Jean-François Grosjean

Antoine Lemenant

Rémy Mougenot

STABLE DOMAINS FOR HIGHER ORDER ELLIPTIC OPERATORS by

Keywords: Capacity, stable, γ m -convergence, shape optimisation

This paper is devoted to prove that any domain satisfying a (δ 0 , r 0 )-capacitary condition of first order is automatically (m, p)-stable for all m ⩾ 1 and p > 1, and for any dimension N ⩾ 1. In particular, this includes regular enough domains such as C 1 -domains, Lipchitz domains, Reifenberg flat domains, but is sufficiently weak to also include cusp points. Our result extends some of the results of Hayouni and Pierre valid only for N = 2, 3, and partially extends the results of Bucur and Zolésio for higher order operators, with a different and simpler proof.

Introduction

Let Ω ⊂ R N be a bounded and open set. Following [START_REF] Henrot | Variation et optimisation de formes. Une analyse géométrique[END_REF][START_REF] Bucur | Variational methods in shape optimization problems[END_REF][START_REF] Bucur | N -dimensional shape optimization under capacitary constraint[END_REF], we say that Ω is (m, p)-stable if W m,p (R N ) ∩ u = 0 a.e. in Ω c = W m,p 0 (Ω).

This notion is related to the continuity of a 2m-order elliptic PDE with respect to domain perturbation. In particular, if Ω is (m, 2)-stable, then it implies that for any sequence of domains (Ω n ) n∈N converging to Ω in a certain Hausdorff sense, one has that (u n ) n∈N converges strongly in H m to u, where u n is the unique solution in H m 0 (Ω n ) for the equation (-∆) m (u n ) = f in Ω n , and u is the solution of the same problem in Ω. It is also equivalent to the convergence of (W m,p 0 (Ω n )) n∈N to W m,p 0 (Ω) in the sense of Mosco (see Section 6). In the literature, a lot of attention has been devoted to the case m = 1 and p = 2 because of its relation to the Laplace operator. On the other hand, very few results are available for the higher order spaces H m 0 (Ω), related to bi-harmonic or more generally poly-harmonic equations, that have a lot of applications. The objective of this paper is to give a short and elementary proof of the fact that any domain which is "regular enough" is always (m, p)-stable for all m, p and all dimensions N .

Notice that in general, the stability for W m,p 0 (Ω) does not simply reduce to the one of W 1,p 0 (Ω). To enlight this fact we recall that for every open set Ω ⊂ R N , we have the characterisation (see for instance [START_REF] Adams | Function spaces and potential theory[END_REF]Chapter 9]) W m,p 0 (Ω) = W m,p (R N ) ∩ ∇ k u| Ω c = 0 (m -k, p) -q.e. for all k ⩽ m -1 , where ∇ k u := (∂ α u) |α|=k and ∂ α u is the (m -k, p)-quasicontinuous representative, which is in particular defined pointwise (m -k, p)-q.e. If Ω is (1, p)-stable, then for any |α| ⩽ m -1 and from the assumption ∂ α u = 0 a.e. in Ω c we would only deduce that ∂ α u = 0 (1, p)-q.e. on Ω c , whereas in order to prove that u ∈ W m,p 0 (Ω) we would need the stronger condition ∂ α u = 0 (m -|α|, p)-q.e. on Ω c . In [START_REF] Hayouni | Domain continuity for an elliptic operator of fourth order[END_REF], Hayouni and Pierre exploited the compact embedding of H 2 into continuous functions in dimensions 2 and 3, in order to get some stability results for the space H 2 0 . In particular, they proved that, in dimension 2 and 3, any (1, 2)-stable domain is automatically (2, 2)-stable (see [START_REF] Hayouni | Domain continuity for an elliptic operator of fourth order[END_REF] or [START_REF] Henrot | Variation et optimisation de formes. Une analyse géométrique[END_REF]). They also proved in the same paper that, in dimensions 2 and 3, any sufficiently smooth domain is a (2, 2)-stable domain.

In the present paper, we show that there is no true restriction on the dimension N to obtain (m, p)-stability. Our main result asserts that any domain that satisfies a variant of the classical (1, p)-capacitary condition, will be automatically (m, p)-stable, in any dimension, and for any m. This includes a large class or "regular" domains such as C 1 -domains, Lipschitz domains, Reifenberg-flat domains, domains satisfying the socalled external corkscrew condition (see Definition 5.1), ε-cone property, or even domains with the segment property which allows domains with cusps, or more generally domains with the so called flat cone property [START_REF] Bucur | Flat cone condition and shape analysis[END_REF].

In the sequel, we restrict ourselves to open subset of a fixed ball D ⊂ R N , and we denote the set of admissible domains by

O(D) := {Ω | Ω ⊆ D is open} .
In the following definition, the precise definition of capacity has no real importance since they are most often equivalent up to a constant. In this paper we will work with the Bessel capacity, that will be precisely defined in the next section.

Definition 1.1. -Let r 0 > 0, δ 0 > 0 and p ∈ (1, +∞). An open set Ω ⊆ R N satisfies the (r 0 , δ 0 )-capacitary condition if for all x ∈ ∂Ω and for all r ⩽ r 0 ,

Cap 1,p 1 r Ω c ∩ B(x, r) -x ⩾ δ 0 , (1.1)
where Cap 1,p is the Bessel capacity of first order. The class of open subset of D having the (r 0 , δ 0 )-capacity condition is denoted by O δ 0 ,r 0 cap (D).

Here is our main statement. Let us provide some comments about the result. One of the main feature and somewhat surprising is that the condition involves only the (1, p)-capacity even if the conclusion yields (m, p)-stability for all m ⩾ 1.

Our main statement reminds the classical one in [START_REF] Bucur | N -dimensional shape optimization under capacitary constraint[END_REF], where Bucur and Zolésio proved that a domain is (1, 2)-stable under a weaker condition with (1, 2)-capacity. More precisely, in [START_REF] Bucur | N -dimensional shape optimization under capacitary constraint[END_REF] the authors prove that the following is enough

∀x ∈ ∂Ω, ∀r ≤ r 0 , Cap 1,2 (Ω c ∩ B(x, r), B(x, 2r)) Cap 1,2 (B(x, r), B(x, 2r)) ⩾ δ 0 . (1.2)
In (1.2), the notation Cap 1,2 (A ∩ B(x, r), B(x, 2r)) refers to the so called "condenser capacity" (see Section 4 for a definition). Actually, it can be proved (see Remark 4.1 in Section 4) that (1.2) is equivalent to

∀x ∈ ∂Ω, ∀r ≤ r 0 , Cap 1,2 1 r (Ω c ∩ B(x, r) -x) ⩾ δ 0 . (1.3)
Observe that (1.3) looks similar to (1.1) but without a bar over Ω, which stands for a substantial difference. Since (1.1) clearly implies (1.3), the main result of [START_REF] Bucur | N -dimensional shape optimization under capacitary constraint[END_REF] is stronger than ours.

The typical example of domain that satisfies (1.3) but not (1.1) is a domain with a crack. For instance the unit ball of R 2 minus a radius, i.e. B(0, 1) \ ([0, 1] × {0}). The capacity of Ω c in a small ball B(x, r) centered at the boundary has a positive (1, 2)-capacity, because it always contains a segment of length at least a radius, so (1.2) is satisfied. On the other hand for a small ball centered on the crack, the complement of Ω in the ball is empty, thus our condition (1.1) is not satisfied.

In other words for the special case of m = 1 and p = 2 we only partially recover the result in [START_REF] Bucur | N -dimensional shape optimization under capacitary constraint[END_REF]. In contrast, under the similar and slightly stronger (1, 2)-capacitary condition (1.1), we obtain (m, 2)-stability for all m ⩾ 1, which is interesting. It is worth mentioning that our proof is different and simpler than the one [START_REF] Bucur | N -dimensional shape optimization under capacitary constraint[END_REF], thus provides an alternative argument for many standard classes of domains (such as Lipschitz domains or domain satisfying the uniform flat cone property, or corkscrew property) which is new even for the standard case m = 1, and works similarly for p ̸ = 2.

Let us further emphasis that the difference between Ω and Ω has a crucial importance. Indeed, under our assumption |∂Ω| = 0, the question of (m, p)-stability is equivalent to asking whether

W m,p (R N ) ∩ {u = 0 a.e. in Ω c } = W m,p 0 (Ω).
For instance for m = 1 and p = 2, as already mentioned before, it is known that

H 1 0 (Ω) = H 1 (R N ) ∩ {u = 0 q.e. in Ω c } ,
thus the main question for stability is whether {u = 0 a.e in Ω c } ⇒ {u = 0 q.e in Ω c }.

If we add a bar on Ω then the simliar question becomes trivial because Ω c is open and therefore we always have, for a precise representative u ∈ H 1 (R N ), {u = 0 a.e in Ω c } ⇔ {u = 0 q.e in Ω c }.

This fact will play a major role in the proof of our main result. Another fact that we use in our proof is the following, valid for any u ∈ W m,p (R N ),

{u = 0 a.e in Ω c } ⇒ {∇ k u = 0 a.e. in Ω c for all k ⩽ m}.
Again, this follows from the fact that Ω c is open and explains why we need a bar over Ω in the capacitary condition (1.1) for our proof to work. Another difference with [START_REF] Bucur | N -dimensional shape optimization under capacitary constraint[END_REF] is the assumption |∂Ω| = 0 that we need in our main statement Theorem 1.1 (here |∂Ω| denotes the Lebesgue measure of ∂Ω). In practice this assumption is not very restrictive since it will be easily satisfied by all standard classes of domains. For instance it holds true as soon as a corckscrew condition is satisfied (see Proposition 5.1). We do not know whether the capacity condition (1.1) directly implies |∂Ω| = 0, in which case this assumption would be redundant.

As a consequence of our main result we get a capacity condition which implies stability for the polyharmonic equation along a Hausdorff converging sequence of domains. We refer to Section 6 for the definition of Hausdorff convergence, Mosco convergence and γ m -convergence, and we give here in the introduction two different statements. In the first one (Corollary 1.1) we assume only the limiting domain Ω to be "regular" while in the second (Theorem 1.2) we assume the whole sequence to be "regular". -→ Ω to obtain the γ m -convergence of a sequence (Ω n ) n∈N . On the other hand they assumed that every term Ω n along the sequence satisfies a capacitary condition with uniform constants. A similar statement will be given later in Theorem 1.2.

Corollary 1.1. -Let Ω ∈ O δ 0 ,
3. It is easy to construct an example of stable domain Ω (even smooth) and a sequence

(Ω n ) n∈N such that Ω n d H c
-→ Ω and (Ω n ) n∈N does not γ m -converges to Ω. This shows that without any other assumption on the sequence, the second assumption Ω n d H -→ Ω is pivotal for the result to hold true. The construction is rather classical : consider the sequence made from an enumeration x i ∈ B(0, 1) of points with rational coordinates. Then define

Ω n := B(0, 2) \ n i=0 {x i }.
It is easy to see that (Ω n ) n∈N converges to Ω := B(0, 2) \ B(0, 1) for the complementary Hausdorff distance, which is clearly a (m, 2)-stable domain because the boundary is smooth. On the other hand, for dimension N ⩾ 2m we know that Cap m,2 ({x i }) = 0, so it is classical that (Ω n ) n∈N does not γ m -converge to Ω (see [10, Section 3.2.6, page 80] for the case m = 1). On the other hand Ω n = B(0, 2) clearly does not Hausdorff converge to Ω = B(0, 2) \ B(0, 1), which explains why Theorem 6.3 does not apply.

Next, in order to get existence of shape optimisation problems for higher order equations under geometrical constraints, the following variant is more usefull. Notice that here we suppose (1.1) on the whole sequence and by this way we can avoid the assumption

Ω n d H -→ Ω but Ω n d H c -→ Ω suffices. Theorem 1.2. -Let Ω ∈ O(D) and (Ω n ) n∈N all belonging to O δ 0 ,r 0 cap (D). If |∂Ω| = 0 and (Ω n ) n∈N d H c -converges to Ω, then (Ω n ) n∈N γ m -converges to Ω, or equivalently, (H m 0 (Ω n )) n∈N converges to H m 0 (Ω) in the sense of Mosco.
Since complementary Hausdorff topology is relatively compact, it is easy to get existence results for shape optimisation problems using Theorem 1.2, with additional geometrical constraints on the domain. This applies to various standard classes of domains such as uniformly Lipschitz domains, Reifenberg-flat, corkscrew, or ε-cone, as described in the last section of the paper (see Theorem 8.1).

Preliminaries

The term domain and the symbol Ω will be reserved for an open and bounded set in the N -dimensional euclidean space R N . We will denote the Lebesgue measure of a set

A ⊂ R N by |A|. The norm of a point x ∈ R N is denoted by |x| := ( N i=1 x 2 i ) 1/2 . If α is a multi-indice, i.e. α ∈ N N , then the norm of α is |α| := N
i=1 α i and we define the partial derivative operator

∂ α := ∂ |α| ∂ α 1 1 • • • ∂ α k N ,
and the vector ∇ k := (∂ α ) |α|=k . The notations ∂Ω and Ω stand for the boundary and the closure of Ω, respectively. Let C ∞ c (Ω) be the space of smooth functions with compact support in Ω. The ball of radius r ⩾ 0 and centered at x ∈ R N is denoted by B(x, r). For m ∈ N and p ∈ (1, +∞), we consider the usual Sobolev space W m,p (Ω) endowed with the norm

∥u∥ W m,p (Ω) := m k=0 ∥∇ k u∥ p L p (Ω) 1/p , where ∥∇ k u∥ p L p (Ω) := Ω |∇ k u| p dx.
Finally, the space W m,p 0

(Ω) is the completion of C ∞ c (Ω) with respect to the norm ∥ • ∥ W m,p (Ω) .
When the dimension N < mp, elements of W m,p (R N ) can be represented as continuous functions. However, if N ⩾ mp, this is no longer the case and the natural way of measuring by how much the functions deviate from continuity is by means of capacity.

In this paper we will work with the Bessel capacity defined for instance in [13, Chap-

ter 2.6]. If K ⊂ R N is any set, then we define the (m, p)-capacity of K by Cap m,p (K) := inf ∥f ∥ p p f ≥ 0 and g m * f ⩾ 1 on K , (2.1)
where g m with m ≥ 1 is the Bessel kernel, defined as being the function whose Fourier transform is ĝα (x) = (2π) -N/2 (1+|x| 2 ) -m/2 . We refer to [START_REF] William | Weakly differentiable functions. Sobolev spaces and functions of bounded variation[END_REF]Chapter 2.6] for more details and several basic properties of the Bessel Capacity.

In this paper, while considering a function u ∈ W 1,p (R N ), we will always tacitly mean that u is a quasicontinuous representative, without mentioning it explicitly (see for instance [START_REF] Adams | Function spaces and potential theory[END_REF] for a definition).

The proof of the main result will use the following Poincaré type inequality that can be found for instance in [

] Let r > 0, p ∈ (1, +∞) and u ∈ W 1,p (B(0, r)). We define Z(u) := {x ∈ B(0, r) | u(x) = 0}. If Cap 1,p (Z(u)) > 0, then B(0,r) |u| p dx ⩽ C r p Cap 1,p (r -1 Z(u)) B(0,r) |∇u| p dx, (2.2)
where C > 0 depends only on p and N .

Proof of Theorem 1.1

Proof of Theorem 1.1. -Let Ω be a bounded domain satisfying the assumptions of Theorem 1.1 and let u ∈ W m,p (R N ) be given satisfying u = 0 almost everywhere in Ω c . To prove the theorem it suffice to prove that u can be approximated in the W m,p (R N ) norm by a sequence of functions in C ∞ c (Ω). To do so we will first truncate u near the boundary of Ω as follows. For all n ∈ N, we consider

K n := x ∈ Ω d(x, ∂Ω) ⩾ 2 -n . The exhaustive family of compact (K n ) n∈N satisfies K n ⊆ K n+1 and Ω = n∈N K n . Then take a test function ρ ∈ C ∞ c (B(0, 1)) such that ρ ⩾ 0 and R N ρ(x)dx = 1.
We define ρ ε (x) := ε -N ρ(x/ε) and

θ n,ε (x) := 1 Kn * ρ ε (x) = ε -N Kn ρ x -y ε dy,
which satisfies Supp (θ n,ε ) ⊆ K n + B(0, ε). We take ε n := 2 n+1 and denote now θ n := θ n,εn so that

θ n ∈ C ∞ c (Ω), θ n = 1 on K n-1 , θ n = 0 on K c n+1 , Supp (∇ k θ n ) ⊆ K n+1 \ Int(K n-1 ).
To prove the theorem it suffices to prove that

u n := uθ n -----→ n-→+∞ u in W m,p (R N ),
because then we can conclude by using the density of C ∞ c (Ω) into W m,p (Int(K n+2 )), and a diagonal argument. Let k ⩽ m be a positive integer. To prove the claim we first estimate the L p norm :

∥u n -u∥ p L p (R N ) ⩽ Ω\K n-1 |u| p dx.
Using the fact that (Ω\K n ) n∈N is a decreasing sequence of Lebesgue measurable sets, and thanks to the condition |Ω \ Ω| = 0, we know that |Ω\K n | -→ 0 as n -→ +∞ and therefore

u n -→ u in L p (R N ).
Next for the norm of gradients we will use a covering of ∂Ω. More precisely, the infinite family (B(x, 2 -(n-2) )) x∈∂Ω is a cover of Supp (∇ k θ n ) and by the famous 5B -covering lemma (see for instance [2, Theorem 2.2.3]) there exists a countably subcover indexed by

(x i ) i∈N ⊆ ∂Ω such that (B(x i , 2 -(n-2) )) i∈N is a disjoint family, Supp (∇ k θ n ) ⊆ i∈N B(x i , 5 • 2 -(n-2) ), and i∈N 1 B(x i ,5•2 -(n-2) ) ⩽ N 0 ,
for a universal constant N 0 ∈ N. In the sequel, we simply write

B n (x i ) instead of B(x i , 5 • 2 -(n-2)
). Afterwards, we estimate

∥∇ k u n -∇ k u∥ p L p (R N ) ⩽ C Ω\K n-1 |∇ k u| p dx + C k=|β|+|γ| γ̸ =0 Ω\K n-1 |∂ β u| p |∂ γ θ n | p dx.
The first term converges to 0 as n -→ +∞ for the same reasons as before. For the other term we use the following estimate

|∂ γ θ n (x)| p ⩽ ε -pN n Kn ε -p|γ| n ∂ γ ρ x -y ε n p dy ⩽ Cε -p|γ| n .
The function u vanishes almost everywhere on the open set Ω c , so

∂ β u is zero in D ′ (Ω c )
and vanishes almost everywhere on this open set. Hence the Poincaré inequality (2.2) applies to all the ∂ β u for |β| < m, and for all ball B n (x i ) such that 2 -(n-2) ⩽ r 0 , thanks to our capacitary condition (1.1) we get

Cap 1,p (5 -1 • 2 n-2 (Z(∂ β u) -x i )) ⩾ Cap 1,p (5 -1 • 2 n-2 (Ω c ∩ B(x i , 5 • 2 2-n ) -x i )) ⩾ δ 0 .
Therefore,

Bn(x i ) |∂ β u| p dx ⩽ Cδ -1 0 ε p n Bn(x i )
|∇∂ β u| p dx, (3.1) and using successively (k -|β|)-times the Poincaré inequality and the covering of ∂Ω, we get

Ω\K n-1 |∂ β u| p |∂ γ θ n | p dx ⩽ Cε -p|γ| n Ω\K n-1 |∂ β u| p dx ⩽ Cε -p|γ| n i∈N Bn(x i ) |∂ β u| p dx ⩽ C i∈N Bn(x i ) |∇ k u| p dx ⩽ CN 0 Ω\K n-5
|∇ k u| p dx and this tends to zero as n -→ +∞ so follows the proof.

Equivalent condition with condencer capacity

In this section we give an equivalent condition to (1.1) with the notion of "condenser capacity" which allows us to compare it with the condition in [START_REF] Bucur | N -dimensional shape optimization under capacitary constraint[END_REF]. More precisely, in [START_REF] Bucur | N -dimensional shape optimization under capacitary constraint[END_REF] the condition involves the condenser capacity defined for any compact subset K ⊂ R N and p ∈ (1, +∞) by

|∇φ| p dx | φ ∈ C ∞ 0 (B(x, 2r)), φ ≥ 1 on K ∩ B(x, r) .
This notion can then be extended to arbitrary sets K by approximation (see for instance [1, Definition 2.2.4]). The difference with the Bessel capacity defined in (2.1) can be seen through the following well known equivalent definition of Bessel Capacity (See for instance [9, Section 2]), for a closed set K,

Cap 1,p (K) = inf R N |φ| p + |∇φ| p dx | φ ∈ C ∞ 0 (R N ), φ ≥ 1 on K .
The next proposition says in particular that our condition implies the one in [START_REF] Bucur | N -dimensional shape optimization under capacitary constraint[END_REF].

Proposition 4.1. -Let Ω ⊂ R N be open. Then the condition ∃δ 0 , r 1 > 0 s.t. ∀x ∈ ∂Ω, ∀r ≤ r 1 , Cap 1,p 1 r Ω c ∩ B(x, r) -x ⩾ δ 0 , (4.1)
is equivalent to the following one

∃δ 0 , r 1 > 0 s.t. ∀x ∈ ∂Ω, ∀r ≤ r 1 , Cap 1,p (Ω c ∩ B(x, r), B(x, 2r)) Cap 1,p (B(x, r), B(x, 2r)) ⩾ δ 0 . (4.2)
Proof. -The condenser capacity enjoys a nice scaling property. Assume for simplicity and without loss of generality, that x = 0. It is easy to prove by a simple change of variables that for all λ > 0, Cap 1,p ((λK) ∩ B(0, λr), B(0, 2λr)) = λ N -p Cap 1,p (K ∩ B(0, r), B(0, 2r)). (4.3)

In particular, Cap 1,p (B(0, r), B(0, 2r)) = C 0 r N -p , with C 0 := Cap 1,p (B(0, 1), B(0, 2)), and it follows that Cap 1,p (Ω c ∩ B(0, r), B(0, 2r))

Cap 1,p (B(0, r), B(0, 2r)) = r N -p Cap 1,p ( 1 r Ω c ) ∩ B(0, 1), B(0, 2)
Cap 1,p (B(0, r), B(0, 2r)) .

= 1 C 0 Cap 1,p ( 1 r Ω c ) ∩ B(0, 1), B(0, 2) .
Therefore, to prove the Proposition it suffices to prove that there exist some constants C 1 , C 2 > 0 such that for all sets K ⊂ B(0, 1),

C 1 Cap 1,p (K) ≤ Cap 1,p (K ∩ B(0, 1), B(0, 2)) ≤ C 2 Cap 1,p (K).
But this is a well known fact about relative capacity. A proof can be found for instance in [10, Proposition 3.3.17].

∃δ 0 , r 1 > 0 s.t. ∀x ∈ ∂Ω, ∀r ≤ r 1 , Cap 1,p 1 r (Ω c ∩ B(x, r) -x) ⩾ δ 0 . (4.5)

Examples of domains satisfying our condition

As we said in the introduction, any smooth enough domain will satisfy our condition. For instance domains satisfying an external corkscrew condition as defined below. -O r 0 seg (D) := {Ω ⊆ D | Ω open and has the r 0 -external segment property}. We say Ω satisfies the r 0 -external segment property if for every x ∈ ∂Ω, there exists a vector y x ∈ S N -1 (0, r 0 ) such that x + ty x ∈ Ω c for t ∈ (0, 1). This notion can also be generalized by the "flat cone" condition as in [5, Definition 5.2] (see also [START_REF] Bucur | Flat cone condition and shape analysis[END_REF]). -O δ 0 ,r 0 Reif flat (D) := {Ω ⊆ D | Ω open and is (ε 0 , δ 0 ) -Reifenberg flat}. We say Ω is (ε 0 , δ 0 ) -Reifenberg flat for ε 0 ∈ (0, 1/2) and δ 0 ∈ (0, 1) if for all x ∈ ∂Ω and δ ∈ (0, δ 0 ], there exists an hyperplan P x (δ) of R N such that x ∈ P x (δ) and

d H ∂Ω ∩ B(x, δ), P x (δ) ∩ B(x, δ) ⩽ δε 0 .
Moreover for all x ∈ ∂Ω, the set It is easy to see that for some fixed parameters we have the inclusions

B(x, δ 0 ) ∩ x ∈ R N d(x, P x (δ 0 )) ⩾ 2δ 0 ε 0 has two connected components: one is contained in Ω, the other one in R N \Ω.
O ε cone (D) ⊆ O a,r 0 corks (D), and
O convex (D) ⊆ O λ Lip (D) ⊆ O δ 0 ,r 0 Reif flat (D) ⊆ O a,r 1 corks (D) ⊆ O δ 1 ,r 2 cap (D). Since a segment has positive (1, p)-Capacity provided p > N -1 (see [6, Proposition 2.5]) we have O r 0 seg (D) ⊆ O δ 0 ,r 1 cap (D) for p > N -1.
Any C 1 domain or Lipschitz domain satisfies an external corkscrew condition. It also follows from porosity estimates that the crokscrew condition implies |Ω \ Ω| = 0, as stated in the following useful proposition.

Proposition 5.1. -If Ω ∈ O a,r 0 corks (D), then |∂Ω| = 0.
Proof. -For all x ∈ ∂Ω and r ⩽ r 0 , there exists y ∈ R N such that

B(y, ar) ⊂ B(x, r) ∩ Ω c ⊂ R N \∂Ω.
In other words, ∂Ω is a σ-porous set in R N , in the sense of [START_REF] Zajíček | Porosity and σ-porosity[END_REF]Definition 2.22], with σ = 2a. In virtue of [START_REF] Zajíček | Porosity and σ-porosity[END_REF] (see the last paragraph at the bottom of page 321 in [START_REF] Zajíček | Porosity and σ-porosity[END_REF], or see also [START_REF] Mera | Porosity, σ-porosity and measures[END_REF]Proposition 3.5]), we conclude that |∂Ω| = 0.

Corollary 5.1. -If Ω belongs to one of the following classes :

O ε cone (D), O convex (D), O λ Lip (D), O δ 0 ,r 0 Reif flat (D), or O a,r 0 corks (D),
then Ω is (m, p)-stable for any m ⩾ 1 and 1 < p < +∞.

Stability with respect to domain perturbation

As before, we consider a fixed bounded domain D ⊂ R N . Let Ω and (Ω n ) n∈N be bounded subdomains of D such that Ω n -→ Ω and D\Ω -→ D\Ω as n -→ +∞ for the Hausdorff convergence. In particular, this implies the convergence "in the sense of compacts" (see [START_REF] Henrot | Variation et optimisation de formes. Une analyse géométrique[END_REF]Section 2.2.4]). In this section we verify that the (m, 2)-stability of Ω implies the Mosco convergence of the sequence (H m 0 (Ω n )) n∈N towards H m 0 (Ω). This will follow from the same argument as for the classical case of H 1 0 (see for instance [START_REF] Henrot | Variation et optimisation de formes. Une analyse géométrique[END_REF]Proposition 3.5.4]), but for the sake of completeness we give here the full details. For this purpose, we first prove the equivalence between γ m -convergence and Mosco convergence (Proposition 6.1). Then we show that (Ω n ) n∈N γ m -converges to Ω when Ω is (m, 2)-stable (Proposition 6.3).

Definition 6.1. -The sequence (Ω n ) n∈N γ m -converges to Ω if for all f ∈ H -m (D), the sequence (u n ) n∈N strongly converges in H m 0 (D) to u, where u n (resp. u) is the unique solution of the Dirichlet problem (-∆) m u n = f (resp. (-∆) m u = f ) in H m 0 (Ω n ) (resp. H m 0 (Ω)). Definition 6.2. -The sequence (H m 0 (Ω n )) n∈N converges to H m 0 (Ω)
in the sense of Mosco if the following holds :

1. If (v n k ) k∈N is a subsequence, where v n k ∈ H m 0 (Ω n k ), and weakly converges to v ∈ H m 0 (D), then v ∈ H m 0 (Ω). 2. For all v ∈ H m 0 (Ω), there exists a sequence (v n ) n∈N , where v n ∈ H m 0 (Ω n ), which strongly converges to v in H m 0 (D). Proposition 6.1. -The sequence (Ω n ) n∈N γ m -converges to Ω if, and only if, (H m 0 (Ω n )) n∈N converges to H m 0 (Ω) in the sense of Mosco.
Proof. -Assume that the sequence (Ω n ) n∈N γ m -converge to Ω. Let (v n k ) k∈N be a subsequence which weakly converges to v ∈ H m 0 (D), where v n k ∈ H m 0 (Ω n k ). Consider the distribution f := (-∆) m v. Then f ∈ H -m (D) and v is the unique solution of the Dirichlet problem in H 1 0 (D). To prove the first item in the definition of Mosco convergence we need to prove that v ∈ H 1 0 (Ω). The γ m -convergence implies that (u n k ) k∈N strongly converges in

H m 0 (D) to u ∈ H m 0 (Ω) where u n k satisfies (-∆) m u n k = f in Ω n k and u satisfies (-∆) m u = f in H 1 0 (Ω). So it suffices to show that u = v. For all k ∈ N, D ∇ m u n k : ∇ m (u n k -v n k ) dx = Ωn k ∇ m u n k : ∇ m (u n k -v n k ) dx = ⟨f, (u n k -v n k )⟩.
Then as k -→ +∞, we use the strong convergence of (u n k ) k∈N and the weak convergence of (v n k ) k∈N to obtain

D ∇ m u : ∇ m (u -v) dx = ⟨f, (u -v)⟩.
Then according to equality f := (-∆) m v, we have

D |∇ m (u -v)| 2 dx = D ∇ m (u -v) : ∇ m u dx - D ∇ m (u -v) : ∇ m v dx = ⟨f, (u -v)⟩ -⟨f, (u -v)⟩ = 0,
thus u = v and the first point of Mosco convergence follows.

The proof of the second item of the Mosco convergence is simpler. Consider v ∈ H m 0 (Ω) ⊆ H m 0 (D) and let f := (-∆) m v. We need to find a recovery sequence u n ∈ H m 0 (Ω n ) that converges strongly to v. It suffice to define u n being the solution of

-∆u n = f in H 1 0 (Ω n ).
By γ m -convergence we directly have u n → v strongly in H m (D). Now we prove the converse. Namely, we suppose that (H m 0 (Ω n )) n∈N converges in the sense of Mosco to H m 0 (Ω). Then we want to prove the γ m -convergence. Consider f ∈ H -m (D) and the associated solutions u n of the Dirichlet problem in Ω n . For all n ∈ N,

D |∇ m u n | 2 dx = Ωn ∇ m u n : ∇ m u n dx = ⟨f, u n ⟩.
We infer that the sequence

(u n ) n∈N is bounded in H m 0 (D) since |⟨f, u n ⟩| ⩽ ∥f ∥ H -m (D) ∥u n ∥ H m 0 (D)
. Let (u n k ) k∈N be a subsequence which weakly converges to a function v ∈ H m 0 (D). Using the Mosco convergence, v ∈ H m 0 (Ω) and for all φ ∈ H m 0 (Ω), there exists a sequence

(φ k ) k∈N , with φ k ∈ H m 0 (Ω n k ), strongly converging to φ in H m 0 (D). Hence, for all k ∈ N, D ∇ m u n k : ∇ m φ k dx = Ωn k ∇ m u n k : ∇ m φ k dx = ⟨f, φ k ⟩,
and using the strong convergence of (φ k ) k∈N and the weak convergence of (u n k ) n∈N as k -→ +∞, we obtain

Ω ∇ m v : ∇ m φ dx = ⟨f, φ⟩.
The uniqueness of the solution of the Dirichlet problem proves u = v. Moreover,

D |∇ m u n k | 2 dx = ⟨f, u n k ⟩ and ⟨f, u n k ⟩ ----→ k→+∞ ⟨f, u⟩ = D |∇ m u| 2 dx.
This yields

∥u n k ∥ H m 0 (D) ----→ k→+∞ ∥u∥ H m 0 (D)
, and the convergence of the subsequence is strong. By uniqueness of the limit, the whole sequence is strongly converging to u in H m 0 (D), and this achieves the proof of the γ mconvergence, so follows the Proposition. By Hausdorff convergence of the complements, there exists n 0 (a) ∈ N such that for all n ⩾ n 0 (a), Ω c n ⊂ {y ∈ R N | dist(y, Ω c ) < a/2}. We deduce from the triangle inequality that inf x∈K dist(x, Ω c n ) > 0 for n large enough and in particular K ⊂ Ω n .

We are now ready to state the following result that will directly imply Corollary 1.1 stated in the introduction.

Proposition 6.3. -Let (Ω n ) n∈N be a sequence in O(D) such that Ω n d H -→ Ω and Ω n d H c -→ Ω where Ω ∈ O(D). If Ω is (m, 2)-stable, then the sequence (Ω n ) n∈N γ m -converges to Ω or equivalently, (H m 0 (Ω n )) n∈N converges to H m 0 (Ω) in the sense of Mosco.
Proof of Proposition 6.3. -Under the assumptions of the Proposition we will prove the γ m -convergence. Consider then f ∈ H -m (D). We know that the sequence (u n ) n∈N of solutions to the Dirichlet problem associated with f in H m 0 (Ω n ), is bounded in H m 0 (D). Therefore there exists a subsequence (u n k ) k∈N that weakly converges to a function v ∈ H m 0 (D). Let φ ∈ C ∞ c (Ω) be a test function. By complementary Hausdorff convergence, there exists an integer k 0 ∈ N such that for all k ⩾ k 0 , Supp(φ) ⊆ Ω n k .

Thus, for all k ⩾ k 0 ,

Ωn k ∇ m u n k : ∇ m φ dx = ⟨f, φ⟩,
and by weak convergence of (u

n k ) k⩾k 0 in H m 0 (D), Ω ∇ m v : ∇ m φ dx = ⟨f, φ⟩. (6.1)
Let us now prove that v ∈ H m 0 (Ω). Up to a subsequence, we can assume that (u n k ) k∈N converges almost everywhere to v. The functions u n k vanishes (m, 2)-quasi everywhere on Ω c n k so almost everywhere. By Hausdorff convergence of the adherence we know that for all compact K ⊂ Ω c then K ⊂ Ω c n for n large enough thus finaly v = 0 almost everywhere in Ω c . Using the definition of (m, 2)-stability, we conclude that v ∈ H m 0 (Ω). Therefore v = u, the unique solution to the Dirichlet problem associated to f in H m 0 (Ω).

Thus let us prove that v ∈ H m 0 (Ω). Since Ω satisfies the (δ 0 , r 0 )-capacitary condition we know from Theorem 1.1 that Ω is a (m, 2)-stable domain. Thus we are left to prove that v = 0 a.e. in Ω c . From here the proof differs from the one of Proposition 6.3 because we do not know anymore that Ω n d H -→ Ω. Instead, we shall benefit from the fact that (1.1) holds true for the whole sequence Ω n and we will use a construction similar to the one used in the proof of Theorem 1.1, but on the functions u n . From now on we will simply denote by n instead of n k for the subsequence u n → v in H m (D) as n -→ +∞. Let K ⊂ Ω c be an arbitrary compact set and let ε > 0 be given. Our goal is to prove that v = 0 a.e. on K. For a general closed set F ⊂ R N and λ > 0 we denote by (F ) λ the λ-enlargement of F , namely,

F λ := x ∈ R N dist(x, F ) ⩽ λ .
By the Hausdorff convergence of Ω c n to Ω c we know that there exists n 0 (ε) ∈ N such that for all n ⩾ n 0 (ε), Ω c ⊂ (Ω c n ) ε , and Ω c n ⊂ (Ω c ) ε . From the above we deduce that The main point being that w n,ε = 0 in (Ω c n ) ε and in virtue of (7.1) we deduce that w n,ε = 0 on K. Now we estimate the difference w n,ε -u n in L 2 (R N ) using a covering of ∂Ω n . More precisely, the infinite family (B(x, 20ε)) x∈∂Ωn is a cover of Ω n \ A n,ε and by the 5B -covering Lemma there exists a countably subcover indexed by

K ⊂ Ω c ⊂ (Ω c n ) ε ⊂ (Ω c ) 2ε . ( 7
(x i ) i∈N ⊂ ∂Ω such that (B(x i , 20ε)) i∈N is a disjoint family, Ω n \ A n,ε ⊂ i∈N B(x i , 100ε), and i∈N 1 B(x i ,100ε) ⩽ N 0 , for a universal constant N 0 ∈ N. Then we can estimate, D |w n,ε -u n | 2 dx ⩽ Ωn\An,ε |u n | 2 dx.
The functions ∂ β u n vanishes almost everywhere on the open set Ω n c , so thanks to our capacitary condition (1.1) we have for ε small enough

(100ε) -(N -p) Cap 1,2 (Z(∂ β u n )) ⩾ Cε -(N -p) Cap 1,2 (Ω n c ∩ B(0, 100ε)) ⩾ Cδ 0 .
Therefore, the Poincaré inequality (2.2) applied to ∂ β u n in all ball B(x i , 100ε) gives

B(x i ,100ε) |∂ β u n | 2 dx ⩽ Cδ -1 0 ε 2 B(x i ,100ε) |∇∂ β u n | 2 dx. (7.2) We deduce that Ωn\An,ε |u n | 2 dx ⩽ i∈N B(x i ,100ε) |u n | 2 dx ⩽ C i∈N ε 2m B(x i ,100ε) |∇ m u n | 2 dx ⩽ CN 0 ε 2m D |∇ m u n | 2 dx ⩽ Cε 2 ,
since the sequence u n is uniformly bounded in H 1 (D). In conclusion we have proved the following : for each ε > 0, we have n 0 (ε) ∈ N such that for all n ⩾ n 0 (ε), there exists w n,ε ∈ L 2 (D) such that ∥w n,ε -u n ∥ L 2 ⩽ Cε and w n,ε = 0 on K. Now for n sufficiently large let ε = 2 -n and let w n := w n 0 (2 -n ),2 -n . We can assume that n 0 (2 -n ) → +∞. The function w n converges to v in L 2 because u n converges to v in L 2 , and w n = 0 on K for all n ∈ N. Therefore, up to a subsequence, w n converges a.e. on K and this shows that u = 0 a.e. on K. Since K is arbitrary, this shows that v = 0 a.e. on Ω c , hence u ∈ H m 0 (Ω) because Ω is (m, 2)-stable. This achieves the proof. Then by lower semi-continuity of the functional we get

F (Ω) ⩽ lim inf n→+∞ F (Ω n ) = inf {F (ω) | ω ∈ O} ,
which finishes the proof.

Theorem 1 . 1 .

 11 -If Ω ∈ O δ 0 ,r 0 cap (D) satisfies |∂Ω| = 0, then Ω is (m, p)-stable for any m ⩾ 1 and p ∈ (1, +∞).

Cap 1 ,

 1 p (K ∩ B(x, r), B(x, 2r)) := inf B(x,2r)

Remark 4 . 1 .

 41 -Of course arguing as in the proof of Proposition 4.1 we can also prove that the condition in [5], ∃δ 0 , r 1 > 0 s.t. ∀x ∈ ∂Ω, ∀r ≤ r 1 , Cap 1,p (Ω c ∩ B(x, r), B(x, 2r)) Cap 1,p (B(x, r), B(x, 2r)) ⩾ δ 0 , (4.4) is equivalent to the following one

Definition 5 . 1 .

 51 -Let Ω ⊂ R N be an open and bounded set, a ∈ (0, 1), and r 0 > 0. We say that Ω satisfies an (a, r 0 )-external corkscrew condition if for every x ∈ ∂Ω and r ⩽ r 0 , one can find a ball B(y, ar) such thatB(y, ar) ⊂ B(x, r) ∩ Ω c .We give a non-exhaustive list of class of domains includes in O(D) : -O convex (D) := {Ω ⊆ D | Ω open and convex}.

-O λ

 λ Lip (D) := {Ω ⊆ D | Ω open and is a Lipschitz domain}.

-

  O ε cone (D) := {Ω ⊆ D | Ω open and has the external ε -cone condition}. We say Ω has the external ε-cone condition if there exists a cone C of angle ε such that for every x ∈ ∂Ω, there exists a cone C x with non empty interior, congruent to C by rigid motion and such that x is the vertex of C x and C x ⊂ Ω c . -O a,r 0 corks (D) := {Ω ⊆ D | Ω open and has the (a, r 0 )-external corkscrew condition}, see Definition 5.1. -O δ 0 ,r 0 cap (D) := {Ω ⊆ D | Ω open and has the (δ 0 , r 0 )-capacity condition (1.1)}.

Definition 6 . 3 .

 63 -For two closed sets A, B ⊂ R N , the Hausdorff distance d H (A, B) is defined by d H (A, B) := max x∈A dist(x, B) + max x∈B dist(x, A).A sequence of closed sets(A n ) n∈N converges to A for the Hausdorff distance if d H (A n , A) -→ 0 as n -→ +∞.In this case, we will writeA n d H -→ A.Next, we define the complementary Hausdorff distance over O(D) byd H c (Ω 1 , Ω 2 ) := d H (D\Ω 1 , D\Ω 2 ),and one can show that the topolgy induced on O(D) is compact. In the sequel we will use the following well known result.Proposition 6.2. -If (Ω n ) n∈N is a sequence in O(D) such that Ω n d H c-→ Ω ∈ O(D), then for any compact set K ⊂ Ω there exists n 0 ∈ N depending on K such that K ⊂ Ω n for all n ⩾ n 0 .Proof. -Since K is compact and Ω is open, we know that inf x∈K dist(x, Ω c ) =: a > 0.

. 1 )

 1 Next, we want to construct a test function in C ∞ c (Ω n ) which is very close to u n in L 2 and equal to 0 on K. Let us consider the following subset of Ω n ,A n,ε := {x ∈ Ω n | d(x, Ω c n ) ⩾ 10ε} ,and the function w n,ε := u n 1 An,ε .

8 .

 8 Existence for shape optimisation problems under geometrical constraints Let D ⊂ R N be a fixed bounded open set and let O D := {Ω ⊆ D | Ω is open} denote all open subsets of D. For a shape functional F : O D -→ R + , it is a natural question to ask if there exists extremal points. In order to answer this question, we introduce a subfamily of O D which is compact for the γ m -convergence and satisfies the capacitary condition (1.1). If F is lower semi-continuous for the γ m -convergence, then we use Theorem 1.2 to conclude. Proposition 8.1. -Any class of the following list is compact for the complementary Hausdorff convergence : O convex (D), O ε cone (D), O a,r 0 corks (D).

Proof. - 1 .

 1 Case O = O convex (D), O ε cone (D). The proof can be founded in [4, Proposition 5.1.1, page 126].

2 .

 2 Case O = O a,r 0 corks (D). Suppose (Ω n ) n∈N is sequence of (a, r 0 )-corkscrew domains which converges to an open set Ω ⊂ D. Let x ∈ ∂Ω and r ⩽ r 0 . By Hausdorff complementary convergence properties, there exists a sequence (x n ) n∈N such that x n ∈ ∂Ω n and x n -→ x as n -→ +∞. By corkscrew conditions, one finds B(y n , ar) ⊂ Ω c n ∩ B(x n , r) with, up to a subsequence, y n -→ y as n -→ +∞. First of all, it is obvious that B(y, ar) ⊂ B(x, r), it remains to prove that B(y, ar) ⊂ Ω c .Let ε > 0, from the enlargement characterisation of Hausdorff convergence, there exists N (ε) ∈ N such that for every n ⩾ N (ε), D\Ω n ⊂ (D\Ω) ε where(D\Ω) ε := x ∈ R N dist(x, D\Ω) ⩽ ε .Thus B(y n , ar) ⊂ (D\Ω) ε and passing to the limit as n → +∞, then taking the intersection in ε, we getB(y, ar) ⊂ ε>0(D\Ω) ε = D\Ω. The ball B(y, ar) is open so we conclude B(y, ar) ⊂ Ω c ∩ B(x, r). Proposition 8.2. -Any class of the following list is compact for the γ m -convergence: O convex (D), O ε cone (D), O a,r 0 corks (D). Proof. -Let O be one of the class of domains listed above and let (Ω n ) n∈N be a sequence in O. Because of the compactness of the complementary Hausdorff convergence in O(D), there exists a subsequence (Ω n ) n∈N denoted by the same indices which d H c -converges to Ω ∈ O(D). Using Theorem 1.2 and Proposition 5.1, it is sufficient to prove that Ω ∈ O, i.e. O is closed for d H c -convergence. Proposition 8.1 concludes the proof. Theorem 8.1. -Let O be a γ m -compact class of subset listed in Proposition 8.2. Let F : O -→ R be a lower semi-continuous functional for the γ m -convergence. There exists Ω ∈ O such that F (Ω) = inf {F (ω) | ω ∈ O} . Proof. -Let (Ω n ) n∈N be a minimising sequence in O, i.e. F (Ω n ) converges to inf {F (ω) | ω ∈ O} as n -→ +∞. Using Proposition 8.2, up to a subsequence there exists an open set Ω ∈ O such that

  r 0 cap (D) and (Ω n ) n∈N be a sequence in O(D). If |∂Ω| = 0, (Ω n ) n∈N d H -converges to Ω, and (Ω n ) n∈N d H c -converges to Ω, then the sequence (Ω n ) n∈N γ m -converges to Ω, or equivalently, (H m 0 (Ω n )) n∈N converges to H m 0 (Ω) in the sense of Mosco.

	Corollary 1.1 follows from gathering together Proposition 6.3 and Theorem 1.1. Let us
	now mention a few remarks.
	1. The interesting feature of Corollary 1.1 is that only the limiting domain Ω is assumed
	to be stable (thus somehow "regular") and nothing is assumed on the sequence
	(Ω n ) n∈N , which could be arbitrary open sets.
	2. It is worth mentioning that in [5] the authors assumed only Ω n	d H c

  13, Corollary 4.5.2, page 195]. To be more precise, we can use the inequality stated for B(0, 1) in [13, Corollary 4.5.2, page 195], and apply it to the function x -→ u(Rx) to get (2.2). A similar statement can also be found in [9, Theorem 4.1]. According to [9], Lemma 2.1 was first proved in [8] by Hedberg in 1978. Lemma 2.1. -[13, Corollary 4.5.2, page 195

To conclude the proof it remains to show the strong convergence in H m 0 (D). By density the equality (6.1) stays true for φ ∈ H m 0 (Ω). In particular for φ = v we get

On the other hand by weak convergence we have

In other words

which together with the weak convergence, proves the strong convergence of u n k to v in H m (D). At the end, from the uniqueness of the possible limit we infer that the whole sequence converges to v, not only a subsequence. This achieves the proof of γ m -convergence.

Proof of Theorem 1.2

In this section we give a proof of Theorem 1.2 stated in the introduction.

Proof of Theorem 1.2. -Let Ω, Ω n ⊂ D be bounded domains as in the statement of Theorem 1.2 that satisfies

and such that (1.1) holds true for all Ω n with the same δ 0 > 0 and r 0 > 0. We want to prove that Ω n γ m -converges to Ω. To this aim we start with a similar argument as in the proof of Proposition 6.3. Consider f ∈ H -m (D). We know that the sequence (u n ) n∈N of the solutions to the Dirichlet problem associated to f in Ω n is bounded in H m 0 (D). There exists a subsequence (u n k ) k∈N which weakly converges to a function v ∈ H m 0 (D). Let φ ∈ C ∞ c (Ω) be test function. By complementary Hausdorff convergence, there exists an integer k 0 ∈ N such that for all k ⩾ k 0 ,

Now argument as in the proof of Proposition 6.3, in order to conclude it suffices to prove that v ∈ H m 0 (Ω). In particular, the strong convergence would then follow by use of the same argument as in the proof of Proposition 6.3.


Paul Veyret
email: paul.veyret@u-bordeaux-montaigne.fr

Nadeem Aslam

Mohsin Hamid

Kamila Shamsie Fragmentation

Kamila Shamsie

Aamir R Mufti

Gilles Deleuze

Félix Guattari

Fractured territories: Deterritorializing the contemporary Pakistani novel in English

Keywords: Pakistani fiction, deterritorialization, ekphrasis, fragmentation

analysing their fiction through

Contemporary Pakistani fiction in English now has an established position not only regionally, as part of the South Asian Anglophone novel, but also as a distinct and identifiable stream within global literature. However, Pakistani literature in English is not exactly a new literature, appearing like "a new star, with all the attendant flash and glory". On the contrary, it is "a long-standing movement with history and depth" (Shah, 2012: 153), with "a long, continuous process from pre-Partition and pre-Raj times to the present day" (M. Shamsie, 2017: 594). The main aim of this paper is to join the "history and depth" and the dynamics of contemporary Pakistani fiction in English, and try to identify the complex ways it articulates the fractured territories and fissured identities of Pakistan's contemporary history in a global, transnational, Anglophone literary form.

from Indian literature as, he speculates, "the violence, extremism, and overall fissiparous social tendencies that are currently tearing the country apart" can "also be understood as the birth pangs of a new cultural independence from Pakistani society's 'Indian ' past" (2016: 177). The Deleuzian imaginary cartography can find here another form of expression in Mufti's description of the Pakistani novel in English as a mode of considering global literature from a new angle: an engagement with "vernacular social and cultural imaginaries" -the Deleuzian lines of flight -and an engagement with the "cultural and social worlds that surround" these novels (Mufti, 2016: 178).

Writing from "extreme edges" [START_REF] Cilano | Writing from extreme edges: Pakistani English-language fiction[END_REF], worlds "colliding" (Waterman, 2016), liminality [START_REF] Chambers | A comparative approach to Pakistani fiction in English[END_REF], lines and border crossing [START_REF] Guignery | Step across this line": Edges and borders in contemporary Indian literature[END_REF]: spatial metaphors abound when South Asian and Pakistani writing is depicted, and the idea of the transgression of borders is associated with both Indian and Pakistani writing. Pakistani fiction occupies shifting ground: it stands as a historically and politically over-determined territory, fashioned by two partitions, and has been at the forefront of contemporary geopolitical turmoil since 9/11 and the state's involvement in Operation Enduring Freedom. In other words, "Pakistan is just like India, except when it's just like Afghanistan" (Granta, 2010); 1 the nation occupies a liminal space between the massive continent of contemporary Indian literature and the political chaos of the Muslim Middle East. Willy-nilly, Pakistan's history has always been international and the fulcrum of global geopolitics since 1947. The initial disjuncture and rupture of history can be considered to have taken place when the signifier "partition" was internalized, deterritorialized, and made into a mode of apprehension as well as a lens of interpretation. Contemporary history haunts Pakistani fiction: the 1947 and 1971 partitions and 9/11 are not only present as defining events but also as critical moments of disjunction and rupture, underlining the centrifugal nature of Pakistan's identity. The country holds a unique hub position in the world: its geographical position along the "spine of the silk roads" (Frankopan, 2015: 477) and at the entry point to Asia from the West, together with its position at the intersection of different imperial expansionisms, make Pakistan a true cultural crossroads between the Islamic world and South and Central Asia.

Although clearly dissimilar in their writing techniques and themes, Aslam's, Hamid's, and Shamsie's writing shares common features. Beyond the commercial and editorial success of their novels, the common denominator for all three is their commitment to a shift in literary affinities which has both internalized the literary discourse about the empire "writing back" -as part of an academic literary curriculum rather than a personal experience of reclaiming -together with the historiographic metafictional practices and self-parodic modes of postmodern fiction. At the same time, they have re-embraced political and social commitment through a form of deterritorialized writing which favours fractures over permanence, fragmentation over continuity. Deterritorialization, according to Gilles Deleuze and Félix Guattari in A Thousand Plateaus, dovetails with the act of writing as tracing "lines of flight " (1987: 32). This act of deterritorialization condenses the metaphor of literature as a space of enclosure and escape, confinement and transgression; the Deleuzian lines of flight articulate both limits and their subversion. Pakistani fiction in English participates in a global movement of both inscription and displacement of borders, suggesting in the writing of its authors a tension between defined lines and their erasure, between rigid frames of reference and their permeability. These authors' fiction stands at the juncture of the paradoxical space expressed by a poetics of deterritorialization, displacement, and fragmentation. In other words, the Deleuzian concepts of lines of flight and deterritorialization suggest the critical avenue to be pursued here: the paramount importance of territories, borders, and maps and the porous nature of dividing lines, the focus placed on the centrifugal forces at work in Pakistani society together account for the strain at the heart of the fiction of these three novelists.

Borders and border-crossing are prominent tropes for South Asian novelists and Pakistani writers in particular. A point of departure for the symptomatic persistence of the image of frontiers and lines is certainly Pakistan's creation history. Manto's 1955 short story "Toba Tek Singh" constitutes one of the fictional and imaginative matrices for the spatial obsession. Even if it was written in Urdu, it nonetheless encapsulates the frictions manifest in contemporary Pakistani Anglophone fiction: the fractured territories of Pakistani fiction are present in this parable about the literal lunacy and absurdity of partition. The vision of the inmates of the Lahore lunatic asylum found in Manto's short story is summed up by the final image of Toba Tek Singh who refuses to pass over the border and dies in a no man's land: "There, behind barbed wire, on one side, lay India and behind more barbed wire, on the other side, lay Pakistan. In between, on a bit of earth which had no name, lay Toba Tek Singh" (Manto, 2001(Manto, /1955: 51: 51). Shamsie's work is also concerned with the tension between the strong presence of borders and divisions on the one hand and, on the other, a strong drive towards liberation from boundaries and definitions. In Kartography (2002), a novel devoted to the illusory cartography of socially and ethnically divided Karachi, Shamsie explores notions of territory, identity, and ethnicity, and how concrete and metaphorical boundaries are created. She interrogates the Pakistani paradigm of building borders and bolstering social divisions: "How many walls can one nation erect and sustain […] Is it possible to circumnavigate one wall without crashing into another?" (2002: 51). Similarly, Salt and Saffron (2000) is constantly haunted by partition and is concerned with the intricacies of a family's past. For Shamsie, the family becomes the symptom of Pakistan's history: the subjective manifestation of a latent, unexpressed conflict. As yet another allegory of the subcontinent's history, in the vein of Rushdie's Midnight's Children, "the family divides at Partition into two sides who hate each other and continually quarrel over whose fault it was" (King, 2004: 284). This is made all the more poignant since the divisions and rancour are "constructed on a past that no one really knows" (King, 2004: 284). For Shamsie the past is both a boon and a burden, and Pakistan is both the place that shapes her imagination and a state whose faults she portrays. Families in Shamsie's fiction are the repositories of the repressed wounds of the past and the silent guardians of the archives of the undisclosed tragedies of personal and national history.

Burnt Shadows (2009) offers a new perspective on the same problematic of the deterritorialization of an author's voice by including a critique of American political hegemony in "the war on terror". Once again, spatial dislocation and temporal erasure generate the contradiction of the novel's plot and themes. The two historical collisions of the narrative, Hiroshima and 9/11, form the alpha and omega of Shamsie's description of trauma over three generations. The shifting horizons of the novel, from Japan to India, Pakistan, Afghanistan, and New York, represent the different geographical points which, once joined together by the narrative, form this novel's lines of flight and contradict the logic of borders. For Shamsie:

[T]he idea of the nation-state as the defining framework of a novel has less and less relevance in this interconnected world. At the very least, I seem to have become incapable of imagining a novel that is restricted within the boundaries of a single nation. (qtd. in Alam et al., 2018: n.p.) Against the homogenization of global culture and the hegemony of the American worldview, Shamsie's novels offer tales of estrangement and reconnection, discontinuity and recombination, betrayal, and atonement. The fragile "lines of flight" presented in Shamsie's novels are essentially historical and geographical: the burden of representation seems to be transferred from one generation to the next and from one location to the other, crossing boundaries and overlapping chronologies.

Shamsie's Home Fire (2017) explores once more the recurring theme of families and their secrets. The political context is the fate of British jihadists gone to fight with ISIS in Syria, but the literary antecedent is Sophocles' Antigone. Through the Greek tragedy framing, Shamsie reads the political cracks and fissures that define migrants' and their descendants' British identity in the aftermath of terror attacks in Europe and wars in the Levant. The characters' names echo those of Sophocles' dramatis personae and the plot revolves around an opposition between the laws of the land which strip a young British jihadist of his nationality, and the blood ties of the family. However, the original tragic quandary of phusis (natural laws) opposed to nomos (man-made laws) is closer to Jean Anouilh's Antigone (1946), written in 1942 during the Nazi Occupation of France. With Shamsie, as with Anouilh, the dilemma is more political and moral than religious: the central importance of the Greek sisters' and brothers' incestuous bloodline and the cathartic conclusion are replaced here by a sense of irreparable loss and unresolved tensions. For Claire Chambers (2018) the novel is about the failure of auditory reception and communication, and where sounds and noises, fragments of communication through WhatsApp messages, are symptomatic of a world which silences the oppressed. By using sound both as a "metaphorical conceit and as a material or embodied experience" (2018: 217), for Chambers, Home Fire is "one of the most plangent and multitonal novels" (2018: 218) yet to be published about terrorist-related violence, as the text itself "can supplement sociological, psychological, and criminological analyses of radicalization" by bringing into discussion "its sensual qualities of visual and aural texture" (Chambers, 2018: 218). Like Hiroko from Burnt Shadows, the characters from Home Fire function "most powerfully as […] agent[s] for the demystification, differentiation and humanisation" (Clements, 2016: 145) of British Muslims. Kamila Shamsie's novels engage here, especially since the publication of Burnt Shadows, not only with the Deleuzian lines of flight but also with the challenges outlined by Mufti, of the "precarious balancing act between writing about the fundamental issues facing Pakistan and South Asian society […] without playing to the metropolitan fascination with the spectre of Islam and stereotyped Muslim sensibilities" (Mufti, 2016: 178). Shamsie's global epics not only destabilize the notions of border and border crossing but also blur the limits of essentialized Muslim identities as they are framed by mainstream Western media. The frisson of delving into the minds of "jihadists" is radically deflated by Shamsie's literary and metaphorical strategies.

In Aslam's and Hamid's fiction, what Deleuze calls the "otherness of language" provides another, more internalized, form of instability and deterritorialization. In Essays Critical and Clinical (1998), Gilles Deleuze delineates his theory on the "otherness of language". According to him, writers discover their voice within the space and experience of writing, which is a form of narrative deterritorialization. The writer therefore invents a language by twisting familiar tropes, defamiliarizing metaphors, and reinventing a voice which is "neither another language nor a rediscovered dialect […] but […] a minoritization of this major language, a delirium carrying it off, a witch's line of flight out of the dominant system" (Deleuze, 1998: 5). Both novelists favour ruptures and displacements, fractures and fissures that point to the deterritorialized lines of flight at work in their writing. But the differences between the two novelists are apparent: whereas Aslam favours a lyrical style based on description and startling analogies, as well as references to Urdu literature, Hamid's writing is more indirectly destabilizing since more often than not minute cracks and fissures appear in the narrative voice, gradually dislocating the reader's expectations.

First, let us consider Aslam's writing. Aslam, himself a painter, is also a keen amateur of icons and miniatures (Aslam, 2017b: n.p.). Paintings play an essential role in his writing, and more specifically his descriptions of painting, or "ekphrasis", a rhetorical device which enables to the writer to introduce long descriptive passages of an inanimate object, like a work of art. With Aslam, the use of these long descriptions has a double function. First, it has import in the narrative as pauses and moments of contemplation, whether it is in the character's musings or the narrator's rumination. Second, it connects the network of images with the main dialectic at work between centrifugal and centripetal forces. Here is an example from The Wasted Vigil:

Marcus takes down Virgil from the shelf. On the cover is a painting of Aeneas fleeing the burning destruction of Troy. The great broken heart of the city in the background. Aeneas is accompanied by his young son -a path to the future -and is carrying his aged father over his shoulder -the reminder of the past. The old man clutches the statues of the household gods in his right hand and because the other hand is out of sight in the folds of his cloak, absent beyond the wrist, Marcus thinks for a moment of himself. If so, then David is Aeneas -he had offered to carry Marcus up the tall minaret in Jalalabad. (Aslam, 2008: 95) The choice of the book whose cover is described here is interesting for two reasons, first as an intertextual reference to a founding myth of Roman identity and second as it illustrates a moment of transition between chaos and hope. Virgil's Aeneid, written between 29 and 19 BCE, is a founding text in Latin literature as it legitimized not only a godly filiation in Rome's inception (Aeneas is the son of Anchises, a mortal, and the goddess Aphrodite), but it also justified Roman identity as both distinct from Greek culture as well as its rightful heir. In other words, the ekphrasis describes not only a flight from a devastated city but also a moment of cultural emancipation. The scene also mirrors Marcus Caldwell's predicament: as he identifies with Anchises, clutching "the household gods in his right hand", he interprets the never-ending war in Afghanistan, where the novel takes place, through the lens of the Latin epic. In a novel focusing on the destruction or survival of ancient works of art and their guardians, the passage depicted by the ekphrasis crystallizes a moment poised between extinction and revival, between a nostalgic fixation on past ruins and promises of rebirth.

Furthermore, the role of ekphrasis in Aslam's work is to mediate the mimetic role of language. Reality -the reality of the border zone between Afghanistan and Pakistan, the reality of the "War on Terror", the reality of religious pogroms -is, therefore, all the better perceived when the work of art frames it. The ekphrasis provides a surface on which character or narrator can project their inner thoughts and contemplations. More often than not the motif of windows -both window frames and window panes -is associated with such moments. The description of the character looking at a painting is echoed by descriptions of outdoor visions or reflections on the surface of a mirror or a window pane. The window motif is therefore dual, both a frame distancing the point of view, but it also acts as a point of insistence, emphasizing the allegory at work. Aslam thus describes the narrator or the character in an in-between position: inside looking out, as if the painted surface or the reflected image act as a zone of contact between inside and outside, and between intimate subjectivity and the outside world. In other words, for Aslam the writer-painter, the ekphrasis performs a connection between, on the one hand, personal emotions and subjectivities, personal history and its catalogue of traumas and, on the other, the stage of history and geopolitics.

The ekphrasis mediates a reality both so violent and, to use Aslam's words, so "beautiful" (Aslam, 2017b: n.p.), that it would otherwise be unbearable. What the ekphrasis also underlines, and this is its second role in Aslam's writing, is the dialectics of centrifugal and centripetal forces. The particularly pervasive trope of fragmentation embodies this dynamic. As the ekphrasis crystallizes the reader's gaze, images of fragments and shattered surfaces abound. It comes as no surprise that paintings, illustrations, and books are often smashed and shattered in Aslam's novels, especially in The Wasted Vigil (2008), The Blind Man's Garden (2013), and The Golden Legend (2017a). In The Wasted Vigil, a mural representing couples is shot to smithereens by the Taliban, irate at the heretic act of representing human figures. One of the tragedies of The Golden Legend is the destruction of the eponymous Golden Legend, whose pages are torn to shreds by philistine authorities. The book's destruction is symbolic for more than one reason. First, as a compendium of Islamic lore and testimony to its humanism and open-mindedness, its being torn to pieces underlines the violence of contemporary Pakistan. Then as a fetish, it condenses Aslam's relation to art and to the lines of fracture that destabilize dominant discourses. At one point, the female protagonist repairs the torn pages, now reduced to fragments and pieces of a puzzle, and the Japanese term kintsugi is used to describe the act of repairing an object and making it more beautiful in the process. In a remark echoing Keats's aphorism which concludes his poem "Ode on a Grecian Urn": "Beauty is truth, truth beauty -that is all | Ye know on earth, and all ye need to know" (Keats, 1909(Keats, /1820: 113): 113), the narratorial voice intimates that "some things were more beautiful and valuable for having been broken" (Aslam, 2017a: 79). The damage, an extended metaphor for the state of affairs in Pakistan and the mental state of the protagonists, replicates the cracks and fissures of the text. By giving a voice to the different subalterns and disposed of Western society -Pakistani immigrants as in Maps for Lost Lovers -or Pakistani society with its strict ethnic, social, and religious stratification, Aslam performs fictional kintsugi. The sutures and fractures of society are metamorphosed into "luminous scars" (Aslam, 2017a: 82), inverting the centrifugal dynamics of Pakistan's history into the centripetal forces of creation.

In an interview Nadeem Aslam uses the metaphor of a helix to describe his identity as an author: "Islam and Marxism are the two strands of my DNA, my double helix" (Aslam, 2017c: n.p.), thus emphasizing the presence of opposite yet complementary forces at work in his writing. Sadia Abbas links the extreme aestheticism of Aslam's writing to what she terms "Cold War Baroque": an echo of seventeenth-century Baroque with which it shares certain formal tendencies and […] signals a longer political durée marked by the ongoing problematic of the restructuring of religion -and the relationship of that reconfiguration with the problem of the nation-state and the status of the citizenship and minority -in modernity. (Abbas, 2014: 151) Specifically, Abbas describes Aslam's writing in Deleuzian terms since "Cold War histories, devotional aesthetics, literary forms are infolded, inverted, revealed to be caught in a cycle of seemingly inescapable yet endlessly torqued iteration" (2014: 185). The hyperaestheticism, the "préciosité" (Abbas, 2014: 185; emphasis in original) of Aslam's writing does indeed have a purpose which he exploits in political terms, meaning that the Baroque image of "torque" is indeed apt. The image of the DNA prolongs Sadia Abbas's explanation seamlessly, together with her use of Baroque aesthetics, since Aslam's notion of identity as a double helix evokes his writing as Deleuzian folds: open-ended and infinite, non-exclusive and unlimited, exterior and interior at the same time.

If the deterritorializing "lines of flight" of Aslam's writing produce a crisis between the centrifugal forces of the present posited against the centripetal forces of art and visual representation, Hamid's deterritorialization lies in his ambiguous story-telling and in the way his novels play with the mimetic conventions of realism. His narratives are spoken in a "minoritized language" within the language of fiction. The narrative voices of Hamid's novels are the ultimate fractured territories mirroring the frictions of the contemporary world. However, unlike Shamsie whose writing is anchored in a specific cultural landscape, and Aslam's mediated and sutured referentiality, Hamid's writing moves towards the generic and the allegorical through his practice of erasure and unlabelling of references.

Hamid's novels have to be read as allegories or even meta-allegories. The novels are the destabilizing metaphors of a shifting world where "fundamentalists" and tourists are indistinguishable, a "pretty girl" may or may not be part of a love subplot in a rags-toriches story in "rising Asia", and where migrants enter a Narnia-like land to escape death. Hamid's writing, from among the three authors taken as representatives of new Pakistani fiction, is certainly the most deterritorialized and the most Deleuzian. Deeply ironic and based on "ludic deception[s]" [START_REF] Chambers | Book review: Mohsin Hamid, How to Get Filthy Rich in Rising Asia[END_REF] n.p.), Hamid's novels deal with Deleuze's otherness of language and the fundamental ambivalence of the narrative voice. Hamid's writing is, therefore, the deterritorialized line of flight which counteracts attempts at stabilization: his novels are places of disjunctions and divisions, ruptures and slippages. In The Reluctant Fundamentalist (2007) and How to Get Filthy Rich in Rising Asia (2013) Hamid opens up within the narrative voice a space of shifting certainties, misleading the reader in a constant movement of "interpellation" and "counter interpellation" [START_REF] Althusser | On ideology[END_REF](Althusser, /1971: 190-97): 190-97). By deterritorializing his narrative voice and playing with its lines of flight, not only does he give a voice to the overlooked and the subalterns of history, the way Shamsie and Aslam do, but he also unfastens ambiguities where certainties rule, disorienting the reader by playing with the conventions of fiction. Each of his four novels, by removing secure standpoints, tends toward the experimental in their use of the narrative voice producing a sense of disorientation.

The opening pages of The Reluctant Fundamentalist (2007), in that respect, are programmatic of the deterritorialized line escaping from the recognizable structure of the novel. The novel is an interpellation which turns the table on the reader's assumptions and prejudices about the Muslim other, making the silent American interlocutor both the silent witness of a confession and a possibly threatening presence. The text is a claustrophobic construct, inspired by Camus' The Fall (1956), 2 and yet that novella's openly binary construction is eventually dismissed and replaced by the shifting grounds of uncertainty. The Reluctant Fundamentalist is carefully built as a moment of enunciation based on the tension between the opposite poles of the enunciating "I" and the enunciated "I", the enunciating "I" is an economics lecturer at the local university in Lahore, known for his political commitment against Operation Enduring Freedom, ensconced in a tea shop in Anarkali addressing his potential assassin, while the enunciated "I" is Changez, the "modern-day janissary" (Hamid, 2007: 173), the remembered "I", whose political awareness is gradual yet violent. The novel's opening sentences illustrate a dual movement of assertion and rebuttal of duality and difference:

Excuse me sir, but may I be of assistance? Ah, I see I have alarmed you. Do not be frightened by my beard: I'm a lover of America. I notice that you were looking for something; more than looking, in fact, you seemed to be on a mission, and since I am both a native of this city and a speaker of your language, I thought I might offer you my services. (Hamid, 2007: 1;emphasis in original) The voice is deceptively straightforward, open and hospitable. The tropes of duality, division, and contradiction abound. Changez the narrator doubles as "Changez", a personable young Pakistani, engaging in a one-sided dialogue with a silent American "tourist" who himself may -or may not -be that paradigm of duplicity, a secret agent on a "licence to kill" mission. The novel's first lines introduce the reader to the fundamental strain at work within the text. Just as Changez presents himself both as a linguistic and cultural go-between bridging "America" and Old Lahore, and a menacing figure whose "beard" is the synecdoche for Islamist terrorism, the narrator's voice equally oscillates between ingratiating and threatening. Changez's voice seems to embody both the Orientalized voice of the Muslim other, at once fascinating and politely threatening, and the worldly cosmopolitan New Yorker, no longer a Pakistani, but a "brown" citizen of the world. For Delphine Munos, The Reluctant Fundamentalist describes how the events surrounding 9/11 are "a symptom […] reveal[ing]the racial melancholia surreptitiously informing today's 'new' versions of the American Dream" (2012: 399). Indeed, the novel points at the hidden fractures of American society after 9/11 and creates spaces of instability within the narrative. For Mandala White, the "ambiguity of the exchange in the frame narrative functions" is allegorical of the "uncertainties in the globalized world, and the narration itself as an allegory of the attempt to combat paranoia by maintaining control in the face of an increasingly uncertain global space" (2017: 14). Hamid is a novelist of allegories and the metafictional, pointing at the constructed nature of identity discourses.

Hamid's lines of flight lie less in what is left unspoken or in his narrator's unreliability, than in Changez's openness, his highly articulate tone, and his almost excessive, parodic, eloquence. Muneeza Shamsie points out that his American lover's name, Erica, echoes the syllables of America and that the company for which he works, Underwood Samson, shares the same initials as the United States (M. Shamsie, 2017: 404). The symbolic and implicit are part of the novel's explicit, involving the reader "in a literary game of misrecognition (and recognition)" (Clements, 2016: 61). The pivotal episode of the 9/11 attacks on the Twin Towers is the occasion for Hamid to describe a radical gesture of deterritorialization: the attacks are not seen as the irruption of the reality of violence disrupting the fiction-making of the American myth, but on the contrary as the irruption of fiction, "the symbolism of it all", into the real: I turned to the television and saw what at first I took to be a film. But I continued to watch, I realized that it was not fiction but news. I stared as one -and then the other -of the twin towers of New York's World Trade Center collapsed. And then I smiled […]. But at that moment, my thoughts were with the victims of the attack -death on television moves me most when it is fictitious and happens to characters with whom I have built relationships over multiple episodes -no, I was caught up in the symbolism of it all, the fact that someone had visibly brought America to her knees. (Hamid, 2007: 83;emphasis in original) 9/11 is perceived, at first, by an incredulous Changez through the mediation of fiction as if the historical event underwent a double process of derealization and dehistoricization before being rehistoricized and re-entering the real as a historicized, meaningful, and resymbolized event. The re-symbolization of 9/11 as Changez perceives it also coincides with its sexualization as America becomes feminine since it is brought to "her" knees, in a gesture of sexual submission and aggression. Moreover, the passage echoes Slavoj Žižek's analysis of 9/11: "this fantasmatic screen image entered our reality. It is not that reality entered an image; the image entered and shattered our reality " (2002: 16). What Hamid offers in this twice-mediated representation of 9/11 -mediated first as a memory recalled by the framing narrator telling the story and then mediated once more by television repeating the same sequences in loops -is a displacement of 9/11 as a fracture of history. It appears here less as a traumatic event than as another allegory of that event, pointing at the metafictional nature of the text and its claim towards the interpretation of the novel as a fable. The self-reflexive nature of The Reluctant Fundamentalist points at Hamid's own interrogation of the world as an allegory, a Borgesian tale founded on the deterritorialization of fiction. Reading and writing as tools of self-invention are central to Hamid's narrative strategy: How to Get Filthy Rich in Rising Asia is another example of the author's tongue-in-cheek approach to the literal interpretation of the capitalist discourse of self-improvement and social mobility. As Angelia Poon demonstrates, the novel is caught in a "dialectic between the individual subject and historical forces" (2017: 149): reading and writing provide counter-forces that balance the univocal and centrifugal pull of neo-capitalism. Indeed, "reading literature creates room for creative human interpretation, and it thereby creates space beyond global capitalism's reach and beyond the narrow, counter-creative logic that global capitalism arguably necessitates" (Naydan, 2016: 100). Alternate spaces are certainly at the centre of Mohsin Hamid's Exit West (2017), a novel of escape, disillusionment, and recreation of identity.

Exit West offers an even more allegorical representation of the present. The fractured territories of his fiction give way to a form of anti-mimetic writing where labels and references are left out as if the signified of language were erased and the signifiers left free-floating. In this narrative about border-crossings and shifting identities, the central problematic is between erasure and inscription, naming and deleting. Hamid's writing leans towards the generic and the allegorical as, once again, the opening sentence is programmatic of the whole novel: "In a city swollen by refugees but still mostly at peace, or at least not yet openly at war, a young man met a young woman in a classroom and did not speak to her" (Hamid, 2017: 1). The indefinite articles and the use of the almost notional plural -these "refugees", who have no origin, no labels -highlight the book's propensity to avoid categorization and grouping. The characters' names, Saeed and Nadia, although Muslim-sounding, do not link them geographically to any particular place on the map: the anonymous city of their youth is located somewhere between the East of the Mediterranean and the Hindu Kush. The reader has to make a not-so-educated guess to situate the plot somewhere between Lebanon, Syria, or Iraq.

Furthermore, Nadia's "flowing black robe" is never named although one might recognize an abaya. The refusal to name and label is even found in descriptive details such as "the blue tattoo of a small mythological bird" (Hamid, 2017: 9) inked on a woman's right ankle. Hamid's refusal to describe certainly underlines his "ludic" approach to mimetic fiction and the power of realistic writing to assign a place. The "exits" westward and the dangerous journeys are not described except as magical moments of border-crossing through mysterious dark doors, and the text even shifts genres and becomes a parody of C. S. Lewis's children's fantasy novel The Lion, the Witch and the Wardrobe (1950), but without the Christian symbolism and the fantasy of the original as the magic, or rather unexplained, element serves as a political allegory. Hamid's uncoupling of obvious signifiers and signified transforms his text into a protean novel, equally difficult to categorize. It's only when languages (Igbo) and a Western city (London) are named, that labels and categories reappear as if once in the relative haven of one of the Western world's most multicultural places, linguistic and religious identities could, paradoxically, be reasserted. Similarly, during the opening chapter, Hamid plays with a series of platitudes: "Location, location, location, the estate agents say. Geography is destiny, respond the historians" (Hamid, 2017: 9). It is as though language as a mimetic tool is itself trapped by the petard of its arbitrary nature: the refusal to label and categorize is Hamid's ironic strategy to cover the fractured territories of reality.

According to Aamir Mufti, the contemporary Pakistani novel in English confirms "the process of international validation" (2016: 172) by various instances such as the British and American publishing industry, the literary prizes circuit, and, of course, last but not least, the academic world. As the author of these lines, a white European male, I am aware of my -albeit modest -position in the endorsement of Pakistani fiction in English. Mufti identifies the situation of contemporary South Asian fiction in English as having jumped out the frying pan of Orientalist essentializing and subalternization into the fire of Anglicist globalization. By apparently breaking new aesthetic ground, South Asian novels have, since Midnight's Children, provided the former colonial powers with new forms of exploitative images and metaphors of the subcontinent. In other words, although the English novel has been metamorphosed by the incorporation of "chutnified", "mongrel", "Hinglish", and Oriental modes of narrative such as the Mahabharata or The Arabian Nights, this "tropicalization" of fiction is only superficial and has complied with the expectations of a largely Western audience. Pakistani fiction in English is the flavour of the month, with the added "frisson" of jihadist Islam. Mufti's book provides a convincing study of the genealogy of "global literature" by tracing its origins to the aesthetics and ideology of Orientalism, and the case he makes of globalization being yet another form of assimilation of exotic elements into mainstream discourse is persuasive. Writers like Hamid, Shamsie, and Aslam, and also Mohammed Hanif and Daniyal Mueenuddin, have demonstrated their awareness of these complex issues at stake first of all with irony, and second by exhibiting precisely a type of aesthetic resistance in their approach to fiction writing. The novels under scrutiny share the same involvement with the politics and history of Pakistan while creating artistic objects wholly alert of their aesthetic tensions.

According to Hamid, "Pakistan is a market-leader […]. The Most Dangerous Place in the World TM " (Granta, 2010: n.p.), while Mohammed Hanif quips that to write about Pakistan the best advice is "Must have mangoes", before enumerating the metaphorical, political, and poetic uses of this typical South Asian fruit. Moreover, the political stereotyping and "framing" of the South Asian Muslim has been addressed repeatedly by these authors in numerous articles and essays, another proof of the authors' commitment to deconstructing Western stereotypes. Indeed, for Peter Morey and Amina Yaqin the image of Muslims in Western media is more akin to a "refraction" of fears and misunderstandings amplified to serve a political agenda, than a "reflection of reality" (2011: 3-4). According to Pascal Zinck, novels such as The Reluctant Fundamentalist and Burnt Shadows offer "an insight into Islamic terrorism, not perceived as merely a response to Islamophobia, but as a reaction to and a by-product of cultural globalisation" (2010: 1). At the same time, the characters represented in the novels -Changez in The Reluctant Fundamentalist and Parvaiz in Home Fire -do not share the writers' predicament and personal trajectories. Indeed, coming from upper-middle-class Anglophone backgrounds, some of them, like Mohsin Hamid and Kamila Shamsie, have had the combined advantages of elite education in Pakistan at the Karachi Grammar School, then an international education on a North American or British university campus, thus placing them in the position of the cosmopolitan novelist. In fact, as Sarah Brouillette sums up, quoting Leela Gandhi on Salman Rushdie's cosmopolitanism, "the figure of the new/postcolonial Indian English novelist is in a deliciously 'win-win' situation, speaking from an 'enviable position' of simultaneous privilege and dissent, articulating a special kind of subversion that comes with social status" (2007: 87). Like their predecessor Salman Rushdie, the new generation of Pakistani novelists renew "the paradigmatic figure of this privileged simultaneity" (Brouillette, 2007: 87) and as Janet Wilson points out, Hamid's narratives underscore the "assimilation of Western and Muslim values into a new cultural synthesis in which global forces sharpen economic divisions while also promoting social mobility and new possibilities for individual self-determination" (2017: 184-85). Indeed, the dominant characteristic of the novelists under scrutiny in this paper is their ability both to educate Western readers "to a certain degree about 'other' realities" (Brouillette, 2007: 60) in keeping with a broadly anti-imperialist liberal discourse, while at the same time staying in congruity with the very type of multinational capitalism these writers sometimes appear to denounce. By rising above the apparent paradox of holding a literary "passport that identifies" them "as being from a region of underdevelopment and pain" (Brennan, 1989: 7) and deconstructing national narratives of belonging, contemporary Pakistani novelists make the most of these apparent contradictions.

In conclusion, the contemporary Pakistani novelists under scrutiny here rise above the pitfalls of the dialectic process of Anglicization and Orientalization identified and denounced by Mufti. By embracing the social and political issues that define South Asia such as the impossibility of laying to rest the ghost of partition, the aftermath of the War on Terror on Pakistani society, or the framing of Muslims in the West by the media and extreme political discourse, Pakistani fiction in English furnishes the complex novels demanded by complex times. Perfectly aware of the paradoxical nature of their marginal position, these authors venture beyond the mere allegorization of some national narrative and work well beyond their assigned location of "postcolonial" authors "writing back". Although postcolonial problematics live on, "the rules of the game have changed" [START_REF] Morey | The rules of the game have changed": Mohsin Hamid's The Reluctant Fundamentalist and post-9/11 fiction[END_REF] and the novels produced by Pakistani novelists have shifted their dialectics from a colonial/postcolonial historical tension towards a colonialism/globalization tension. Indeed, according to Paul Jay, Mohsin Hamid's Moth Smoke (2000) offers a new paradox. Although the novel focuses on the disruptive consequences of the forces of economic globalization on an emerging Pakistani middle class struggling for lack of the right social connections it "ends up obfuscating the relationship between these forces and those Pakistan experienced under British rule" (Jay, 2010: 106). Indeed, Hamid's perceived ambivalence about globalization only underlines the dialectical relationship that binds colonialism and globalization since these two histories are identical (Jay, 2010: 116). And this dialectic defines Pakistani fiction in English. Whether, according to Mufti, Pakistani novelists remain caught up in postcolonial exotic and Orientalization demands remains to be seen, but with the balance of economic power tilting East, and with the spectre of religious nationalism rising in India, Pakistan finds itself, thanks to the forces of history, at the centre of a fine balance. Shamsie's, Aslam's, and Hamid's latest publications seem to have proved that they have transcended the Orientalist/ Anglicist oscillation Mufti feared and to have combined entirely in their writing the vernacular traditions of Pakistan and South Asia with the protean form of the global novel. The relationship with English literature is neither agonistic nor subaltern; it is instead, a Baroque, Deleuzian line of flight without origin or centre, a rhizome. Midnight's children have thrived and multiplied, and now midnight's grandchildren are the new voices from the Land of the Pure.

The Journal of Commonwealth Literature 00(0)

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Notes

1. This quotation is taken from "How to Write About Pakistan", in the online material to accompany Granta's Pakistan issue. Short pieces were contributed anonymously by four novelists, Mohsin Hamid, Mohammed Hanif, Daniyal Mueenuddin, and Kamila Shamsie. This is taken from Part II, written by Shamsie (personal conversation). 2. For the connection with Albert Camus, see [START_REF] Hutton | The Janus and the janissary: Reading into Camus's La Chute and Hamid's The Reluctant Fundamentalist[END_REF].