (Q 1 ) ,
where C > 0.

Step 4: Enforcing the divergence free constraint. By admissibility, u is divergence free in the sense of distributions in Q 1 , so that the trace of e(u) is zero in Q 1 , while (7.15)

∂Qr u • ν dH 1 = 0,
where ν is the outward normal vector of Q r , and u denotes the trace on ∂Q r (J does not intersect ∂Q r by construction).

Recalling that w ∈ H 1 (Q r ), we may write thanks to (7.13) div w L 2 (Qr) = Tr(e(w)) L 2 (Qr) = Tr(e(w) -ϕρ δ * e(u)) L 2 (Qr) δ

1 6 e(u) L 2 (Q 1 )
. By (7.12) the trace of u on ∂Q r coincides with that of w, so that from (7.15) we deduce Qr div w dx = 0.

Using a classical result (recorded at the end of this proof in Lemma 7.3), there exists a vector field q ∈ H 1 0 (Q r ) such that (7.16) div q = div w and ∇q L 2 (Qr) div w L 2 (Qr) δ

1 6 e(u) L 2 (Q 1 ) . Let v := w -q in Q r u in Q 1 \ Q r ,
and let us check that v satisfies the conclusions of the lemma.

The choice of r given by Step 2 yields immediately point (a). Clearly v ∈ SBD(Q 1 ) ∩ H 1 (Q r ) with {v = u} ⊆ Q r . Moreover, since the trace of w -q and u coincide on ∂Q r , we get div v = 0 in the sense of distributions in Q 1 , so that point (b) is proved. Points (c) and (d) follow from the corresponding properties for w (see (7.13) and (7.14)) taking into account that the correction term q has a small gradient norm of the order δ 1 6 as estimated in (7.16).

Step 5: Proof of the claims (7.10), (7.11) and (7.12). In order to conclude the proof, we need to check the claims on the function w contained in Step 3.

Let us start by noticing that the oscillation of the maps a k,j on intersecting squares can be estimated. Indeed as soon as q k,j and q p,i intersects, then

|q k,j ∩ q p,i | max(|q k,j |, |q p,i |),
and since (see (7.9))

|(q k,j ∩ q p,i ) ∩ (ω k,j ∪ ω p,i )| |q k,j ∩ q p,i | and a j,k , a i,p are affine, then using [START_REF] Chambolle | Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy[END_REF]Lemma 3.4] and (7.7) we deduce (7.17) a k,j -a p,i L 4 (q k,j ∩q p,i ) a k,j -a p,i L 4 ((q k,j ∩q p,i )\(ω k,j ∪ω p,i ))

≤ a k,j -u L 4 (q k,j \ω k,j ) + a p,i -u L 4 (q p,i \ω p,i ) δ

1 2 k e(u) L 2 (q k,j ) + δ 1 2 p e(u) L 2 (q p,i ) δ 1 2
k e(u) L 2 (q k,j ∪q p,i ) , as δ k and δ p are comparable.

Let us come to the claims. Clearly For the first term of the right hand side, we have thanks to (7.8) k,j ϕ k,j e(w k,j ) -ρ δ k * e(u)

2 L 2 (Qr) k,j ϕ k,j e(w k,j ) -ρ δ k * e(u) 2 L 2 (Qr) ≤ k,j e(w k,j ) -ρ δ k * e(u) 2 L 2 (q k,j ) ≤ δ 1 3 k,j e(u) 2 L 2 (q k,j ) δ 1 3 e(u) 2 L 2 (Qr) , (7.19)
where we used the finite overlapping of the squares q k,j for the first and last estimates.

Let us estimate the second term on the right hand side of (7.18). Notice that we may write k,j ∇ϕ k,j w k,j = q k,j ∩q p,i =∅ ∇ϕ k,j (w k,j -w p,i ) on q p,i since k,j ∇ϕ k,j = 0.

(a1) If q p,i Q r -1 , then q j,k ∩ q i,p = ∅ means that δ k = δ p = δ, k = p = 0, and we may rewrite the term as q 0,j ∩q 0,i =∅ ∇ϕ 0,j (w 0,j -w 0,i )

We get (7.20) q 0,j ∩q 0,i =∅ ∇ϕ 0,j (w 0,j -w 0,i ) 2 L 2 (q 0,i ) q 0,j ∩q 0,i =∅ 1 δ 2 w 0,j -w 0,i 2 L 2 (q 0,j ∩q 0,i ) .

Now

w 0,j -w 0,i L 2 (q o,j ∩q 0,i ) = ρ δ * (v 0,j -v 0,i ) L 2 (q 0,j ∩q 0,i ) ≤ v 0,j -v 0,i L 2 (q 0,j ∩q 0,i ) .

Since v 0,j -v 0,i L 2 (q 0,j ∩q 0,i ) ≤ (a 0,j -a 0,i )1 ω 0,j ∪ω 0,i L 2 (q 0,j ∩q 0,i ) + (u -a 0,j )1 ω 0,i L 2 (q 0,j \ω 0,j )

+ (u -a 0,i )1 ω 0,j L 2 (q 0,i \ω 0,i )

≤ (a 0,j -a 0,i ) L 4 (q 0,j ∩q 0,i ) |ω 0,j ∪ ω 0,i | 1 4 + (u -a 0,j ) L 4 (q 0,j \ω 0,j ) |ω 0,i |

+ (u -a 0,i ) L 4 (q 0,i \ω 0,i ) |ω 0,j | 1 4 , recalling (7.6), (7.7) and (7.17) we get w 0,j -w 0,i L 2 (q o,j ∩q 0,i ) ≤ v 0,j -v 0,i L 2 (q 0,j ∩q 0,i ) ≤ δ 1+ 1 4 e(u) L 2 (q 0,j ∪q 0,i ) .

Coming back to (7.20) we infer k,j ∇ϕ k,j w k,j 2 L 2 (q 0,i ) ≤ q 0,j ∩q 0,i =∅ ∇ϕ 0,j (w 0,j -w 0,i )

2 L 2 (q 0,i ) δ 1 2
q 0,j ∩q 0,i =∅ e(u) 2 L 2 (q 0,j ∪q 0,i ) .

(7.21) (a2) If q p,i Q r -1 , then for q k,j ∩ q p,i = ∅, we decompose

w p,i -w k,j = ρ δp * (v p,i -a p,i ) -ρ δ k * (w k,j -a k,j ) + (a p,i -a k,j ).
Notice the crucial step that ρ δ k * a k,j = a k,j due to the fact that a k,j is harmonic (since it is affine). Then we have thanks to (7.7) and (7.17)

ρ δ k * (v p,i -a p,i ) L 2 (q k,j ∩q p,i ) ≤ v p,i -a p,i L 2 (q p,i ) δ p e(u) L 2 (q i,p ) ρ δ k * (v k,j -a k,j ) L 2 (q k,j ∩q p,i ) ≤ v k,j -a k,j L 2 (q k,j ) δ p e(u) L 2 (q k,j )
a p,i -a k,j L 2 (q k,j ∩q p,i ) δ

1+ 1 4 p e(u) L 2 (q k,j ∪q p,i ) ,
where we also used the fact that δ p and δ k differ from at most a factor 2. And so we obtain with the same computations as the previous point that (7.22) k,j ∇ϕ k,j w k,j 2 L 2 (q p,i ) ≤ q k,j ∩q p,i =∅ e(u) 2 L 2 (q k,j ∪q p,i ) .

Gathering (7.21) and (7.22), and in view of the choice of r which satisfies (7.3), we deduce

k,j ∇ϕ k,j w k,j 2
L 2 (Qr) ≤ p,i k,j ∇ϕ k,j w k,j 2
L 2 (q p,i ) δ 1 2 e(u) 2 L 2 (Qr 1 ) + e(u) 2 L 2 (Qr\Qr -2 ) δ 1 2 e(u) 2 L 2 (Q 1 ) . (7.23)
Coming back to (7.18), in view of (7.19) and (7.23) we deduce that e(w) -

k,j ϕ k,j ρ δ k * e(u) L 2 (Qr) δ 1 6 e(u) L 2 (Q 1 )
, so that claim (7.11) follows.

In particular we get also that w ∈ H 1 (Q r ). Claim (7.12) concerning the traces follows by the construction which involves convolutions whose radius becomes finer and finer as we approach ∂Q r as detailed in [START_REF] Chambolle | Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy[END_REF]. Finally we deduce that w ∈ SBD(Q 1 ), and that claim (7.10) holds true.

In the proof of Proposition 7.2 we made use of the following lemma due to Nečas ( see [START_REF] Boyer | Mathematical tools for the study of the incompressible Navier-Stokes equations and related models[END_REF]Theorem IV.3.1], or also [START_REF] Bogovskiȋ | Solution of the first boundary value problem for an equation of continuity of an incompressible medium[END_REF]). We consider the following (very weak) notion of local minimality.

Definition 7.4 (Quasi minimizers). Let Λ, r > 0. We say that (J, u) ∈ V(Ω) (recall Definition 7.1) is a (Λ, r) quasi minimizer of G on V(Ω) if G(J, u, ω) < +∞ for any open set ω Ω, and for any square Q x,r Ω with r ∈ (0, r), H 0 (J ∩ ∂Q x,r ) = 0 and

lim sup s→0 + 1 s H 1 (J ∩ (Q x,r \ Q x,r-s )) < 1,
and for any function v ∈ H 1 (Q x,r ; R 2 ) with div v = 0 and v = u on ∂Q x,r , we have

(7.24) Qx,r |e(u)| 2 dx + H 1 (J ∩ Q x,r ) ≤ Qx,r |e(v)| 2 dx + Λr 2 .
Remark 7.5. Notice that under the assumption of the previous definition, we have (J \Q x,r , v) ∈ V(Ω), where we extended v to the entire Ω by setting v = u in Ω \ Q x,r , and inequality (7.24) may be written as

G(J, u, Q x,r ) ≤ G(J \ Q x,r , v, Q x,r ) + Λr 2 .
The local minimality property involves thus a comparison between (J, u) and very special competitors: the Sobolev function v is obtained by "smoothing out" the jumps of u inside suitable squares Q x,r , so that it can be paired with the rectifiable set J \ Q x,r , yielding the admissible pair (J \ Q x,r , v). Such competitors are provided by the Smoothing Lemma 7.2, for which the dimension two is essential. A somehow related weak notion of minimality involving Sobolev competitors, still in dimension two, has been investigated in [START_REF] Bucur | Local minimality results for the Mumford-Shah functional via monotonicity[END_REF] (minimality with respect to its own jump set) for the (scalar) Mumford-Shah functional.

Remark 7.6. The notion of minimality is weak enough to include any local minimizer of a functional of the form

F (u, A) := A |e(u)| 2 dx + Ju∩A Θ(ν u , u + , u -)dH 1
where Θ is a measurable function such that inf(Θ) ≥ 1 (or, inf(Θ) > 0 up to scaling).

The following result is the key ingredient for obtaining regularity. Proposition 7.7 (Decay estimate). Let Λ > 0. There exists a universal constant τ ∈ (0, 1) such that for every τ ∈ (0, τ ) there exist ε = ε(τ ) and r = r(τ ) with the property that for any (Λ, r)-quasi minimizer (J, u) of G on V(Ω), if for r < r

G(J, u, Q r ) ≥ r 3/2 and H 1 (J ∩ Q r ) ≤ εr, then G(J, u, Q τ r ) ≤ τ 3/2 G(J, u, Q r ).
Proof. By contradiction assume that for τ sufficiently small there exist ε n → 0, rn → 0, 0 < r n < rn , and a sequence (K n , w n ) of (Λ, r n )-minimizers for such that for every n

G(K n , w n , Q rn ) ≥ r 3/2 n , H 1 (K n ∩ Q rn ) ≤ ε n r n , and G(K n , w n , Q τ rn ) > τ 3/2 G(K n , w n , Q rn ). Let g n := G(K n , w n , Q rn ), J n := K n r n and u n (x) := w n (r n x) √ g n . Then (J n , u n ) is a (Λ √ r n , 1)-minimizer of G n (•, •, Q 1 ), where G n (J, u, A) := A |e(u)| 2 dx + r n g n H 1 (J ∩ A), with (7.25) G n (J n , u n , Q 1 ) = 1, G n (J n , u n , Q τ ) > τ 3/2 and H 1 (J n ∩ Q 1 ) = ε n .
Let us apply the Smoothing Lemma 7.2: if

δ n = ε 1 2 n , let Q sn with 1 -δ 1 2
n < s n < 1 be the square on which the jumps of u n are smoothed out giving raise to the function v n , associated to an admissible pair

(J \ Q sn , v n ) ∈ V(Q 1 ). In particular (7.26) e(v n ) L 2 (Q 1 ) ≤ (1 + Cδ 1 6 n ) e(u n ) L 2 (Q 1 ) with e(u n ) L 2 (Q 1 ) ≤ 1, and
(7.27) e(v n ) -ϕ n ρ δn * e(u) L 2 (Qs n ) ≤ Cδ 1 6 n e(u n ) L 2 (Q 1 ) ,
where C > 0 is independent of n and

ϕ n ∈ C ∞ (Q sn , [0, 1]) is such that ϕ n = 0 on Q sn \ Q sn-δn , ϕ n = 1 on Q sn-4δn
. Since v n is divergence free and Sobolev on Q sn we have (7.28)

∂Qs n v n • ν dH 1 = 0.
By the classical Korn inequality on Q sn there is an antisymmetric affine function a n such that

Qs n (v n -a n ) dx = 0 and Qs n |∇(v n -a n )| 2 dx ≤ C 1 Qs n |e(v n )| 2 dx
for some C 1 > 0 independent of n. We infer that (v n -a n ) is bounded in H 1 (Q sn ). Since s n → 1, we can assume, up to extracting a further subsequence,

(7.29) v n -a n w weakly in H 1 loc (Q 1 ; R 2 ) for some w ∈ H 1 (Q 1 )
. Since every v n -a n has zero divergence, then so does w. Moreover e(w)

L 2 (Q 1 ) ≤ 1. Let ψ ∈ C ∞ c (Q 1 ; R 2 ) have zero divergence, and let η ∈ C ∞ c (Q 1 , [0, 1]) be a cut-off function such that {ψ = 0} {η = 1}. Let us consider z n := P Qs n (1 -η)v n + η(a n + w + ψ) in Q sn u n in Q 1 \ Q sn ,
where P Qs n denotes the projection on divergence free H 1 (Q sn ) vector fields which preserves the trace obtained according to Lemma 7.3 by considering

P Qs n (u) := u -Φ Qs n (div u) for any u ∈ H 1 (Q sn ; R 2 ) with a zero mean divergence. Note that z n is well defined as (1 -η)v n + η(a n + w + ψ) = v n on ∂Q sn
for n large enough, and so its divergence has zero mean thanks to (7.28). Since (J n \ ∂Q sn , z n ) is an admissible competitor for (J n , u n ) according to Definition 7.4, we obtain

G n (J n , u n , Q sn ) ≤ e P Qs n (1 -η)v n + η(a n + w + ψ) 2 L 2 (Qs n ) + Λ √ r n ≤ e ((1 -η)v n + η(a n + w + ψ)) L 2 (Qs n ) + C div((1 -η)v n + η(a n + w + ψ)) L 2 (Qs n ) 2 + Λ √ r n ≤ e ((1 -η)v n + η(a n + w + ψ)) L 2 (Qs n ) + C ∇η • (w + a n -v n ) L 2 (Qs n ) 2 + Λ √ r n . Since ∇η • (w + a n -v n ) L 2 (Qs n ) → 0 and (recall that {ψ = 0} {η = 1}) e ((1 -η)v n + η(a n + w + ψ)) L 2 (Qs n ) = (1 -η)e(v n ) + ηe(w + ψ) + ∇η (w + a n -v n ) L 2 (Qs n ) ≤ (1 -η)e(v n ) + ηe(w + ψ) L 2 (Qs n ) + o n
where o n → 0, we infer thanks to (7.26) (and since e(a n ) = 0)

(7.30) G n (J n , u n , Q sn ) ≤ (1 -η)e(v n -a n ) + ηe(w + ψ) 2 L 2 (Qs n ) + o n . Now, still using (7.26) we may write Qs n |e(v n -a n )| 2 dx ≤ (1 + Cδ 1 6 n ) 2 Qs n |e(u n )| 2 dx ≤ (1 + Cδ 1 6 n ) 2 G n (J n , u n , Q sn ) + o n ,
and so coming back to (7.30) we deduce

e(v n -a n ) 2 L 2 (Qs n ) ≤ (1 -η)e(v n -a n ) + ηe(w + ψ) 2 L 2 (Qs n ) + o n .

This yields

Qs

n 1 -(1 -η) 2 |e(v n -a n )| 2 dx ≤ Qs n 2η(1 -η)e(v n -a n ) : e(w) + η 2 |e(w + ψ)| 2 dx + o n
so that in view of (7.29)

Q 1 1 -(1 -η) 2 |e(w)| 2 dx ≤ lim sup n→∞ Q 1 1 -(1 -η) 2 |e(v n -a n )| 2 dx ≤ Q 1 2η(1 -η)e(w) : e(w) + η 2 |e(w + ψ)| 2 dx.
Notice that by choosing ψ = 0 and letting η localize on characteristic functions of open sets, we infer that

(7.31) e(v n -a n ) → e(w) strongly in L 2 loc (Q 1 ; M 2×2 sym ). In particular we get Q 1 |e(w)| 2 dx ≤ Q 1 |e(w + ψ)| 2 dx,
which means that w is a local minimizer of the energy z → e(z) 2 L 2 (Q 1 ) on H 1 functions with zero divergence. This yields ∆w = ∇p for some p ∈ L 2 (Q 1 ). Using the Lemma 7.8 below, we have

Qτ |e(w)| 2 dx ≤ 1 2 τ 3 2 Q 1 |e(w)| 2 dx ≤ 1 2 τ 3 2 .
Taking into account (7.31) we deduce

(7.32) e(v n ) 2 L 2 (Q τ +δn ) ≤ 1 2 τ 3 2 + o n .
By minimality we have

G(u n , J n , Q sn ) ≤ e(v n ) 2 L 2 (Qs n ) + Λ √ r n = e(v n ) 2 L 2 (Q τ +δn ) + e(v n ) 2 L 2 (Qs n \Q τ +δn ) + Λ √ r n
while thanks to (7.27)

e(v n ) L 2 (Qs n \Q τ +δn ) ≤ e(v n ) -ϕ n ρ δn * e(u n ) L 2 (Qs n \Q τ +δn) + ϕ n ρ δn * e(u n ) L 2 (Qs n \Q τ +δn ) ≤ o n + ρ δn * e(u n ) L 2 (Q sn-δn \Q τ +δn ) ≤ o n + e(u n ) L 2 (Qs n \Qτ ) .
In view of (7.32) we infer

G(u n , J n , Q sn ) ≤ e(v n ) 2 L 2 (Q τ +δn ) +[o n + e(u n ) L 2 (Qs n \Qτ ) ] 2 +Λ √ r n ≤ 1 2 τ 3 2 +õ n + e(u n ) 2 L 2 (Qs n \Qτ ) so that G(u n , J n , Q τ ) ≤ 1 2 τ 3 2 + õn .
In conclusion, taking into account (7.25), if n is large enough we get

τ 3 2 < G(u n , J n , Q τ ) ≤ 1 2 τ 3 2 + õn ,
which is a contradiction.

In the preceding proof, we made use of the following result.

Lemma 7.8. There exists a constant C 0 > 0 such that for any divergence-free vector field u ∈ H 1 (Q 1 ; R 2 ) such that ∆u = ∇p for some pressure p ∈ L 2 (Q 1 ), we have ∀τ ∈ (0, 1/2] :

Qτ |e(u)| 2 dx ≤ C 0 τ 2 Q 1 |e(u)| 2 dx.
In particular, for any 0

< τ ≤ τ := 1 4C 2 0 ∧ 1 2 we have Qτ |e(u)| 2 dx ≤ 1 2 τ 3/2 Q 1 |e(u)| 2 dx.
Proof. Notice that e(u) is invariant by the addition of an asymmetric affine function a. Up to a translation by such a function, Korn's inequality tells us that

Q 1 u 2 dx ≤ C Q 1 |e(u)| 2 dx.
The equations verified by u are equivalent to the existence of ϕ ∈ H 2 (Q 1 ) such that ϕ(0) = 0, u = ∇ ⊥ ϕ, and ∆ 2 ϕ = 0. By elliptic regularity there is a constant C such that sup

Q 1/2 ∇ 2 ϕ 2 ≤ C Q 1 |∇ϕ| 2 dx
and so for any

τ ≤ 1/2, Qτ |e(u)| 2 dx ≤ 4|Q τ | sup Q 1/2 ∇ 2 ϕ 2 ≤ 4CC |Q 1 |τ 2 Q 1 |e(u)| 2 dx.
The decay estimate can be iterated as follows.

Lemma 7.9 (Iteration of the decay). Let Λ > 0 and, according to Proposition 7.7, let τ 0 be small enough such that the decay estimate applies with ε 0 = ε(τ 0 ) and r0 = r(τ 0 ), and let τ 1 ∈ (0, ε 2 0 ) be small enough that the decay property applies with ε 1 , r1 . Finally, let

r := min r0 , r1 , ε 2 0 τ 2 1 , ε 2 0 τ 3 0 τ 1 .
Suppose that (J, u) is a (Λ, r)-quasi minimizer of G on V(Ω) and G(J, u, Q x,r ) ≤ ε 1 r for some r ∈ (0, r).

Then for all k ∈ N,

G(J, u, Q x,τ k 0 τ 1 r ) ≤ ε 0 τ 3 2 k 0 τ 1 r.
Proof. Let us prove the statement by induction on k. In the following, we write g(r) = G(J, u, Q x,r ), so that we need to check that if g(r) ≤ ε 1 r, then for every k ∈ N

(7.33) g(τ k 0 τ 1 r) ≤ ε 0 τ 3 2 k
0 τ 1 r. The inequality is true for k = 0. Indeed we have the following alternatives:

(a) If g(r) > r 3/2 then g(τ 1 r) ≤ τ 3/2 1 g(r) ≤ √ τ 1 τ 1 ε 1 r ≤ ε 0 τ 1 r by definition of τ 1 .
(b) If g(r) ≤ r 3/2 , then g(τ 1 r) ≤ g(r) ≤ r 3/2 ≤ ε 0 τ 1 r by definition of r. Assume now that (7.33) holds. Notice that by definition of G we have (since τ 0 < 1)

H 1 (J ∩ Q τ k 0 τ 1 r ) ≤ g(τ k 0 τ 1 r) ≤ ε 0 τ 3 2 k 0 τ 1 r ≤ ε 0 τ k 0 τ 1 r
, so the decay property of Proposition 7.7 may be applied. Again we have two alternatives. In both cases, (7.33) follows for the choice k + 1, so that the induction step is proved.

If we want to draw some conclusions on the regularity of quasi minimizers (J, u), we need somehow to bound the freedom connected to the choice of J: notice indeed that any pair (J∆N, u) with H 1 (N ) = 0 is essentially equivalent to (J, u), where A∆B denotes the symmetric difference of sets.

We set (7.34)

J + := x ∈ Ω : lim sup r→0 H 1 (J ∩ Q x,r ) r > 0 .
J + is a sort of normalized version of J, where points of density zero have been erased. By standard properties of rectifiable sets we have

H 1 (J∆J + ) = 0.
As a consequence if (J, u) ∈ V(Ω), then also (J + , u) ∈ V(Ω) with G(J, u, A) = G(J + , u, A) for every Borel set A ⊆ Ω.

Proposition 7.10. Given Λ > 0, there exist ε, r > 0 such that of any (Λ, r)-quasi minimizer (J, u) of G on V(Ω), if G(J, u, Q x,r ) ≤ εr for some Q x,r Ω with r < r, then J + ∩ Q x, r 2 = ∅. Proof. Let ε 0 , ε 1 , τ 1 , τ 2 , r be given according to Lemma 7.9. Notice that if G(J, u, Q x,r ) ≤ ε 1 r with r < r, then for any ρ ∈ (0, r) Let us set ε := 1 2 ε 1 , and assume G(J, u, Q x,r ) ≤ εr. Notice that for any y ∈ Q x, r 2 , we have G(J, u, Q y, r

2 ) ≤ G(J, u, Q x,r ) ≤ εr = ε 1 r 2 so that from (7.35)

0 = lim ρ→0 + G(J, u, Q y,ρ ) ρ ≥ lim sup ρ→0 + H 1 (J ∩ Q y,ρ ) ρ , which yields J + ∩ Q x, r 2 = ∅.
Proposition 7.11 (Regularity for quasi minimizers). Let Λ, r > 0. Then for any (Λ, r)-quasi minimizer (J, u) of G on V(Ω) we have that J + (see (7.34)) is essentially closed in Ω, i.e., H 1 Ω ∩ (J + \ J + ) = 0, while u ∈ C ∞ (Ω \ J + ).

Proof. Since the functional G coincides with a volume integral outside J, there exists a H 1 -negligible set N ⊂ Ω \ J such that for every x ∈ Ω \ (J ∪ N ) we have lim ρ→0 G(J, u, Q x,ρ ) ρ = 0.

Thanks to Proposition 7.10 we infer Ω ∩ J + ⊂ J ∪ N ⊂ J + ∪ (J \ J + ) ∪ N.

Since the last two sets are H 1 -negligible, we infer H 1 (Ω ∩ (J + \ J + )) = 0. Since H 1 (J u \ J + ) ≤ H 1 (J \ J + ) = 0, we get that u is locally H 1 on Ω \ J + (thanks to Korn's inequality): smoothness then follows from the regularity theory for solutions to Stokes equation (see e.g. [6, Theorem IV.5.8]).

7.3. Proof of Theorem 4.10. We are now in a position to prove the regularity result given by Theorem 4.10.

Let (E, u) be a minimizer of J and let us set Λ := 4Lip(f ) and J := J u ∪ ∂ * E.

We also assume (up to multiplying u by c -1 2 ) that the constant c of (4.2) is 1. We first prove that (J, u) is a (Λ, 1) quasi minimizer of the Griffith functional G on V(Ω) according to Definition 7.4. Indeed, let Q x,r Ω with r < 1 be a square as in Definition 7.4, with associated competitor (J \ Q x,r , v). We claim that either (7.36) H 1 (∂Q x,r \ E (1) ) = 0 or H 1 (∂Q x,r \ E (0) ) = 0.

In the first case, from the minimality inequality J (E, u) ≤ J (E, u1 Ω\Qx,r )

we deduce u = 0 a.e. on Q x,r and H 1 (∂ * E ∩ Q x,r ) = 0, so that the inequality to check for quasi minimality is trivially satisfied. Notice that admissibility of (E, u1 Ω\Qx,r ) for the main problem follows from the fact that the trace of u on ∂Q x,r is zero, being that boundary composed of points of density one of the set E on which u vanishes.

If the second possibility in (7.36) holds true, then the relations (see [START_REF] Maggi | Sets of finite perimeter and geometric variational problems[END_REF]Theorem 16.3] and recall that H 0 (∂Q x,r ∩ ∂ * E) = 0 by the properties of Q x,r ) so that the quasi minimality of (J, u) follows. By Proposition 7.11, we get that the normalized set J + (see (7.34)) is essentially closed in Ω, i.e., H 1 (Ω ∩ (J + \ J + )) = 0, and u is smooth on Ω \ J + , so that Ω ∩ J u ⊆ J + .

J (E, u) ≤ J (E \ Q x,r , v) and ∂ * (E \ Q x,r ) = ∂ * E \ Q x,r = ∂ * E \ Q x,
On the other hand, in view of the general properties of the reduced boundary of sets of finite perimeter (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]Theorem 3.59] or [START_REF] Maggi | Sets of finite perimeter and geometric variational problems[END_REF]Theorem 15.5]) we have ∂ * E ⊆ J + . Taking into account that H 1 (J + ∆J) = 0 (where ∆ denotes the symmetric difference of sets) we infer

H 1 (Ω ∩ J u ∪ ∂ * E \ (J u ∪ ∂ * E)
) ≤ H 1 (Ω ∩ J + \ J) ≤ H 1 (Ω ∩ J + \ J + ) + H 1 (J + \ J) = 0 so that the conclusion follows.

In order to complete the proof, we need to check claim (7.36).

Assume by contradiction that the claim is false. Then there exists p ∈ E (1) ∩∂Q x,r and q ∈ E (0) ∩∂Q x,r that are not in one of the corners. Without loss of generality we suppose p, q ∈ {x -re 2 + Re 1 } with p 1 < q 1 , the case when both are in different sides being analog. We let for s > 0 small • C p := p + [-s, 0] × [0, s] and C q := q + [0, s] 2 ,

• g s : [p 1 -s, q 1 + s] → [0, 1] be zero at the extremes, affine on [p 1 -s, p 1 ] and [q 1 , q 1 + s] and equal to 1 on [p 1 , q 1 ], • f s ∈ C 1 c (]0, s[) with 0 ≤ f s ≤ 1, • ϕ s (x) = g s (x 1 )f s (x 2 + r). Then

H 1 (J ∩ (Q x,r \ Q x,r-s )) ≥ ∂ * E ϕ s (ν E ) 1 dH 1 = E ∂ 1 ϕ s dH 1 =
1 s E∩Cp f s (y 2 + r)dy -1 s E∩Cq f s (y 2 + r)dy so that, letting f s 1 we get

H 1 (J ∩ (Q x,r \ Q x,r-s )) s ≥ |E ∩ C p | |C p | - |E ∩ C q | |C q | .
Since as s → 0 + , by assumption on the density properties of p and q, we have

|E ∩ C p | |C p | → 1 and |E ∩ C q | |C q | → 0, we infer lim sup s→0 H 1 (J ∩ (Q x,r \ Q x,r-s )) s ≥ 1
which is against the assumption on r in Definition 7.4 of quasi minimality. The proof is thus concluded.

7.4. Some remarks on a "strong" formulation of the problem. In this section we elaborate on a different relaxation of the drag minimization problem which involves topologically closed (but not necessarily regular) obstacles F in the channel Ω and velocity vector fields which are H 1 loc on Ω \ F . Within this perspective, given Ω ⊂ R d open and bounded, it is natural to start with pairs (F, u) such that (7.37) F ⊆ Ω is relatively closed, Ω ∩ ∂F is rectifiable, H d-1 (Ω ∩ ∂F ) < +∞ and (7.38) u ∈ H 1 loc (Ω \ F ; R d ), div u = 0 in Ω \ F , e(u) ∈ L 2 (Ω \ F ; M d×d sym ). Notice that, as for the relaxation studied in the previous sections, ∂F may contain "lower dimensional" parts. The set Ω \ F is open, so that the space H 1 loc (Ω \ F ; R d ) is well defined. It is not clear how to talk about traces on ∂(Ω\F ), which are fundamental to formulate the tangency constraint, as the set is in general not regular. It turns out that velocities admit a well defined trace on H d-1 almost every point ∂F even if this set is not assumed to be only rectifiable and not regular. This is a consequence of the following result which involves the space GSBD of Generalised Functions of Bounded Deformations introduced in [START_REF] Dal | Generalised functions of bounded deformation[END_REF]. Let us set (7.39) ũ := u in Ω \ F 0 in F.

Lemma 7.12. Let Ω ⊆ R d be a bounded open set, and assume that the pair (F, u) satisfies (7.37) and (7.38). Then ũ ∈ GSBD(Ω) with H d-1 (J ũ \ ∂F ) = 0.

Theorem 2 . 2 (

 22 One dimensional sections of SBD-functions). Let Ω ⊆ R d be open, ξ ∈ R d with |ξ| = 1 and let u ∈ SBD(Ω). Then for H d-1 -a.e. y ∈ Ω ξ we have ûξ y ∈ SBV (Ω ξ y ) with (û ξ y ) (t) = (e(u)ξ • ξ)(y + tξ) for a.e. t ∈ Ω

3. 1 .

 1 The flow around the obstacle. Let Ω ⊂ R d be an open bounded set with Lipschitz boundary, and let V ∈ C 1 (R d ; R d ) be a divergence free vector field. Given E Ω open and with a Lipschitz boundary, let us consider the stationary flow for a viscous incompressible fluid around E with boundary conditions on ∂Ω given by V , and with Navier boundary conditions on ∂E. More precisely, if u : Ω\E → R d is the velocity field, we require that the following items hold true. (a) Incompressibility: div u = 0 in Ω \ E. (b) Boundary conditions: we have u = V on ∂Ω and the non-penetration condition u • ν = 0 on ∂E, where ν denotes the exterior normal to E. (c) Equilibrium: considering the stress (3.1) σ := -pI d + 2µe(u),

  (a) The flow is divergence free: div u = 0 in the sense of distributions in Ω . (b) External boundary conditions: u = V a.e. on Ω \ Ω. (c) Non-penetration condition on the obstacle:

Remark 4 . 6 .

 46 Let E Ω be open and with a Lipschitz boundary. Then we can find

Theorem 4 . 10 (

 410 Regularity in dimension two). Let Ω ⊆ R 2 be a bounded open set with Lipschitz boundary, V ∈ C 1 (R 2 ; R 2 ) a divergence-free vector field, and f : [0, |Ω|] → [0, +∞[ a Lipschitz function.

5. 1 .

 1 An immersion result. The following embedding result holds true. Theorem 5.1. Let Ω ⊆ R d be a bounded open set, and let u ∈ SBD(R d ) be supported in Ω such that

y

  ] L 1 → 0 uniformly for y ∈ H - ε , and

  which proves the initialisation.Assume now that the result is true for d and let us check it for d + 1consider N × M sets that we classify into N groups of M sets, written (E k,i ) 1≤k≤N,1≤i≤M .

2 ε . Remark 5 . 6 .

 256 Let us detail the blow up argument used in the proof of Theorem 5.4. If we set

1 2 , 1 2

 121 the mismatch does not affect the validity of the conclusion. For r ∈]1 -δ , 1[ and each k ≥ -2, let us set

  j ρ δ k * e(u) = k,j ϕ k,j e(w k,j ) -ρ δ k * e(u) + k,j ∇ϕ k,j w k,j .

Lemma 7 . 3 . 2 .

 732 Let Ω be a bounded, connected open set with Lipschitz boundary, and let L 2 0 (Ω) be the set of zero-average L 2 -functions. Then there is a continuous linear map Φ :L 2 0 (Ω) → H 1 0 (Ω; R d ) such that div • Φ = Id L 2 0 (Ω) . 7.Regularity for quasi minimizers of the Griffith energy. Let Ω ⊆ R 2 be an open set. In all the following, we will consider the Griffith functionalG(J, u, B) := B |e(u)| 2 dx + H 1 (J ∩ B),where B ⊆ Ω is a Borel set.

  (a) If g(τ k 0 τ 1 r) > (τ k 0 τ 1 r) 3/2, by the decay property we have, using (7.33), If g(τ k 0 τ 1 r) ≤ (τ k 0 τ 1 r) 3/2 then by the definition of r

( 7 . 35 )

 735 G(J, u, Q x,ρ ) ≤ C 0 r -

  r yield in particular Qx,r |e(u)| 2 dx + H 1 (J ∩ Q x,r ) ≤ Qx,r |e(v)| 2 dx + Λr 2 ,

  Theorem 2.1. Let Ω ⊆ R d be open, bounded and with a Lipschitz boundary, and let (u n ) n∈N be a sequence in SBD(Ω) such that sup Then there exists u ∈ SBD(Ω) and a subsequence (u n k ) k∈N such that

	u n k → u	strongly in L 1 (Ω; R d ),
	e(u n k )	e(u)	weakly in L p (Ω; M d×d sym ),
	and
	H d-1 (J u ) ≤ lim inf k→+∞	H d-1 (J un k ).
	We will need also some properties of the sections of SBD-functions. If Ω ⊆ R d is open and u ∈
	SBD(Ω), let us consider the scalar function on Ω ξ y given by
	(2.3)	ûξ y (t) := u(y + tξ) • ξ
	and the set
	(2.4)

n |Eu n |(Ω) + u n L 1 (Ω;R d ) + e(u n ) L p (Ω;M d×d sym ) + H d-1 (J un ) < +∞

for some p > 1.

D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON

We can thus write, in view of the continuity of φ

where ηn → 0, so that, taking into account (5.26) and (5.34)

The conclusion follows by letting ε → 0.

In the proof of Theorem 5.4 we made use of the following abstract lemma.

Proof. Since H d-1 (Ω ∩ ∂F ) < ∞, for every ε > 0 we may find some covering of ∂F through a finite union of balls of radius less than ε, denoted (B ε i ) 1≤i≤N ε , such that

for some C > 0 that does not depend on ε. Let B ε be the union of these balls -which is a Lipschitz set up to a small perturbation of the radii -and let u ε := u1 Ω\B ε . Then u ε ∈ SBD(Ω) with

We apply [START_REF] Chambolle | Compactness and lower semicontinuity in GSBD[END_REF]Theorem 1.1] to (u ε ): since ũ is finite almost everywhere, we directly identify ũ with the limit that is obtained, and we infer ũ ∈ GSBD(Ω); moreover up to a H d-1 -negligible set, J ũ ⊂ ∂F by construction, and the result follows.

Coming back to configurations (F, u) satisfying (7.37) and (7.38), up to a choice of orientation of the rectifiable set Ω ∩ ∂F , there is no ambiguity in defining the traces u ± |∂F of u on H d-1 -almost all points of Ω ∩ ∂F .

In addition to the previous items, we thus require also for (F, u) the non-penetration condition

Given an admissible configuration (F, u), we can consider the following energy (all the constants have been normalized to 1)

where ∂ e F denotes the measure theoretical boundary of F , and f is the penalization function introduced in the previous sections (see (4.1)).

Configuration with finite energy are linked to admissible configurations of our main relaxed problem by the following result. Lemma 7.13. Let Ω ⊆ R d be open and bounded, and let (F, u) satisfy (7.37) and (7.38) with J(F, u) < +∞. Then the function ũ defined in (7.39) is such that ũ ∈ SBD(Ω).

Proof. It suffices to note that for every direction ξ ∈ S d-1 we have

Dealing with boundary conditions yields to the same problem highlighted in our main relaxation. Assume Ω has a Lipschitz boundary, and let us write simply u in place of ũ. We have that u ∈ SBD(Ω) so that the trace on ∂Ω is well defined. Given a divergence free vector field V ∈ C 1 (R d ; R d ), we can deal with the relaxation of the boundary condition by considering the set Γ u,V := {x ∈ ∂Ω : u(x) = V (x)}, and enforcing the non-penetration constraint leading to (7.41) u • ν ∂Ω = 0 and V • ν ∂Ω = 0 H d-1 -a.e. on Γ V,∂Ω .

So for a configuration (F, u) satisfying (7.37), (7.38), (7.40) and (7.41), we can consider the energy

The minimization of J strong on admissible configurations is a different possible relaxation of the original drag minimization problem. We clearly have min

Equality is reached in dimension two thanks to the regularity result given by Theorem 4.10. Indeed, if (E, u) is a minimizer for J , we know that ∂ * E ∪ J u is essentially closed, so that an admissible relatively closed set F arises by considering the complement of the union of the connected components of Ω \ ∂ * E ∪ J u on which u does not vanish identically. The function u is smooth outside F , so that the pair (F, u) is strongly admissible with J strong (F, u) = J (E, u). As a consequence, in dimension two the relaxed problem min (F,u)

is indeed well posed.