Cloaking for a quasi-linear elliptic partial differential equation

In this article we consider cloaking for a quasi-linear elliptic partial differential equation of divergence type defined on a bounded domain in $\mathbb{R}^N$ for $N=2,3$. We show that a perfect cloak can be obtained via a singular change of variables scheme and an approximate cloak can be achieved via a regular change of variables scheme. These approximate cloaks though non-degenerate are anisotropic. We also show, within the framework of homogenization, that it is possible to get isotropic regular approximate cloaks. This work generalizes to quasi-linear settings previous work on cloaking in the context of Electrical Impedance Tomography for the conductivity equation.


Introduction and Preliminaries
The topic of cloaking has long been fascinating and has recently attracted a lot of attention within the mathematical and broader scientific community. A region of space is said to be cloaked if its contents along with the existence of such a cloak are invisible to wave detection. One particular route to cloaking that has received considerable interest is that of transformation optics, the use of changes of variables to produce novel optical effects on waves or to facilitate computations.
A transformational optics approach to cloaking using the invariance properties of the conductivity equation was first discovered by Greenleaf, Lassas and Uhlmann [1,23] in 2003. Pendry, Schuring and Smith in 2006 [38] used a transformation optics approach, using invariance properties of the governing Maxwell's equations to design invisibility cloaks at microwave frequencies. Leonhardt in 2006 [32] discusses an optical conformal mapping based cloaking scheme. For the ideal/perfect invisibility cloaking considered in [38,32], it is a singular 'blow-up-a-point' transformation. The cloaking media achieved in this way inevitably have singular materials parameters and require design of metamaterials. The singularity poses much challenge to both theoretical analysis and practical construction. While several proof-of-concept prototypes have been proposed as cloaks, several challenges still remain in developing fully functional devices capable of fully cloaking objects. A lot of current academic and industrial research in material science is focused on development of such metamaterials from proof-of-concept prototypes to practical devices. See [15] for more details on this topic.
In order to avoid the singular structure, it is natural to introduce regularizations into the construction, and instead of the perfect cloak, one considers the approximate cloak or nearcloak. In order to handle the singular structure from the perfect cloaking constructions, the papers [19,18,39] used this truncation of singularities methods to approach the nearly cloaking theory, whereas other papers regularize the 'blow-up-a-point' transformation to 'blow-up-asmall-region' transformation. The small-inclusion-blowup method was studied in [4,29] for the conductivity model.
The papers [1,23] considered the case of electrostatics, which is optics at frequency zero. These papers provide counter examples to uniqueness in Calderón Problem, which is the inverse problem for electrostatics which lies at the heart of Electrical Impedance Tomography [EIT]. EIT consists of determining the electrical conductivity of a medium filling a region Ω by making voltage and current measurements at the boundary ∂Ω and was first proposed in [11]. The fundamental mathematical idea behind cloaking is using the invariance of a coordinate transformation for specific systems, such as conductivity, acoustic, electromagnetic, and elasticity systems. We refer readers to the article [45] for a nice overview of development in EIT and cloaking for electrostatics. We also refer the readers to [21,22,26,35,34,19,33] for the theory behind cloaking in various systems and related developments.
Under the above conditions, we show in Theorem A.1 that the following boundary value problem has a unique solution u ∈ H 1 (Ω), The Dirichlet-to-Neumann map for the boundary value problem (1.2) is defined formally as the map f :→ Λ A f where ν is the outer unit normal to ∂Ω. It is shown in Appendix A that one can define the Dirichlet-to-Neumann (DN) map for the equation (1.2) in a weak sense as follows.
The inverse problem is to recover the quasi-linear co-efficient matrix A(x, t), also called the conductivity from the knowledge of Λ A . Before we provide the definition of cloaking, since the problem of cloaking is essentially that of non-uniqueness, we digress a bit and mention some previous work regarding uniqueness in the inverse problem for the equation considered in (1.2).
In the isotropic case, that is, A(x, t) = a(x, t)I where I denotes the identity matrix and a is a positive C 2,γ (Ω × R) function having a uniform positive lower bound on Ω × [−s, s] for each s > 0, the Dirichlet to Neumann map Λ a determines uniquely the scalar coefficient a(x, t) on Ω × R. This uniqueness result was first proved for the linear case (i.e when a is a function of x alone) in the fundamental paper [43] for N ≥ 3 for and in [37] for N = 2; and in [40] for the quasilinear case. Subsequent work on unique identification of less regular a in the isotropic linear case has been done in [25,24,12] among others for dimensions 3 and higher and in [10,5] for dimension 2.
For the anisotropic/ matrix valued case, it is well known that one cannot recover the coefficient A(x, t) itself because of the following invariance property for the DN map. Choose a smooth diffeomorphism Φ : Ω → Ω such that Φ = Id on ∂Ω and define We make the change of variables y = Φ(x) in (1.2) to get We can write this more compactly as where Φ * A is as defined in (1.3).
Since Φ is identity at ∂Ω, the change of variables does not affect the Dirichlet data and we obtain Thus, for matrix valued co-efficients A(x, t), one can expect uniqueness only modulo such a diffeomorphism. For dimension 2, in the linear case, such uniqueness up to difeomorphsim has been proved in [42,6] and for dimension 3 and higher in [31]. For the quasilinear case, Sun and Uhlmann in [41] showed uniqueness up to diffeomorphism in dimension 2 assuming C 2,γ , 0 < γ < 1 smoothness of the coefficients A. They also proved uniqueness up to diffeomorphism for N ≥ 3 for real analytic coefficients A(x, t). Whether uniqueness up to diffeomorphism can be shown for less regular anisotropic quasi-linear coefficient A(x, t) is an interesting question and remains open.
Equations of the form (1.1) are important and arise in many applications (eg, the stationary form of Richards equation [7], the modeling of thermal conductivity of the Earth's crust [2] or heat conduction in composite materials [28]). One of the goals of our work is to extend the result obtained in [1,23,29] to the quasi-linear elliptic equation (1.2). We propose a change of variable scheme, similar to the one in [1,23] and show how one can, in principle, obtain perfect cloaking, in the context of the equation considered in (1.2), using singular change of variables and approximate cloaking using a regular change of variables. For approximate cloaking we use the small inclusion blow up method as in [29,4]. The singularity and extreme anisotropy resulting from a singular change of variables pose a great challenge in manufacturing invisibility devices. The construction of approximate cloaks using regular change of variables is more tractable. However, these approximate cloaks, though non-singular are still anisotropic. The other major goal of our paper is to construct approximate isotropic cloaks. This will be accomplished using techniques of homogenization. First we construct approximate anisotropic cloaks using regular change of variables. Next, within the framework of homogenization, we approximate each approximate regular anisotropic cloak by a sequence of regular isotropic cloaks. Homogenization process for constructing isotropic regular approximate cloaks has been considered for a linear equation in [20]. We wish to extend the construction in [20] to the quasilinear equation considered in (1.1). To the best of our knowledge, construction of approximate cloaks within the framework of homogenization for a quasi-linear elliptic equation has been done for the first time here.
1.1. Definition of Cloaking. We now provide a mathematical definition of cloaking for the quasi-linear elliptic partial differential equation considered in (1.2). Definition 1. Let E ⊂ Ω be fixed and let σ c : Ω \ E × R be a non negative matrix valued function defined on Ω \ E × R. We say σ c cloaks E if its any extension across E of of following form produces the same Dirichlet-to-Neumann map as a uniform isotropic region irrespective of the choice of A(x, t) ∈ M(α, β, L; Ω × R).
That is, σ c cloaks E in the sense of Definition 1 if Λ σ A = Λ 1 regardless of the choice of the extension A(x, t).
Suppose σ c (x, t) cloaks E in the sense of Definition 1 and let Ω ′ be any domain containing Ω.
Then the Dirichlet-to-Neumann map for This paper is organized as follows. In section 2, we introduce a regular change of variables scheme which will give us the desired approximate cloaking. In section 3, we introduce a singular change of variables and show how perfect cloaking can be achieved. The analysis in this section is essentially a simple extension of the arguments in [1,23,29]. Section 4 is devoted to using homogenization techniques for constructing regular isotropic cloaks. We begin this section by recalling the basic notions of H-convergence in the linear case. Following that we perform the periodic homogenization in the quasi-linear settings. This enables us to construct regular isotropic cloak in the sense made precise in Section 4. In Appendix A, we prove existence and uniqueness for the boundary value problem (1.2) and show that is possible to define, in a weak sense, the DN map associated with (1.2). Moreover, we also state a result on higher regularity for the solutions to (1.1) which will be used in Section 4. Henceforth, we consider the physical dimensions N = 2, 3.

Regular Change of Variables
In this section, we apply a regular change of variables and nearly cloak E in the sense made precise below. For simplicity, we let Ω = B 2 and restrict our attention the case when B 1 = E needs to be nearly cloaked. We extend the result to non-radial domains later.
The basic premise is as follows. Consider a small ball of radius r, B r centered at 0 where r < 1. Construct a map F r (x) : B 2 → B 2 with the following properties.
1) F r is continuous and piecewise smooth.
2) F r expands B r to B 1 and maps B 2 to itself.
It is easy to see that the following candidate for F r satisfies the above properties.
Note that F r is continuous, piecewise smooth and non-singular and (F r ) −1 is also continuous and piecewise smooth. Consider where A(x, t) ∈ M(α, β, L; B 2 × R). By nearly cloaking, we mean that the following must hold where the o(1) term is independent of f and g.
(2.2) is equivalent to By (1.4), the DN map for σ r A is identical to that of (F r ) −1 * σ r A . We will show where the o(1) term is independent of f and g and where Let us now explicitly calculate A r (x, t). We note that for Φ(x) = (F r ) −1 (x), DΦ(x) = rI for x ∈ B 1 . This implies that |DΦ| = r N . From (1.3), it follows that . This ultimately implies that for r ≪ 1, (Unique solution to (2.4) is indeed guaranteed by the existence and uniqueness result proved in Theorem A.1).
Note that u r,f − v f ∈ H 1 0 (B 2 ). Using coercivity for (F r ) −1 * σ A gives us We note that || A r (x, t)|| L ∞ (Br) ≤ C r N−2 where the constant C is independent of r. We apply Hölder's inequality to the last line in (2.7) to obtain where C is independent of r and 1 p 1 + 1 p 2 = 1 2 . We thus have By Poincare's inequality we can say that By Corollary 6.3 in [17], we can say that where C is independent of r and f .
We now use [17,Theorem 8.24] to conclude that where C is independent of r and f . (2.9) and (2.10) together imply (2.11) (2.8) and (2.11) hence give us where C is independent of r and f also.
Let g ∈ H 1 2 (∂B 2 ) be arbitrary. We know that there exists a unique v g ∈ H 1 (B 2 \ B 1 ) which solves the following boundary value problem where C is independent of g.

Faster decay.
In this subsection we derive an improved rate of convergence in (2.15) at the cost of choosing smoother boundary data. Fix f ∈ H 3 2 (∂B 2 ).
Since u r,f solves (2.4) and v f solves (2.5), we get, for any where C is independent of r.
Note that we have the following conditions Then by using (2.18) we have where, 1 p ′ = 1 2 + 1 s . Now by using (2.9), (2.10) and (2.12) together give us or, ). This is of the same order as in (2.12) for N = 3. However, from (2.20) and (2.19), we can conclude that For N = 2, the best possible decay is O(r 2− 2 q ) for any q > 2. This decay rate is faster than the one we obtained in (2.12). where The decay estimates for nearly cloaking here are weaker than the one in [29] where a decay of O(r N ) is obtained. In our case, though we have a slower decay rate, it is sufficient to show approximate cloaking.
So far, we focused on the radial setting because of its explicit character. A similar argument as to the one provided in this section in fact proves (1) G r expands B r to E, . Then the following holds In other words, G r * 1 approximately/nearly cloaks E.

Perfect cloaking
We now show how E ⊂ Ω can be perfectly cloaked using a singular change of variables. For simplicity we first take E = B 1 and Ω = B 2 . The analysis in this section mirrors that in Section 4 of [29].
Let us define F has the following properties 1) F is smooth except at x = 0.
2) F expands 0 to B 1 and maps B 2 to itself.
Our candidate for perfect cloaking will be F * 1. We first calculate F * 1 explicitly. Note that for |x| > 0, where I is the identity matrix andx = x |x| and DF is a symmetric matrix such that a)x is an eigenvector with eigen-value The determinant is thus Hence, whenever 1 < |y| < 2, we have where the right hand side is evaluated at As |y| → 1, F * 1 becomes singular. In fact, following the same arguments as in [29], we get * When N = 2, one eigenvalue of F * 1 → 0 and the other tends to ∞ as |y| → 1 * For N = 3, one eigenvalue goes to 0 while others remain finite as |y| → 1.
Consider σ A defined as Since F * 1 is degenerate near |y| = 1, it is not immediately clear if (3.7) has a weak solution.
We need to put some constraints to get a unique weak solution for (3.7) since σ A (y, t) is not uniformly elliptic. As F * 1 is smooth for |y| > 1, by elliptic regularity v will be uniformly bounded in any compact subset of B 2 \ B 1 . Since F * 1 becomes degenerate near |y| = 1, we ask that any solution v(y) not diverge as |y| → 1. That is, we ask that for some finite C and 1 < ρ < 2.
We first prove a lemma which identifies the value of any solution v to (3.7) on B 2 \ B 1 .
where x = F −1 (y) and u us the harmonic function on B 2 with the same Dirichlet data as v.
Proof. For any compactly supported test function φ in B 2 \ B 1 , by change of variables, we have We thus see that v(F (x)) is weakly harmonic in the punctured ball B 2 \ {0}. Elliptic regularity implies v(F (x)) is strongly harmonic in the punctured ball.
Since, we demand that v satisfy (3.8), u(x) = v(F (x)) has a removable singularity at 0. Thus u(0) is determined by continuity and the extended u is harmonic in the entire ball B 2 .
Clearly, since F = x on ∂B 2 , u has the same Dirichlet data as v on ∂B 2 . The conclusion of the lemma thus holds.
We now show that v as defined in (3.6) solves (3.7).
where u is the harmonic function on B 2 with Dirichlet data f . Proof.
(1) We first show that |∇v| is uniformly bounded in B r for every r < 2. To see this, note that by chain rule and symmetry of DF , we have The matrix DF −1 is uniformly bounded by (3.2) and so is ∇ x u, except perhaps near ∂B 2 as u is harmonic in whole B 2 . Hence |∇ y v| is bounded on B r \ B 1 for any 1 ≤ r < 2. Moreover v is constant on B 1 , and continuous across ∂B 1 . Thus |∇v| is uniformly bounded on B r for r < 2.
(2) Next we prove that |σ A (y, v(y))∇ y v| is also uniformly bounded on B r for every r < 2.
For 1 < |y| < 2, using definition of σ A and chain rule and symmetry of DF , we have The symmetric matrices F * 1 and (DF ) −1 have the same eigenvectors, namelyx andx ⊥ . Taking N = 2, we see that the eigenvalue of F * 1 in directionx ⊥ behaves like |x| −1 , while that of (DF ) −1 behaves like |x|. The eigenvalues of both matrices in direction x are bounded. Thus the product F * 1(DF ) −1 is bounded. This proves Step (2), since ∇ x u is bounded away from ∂B 2 and σ A (y, v)∇v = 0 for y ∈ B 1 as v is defined to be a constant in B 1 . For N = 3, this follows directly from the above proved fact that |∇v| is uniformly bounded on B r for every r < 2. Since F * 1 is uniformly bounded and by (3.2), DF −1 is uniformly bounded we get that σ A (y, v(y))∇ y v is uniformly bounded in B r for every r < 2.
(3) Next we show σ A (y, v)∇ y v · ν → 0 uniformly as |y| → 1 where ν is the unit outer normal to ∂B 1 . We have, y |y| = x |x| =x and |y| → 1 ≡ x → 0. We thus need to show thatx component of (3.9) goes to 0 as |x| → 0. Since F * 1(DF −1 ) is symmetric andx is an eigenvector, it is enough to show that the corresponding eigenvalue tends to 0. We have, from (3.3) and (3.4), that the eigenvalue corresponding tox is which tends to zero for any N ≥ 2 as x → 0 (4) We will use the fact that a bounded vector field X is weakly divergence free on B 2 iff it is weakly divergence free on B 2 \ B 1 and B 1 and the normal flux ν · X is continuous across ∂B 1 to show that σ A ∇v is divergence-free. We have shown in Step (2) above that the vector field σ(y, v)∇ y v is uniformly bounded away from ∂B 2 and by Step (3), We have shown in Lemma 3.2 that v as defined in (3.6) solves (3.7). We have also identified in However, such a possibility does not arise here as the degeneracy for σ A is only near ∂B 1 .
To show that v = u(0) in B 1 , we need to restrict further the class in which v belongs. We assumed earlier that v is uniformly bounded near ∂B 1 . We need a condition to make v continuous across ∂B 1 and a hypothesis on σ A (y, v(y))∇v(y) for the PDE (3.7) to make sense. We thus further assume ∇v ∈ L 2 (B 2 ) and σ A (y, v)∇v ∈ L 2 (B 2 ) (3.10) If v is a weak solution of (3.7) which satisfies (3.8) and (3.10), then v must be given by (3.6).
Since ∇v ∈ L 2 (B 2 ) by assumption, the trace of v on ∂B 1 is well defined. Since v(y) → u(0) as y approaches ∂B 1 from outside, the restriction of v on ∂B 1 must be equal to u(0). By the uniqueness result in Theorem A.1 for the boundary value problem 3.1. Equality of DN maps. We now show that the singular cloak F * 1 cloaks B 1 in the sense of Definition 1.
Proof. By Lemma 3.1, Lemma 3.2 and Lemma 3.3, the Dirichlet-to-Neumann map Λ σ A is well defined. Let f, g ∈ H 1 2 (∂B 2 ). Let φ g ∈ H 1 (B 2 ) be such that φ g | ∂B 2 = g. We have This completes the proof for perfect cloaking.
We have focused on the radial setting because of its explicit character. The analysis extends similarly to non-radial domains.
Proof. The proof in Theorem 4 in [29] goes through, almost word by word, for the quasi-linear equation (1.2). For brevity, we omit the details and refer the reader to Theorem 4 in [29].
Let us remark that (3.13) does not contradict the uniqueness up to diffeomorphism result of Sun and Uhlmann in [41]. The result in [41] crucially depends on the ellipticity of the matrix A(x, t) in (1.2) which we violate by considering a singular change of variables that makes A(x, t) degenerate. In other words, non-uniqueness for the Calderón problem for the quasi-linear elliptic equation (1.2) is possible if we allow A(x, t) to be degenerate.

Homogenization Framework
In Section 3, we showed how it is possible to nearly cloak E ⊂ Ω. The approximate cloak, though non-degenerate, is anisotropic. What we would like to do in this section is to construct near cloaks which are isotropic. This will be done within the framework of homogenization. This section is organized as follows. In section 4.1, we develop the tools needed to prove homogenization for the quasi-linear PDE (1.2) with inhomogeneous boundary conditions for locally periodic microstructures. In section 4.2, we use the results of section 4.1 to construct explicit isotropic regular approximate cloaks for radial domains.
The main idea of homogenization process [3,44] is to provide a (macro scale) approximation to a problem with heterogeneities/microstructures (at micro scale) by suitably averaging out small scales and by incorporating their effects on large scales. These effects are quantified by the so-called homogenized coefficients.
Here we are concerned with the notion of H-convergence for quasi-linear PDEs of the form −div (A ǫ (x, u ǫ (x))∇u ǫ (x)) = 0 in Ω u ǫ = f on ∂Ω. (4.1) We begin by recalling the notion of H-convergence [3,44]  Let A ǫ and A * belong to M(α, β; Ω). We say A ǫ H − → A * or H-converges to a homogenized matrix A * , if A ǫ ∇u ǫ ⇀ A * ∇u in L 2 (Ω) N weak, for all test sequences u ǫ satisfying In particular, we consider the homogenization of linear PDEs of the form and extend it to the whole R N by ǫ-periodicity with a small period of scale ǫ via scaling the coordinate y = x ǫ . The restriction of A ǫ on Ω is known as periodic micro-structures. In this classical case, the homogenized conductivity A * = [a * kl ] is a constant matrix and can be defined by its entries (see [3,8,14]) as where we define the χ k through the so-called cell-problems. For each canonical basis vector e k , consider the following conductivity problem in the periodic unit cell : Let us generalize the above case and consider a locally periodic function A : Ω × Y → R N ×N defined as A(x, y) = [a kl (x, y)] 1≤k,l≤N ∈ M(α, β; Ω × Y ) such that a kl (·, y) are Y -periodic functions with respect to the second variable ∀k, l = 1, 2.., N and for almost every x in Ω. Now we set Then the homogenized conductivity A * (x) = [a * kl (x)] is defined by its entries (see [8,27]) where χ k (·, y) ∈ H 1 (Y ) solves the following cell problem for almost every x in Ω: We end the discussion on homogenization for the linear case by mentioning the following localization result [3].
Let ω be an open subset of Ω. Then A ǫ | ω (restrictions of A ǫ to ω) H-converge to A * | Ω Based on the above localization result, we present the following example.
Let Ω be a domain which is subdivided into domains Ω z , z = 1, 2..., m with Lipschitz boundaries. Let A z (x, y) be periodic functions in y variable with periods Y z for z = 1, 2, ..., m. Let us define for any ǫ > 0, Then where χ z k (·, y) ∈ H 1 (Y z ) solves the following cell problem for almost every x in Ω z : Let us now turn our attention to the equation (4.1). For each fixed ǫ > 0, we consider A ǫ (x, t) ∈ M(α, β, L; Ω × R) where α, β, L are positive, finite and independent of ǫ.
It is shown in Theorem A.1, that, for all fixed ǫ > 0, the weak form of (4.1) with f ∈ H 1 2 (∂Ω) has a unique solution u ǫ ∈ H 1 (Ω) satisfying the estimate where C = C(N, Ω, α, β, L) is independent of ǫ. Thus, standard compactness arguments imply that up to a subsequence (still denoted by ǫ) u ǫ ⇀ u weakly in H 1 (Ω).
Our goal is to get the limiting equation for u ∈ H 1 (Ω). We remain in the class of periodic microstructures and derive the homogenization result in the quasi-linear settings.

We now set
, (x, t) ∈ Ω × R and this is known as periodic micro-structures in quasi-linear settings.
We will show that in this case the homogenized conductivity A * (x, t) = [a * ij (x, t)] ∈ M( α, β, L; Ω × R) can be defined by its entries (see [36,16,9]) where χ k (x, y, t) ∈ H 1 # (Y ) for almost every (x, t) ∈ Ω × R are the solutions of the so-called cell-problems: For each canonical basis vector e k ∈ R N , χ k (x, y, t) satisfy the following problem for y ∈ Y , where Y is the periodic unit cell and for almost every (x, t) ∈ Ω × R: (4.6) The above problem has a unique solution in Note that, from (4.5) it follows that a * kl = a * lk for k, l = 1, .., N and there exist 0 < α < β < ∞ such that, A * (x, t) ∈ M( α, β, L; Ω × R). We will in fact show that t → A * (x, t) is uniformly Lipschitz in Ω.
Before proving the homogenization result, we first discuss few properties of the expected homogenized matrix A * (x, t) defined in (4.5).
and this holds for any t 1 , t 2 ∈ R, where C L is a constant independent of x, y, t 1 , t 2 .
Let us analyze the first term in the above formula. We write Then by using the fact that t → a il (x, y, t) and t → χ k (x, y, t) are uniformly Lipschitz functions, it follows that t → a * il (x, y, t) is uniformly Lipschitz for almost every (x, y) ∈ Ω × Y . Hence we have (4.11).
Next we present the local characterization in the quasi-linear settings analogous to local case in the linear setting mentioned in Proposition 4.1.
Then the homogenized limit A * (x, t)| ω×R is also independent of t, i.e.
Proof. The proof is straightforward from the locality of the second order PDE (4.6) satisfied by χ k (x, y, t); as under the assumption χ k (x, y, t) = χ k (x, y) whenever (x, y, t) ∈ ω × Y × R and by using (4.5) we can conclude that A * (x, t) = A * (x) whenever (x, t) ∈ ω × R.
Now we are going to prove homogenization result for the quasi-linear PDE with inhomogeneous boundary conditions for locally periodic microstructures. Similar result for example, with homogeneous boundary condition and globally periodic microstructures can be found in [36,16]. We first choose Ω to be a smooth enough domain and later relax the assumptions and let Ω be a Lipschitz domain.
Let Ω be a bounded domain in R N with C 2,γ boundary where 0 < γ < 1. Let the matrix A(x, y, t) satisfy: where α, β, L are independent of x, y, t.
We say A ǫ (x, t) in Ω × R, where the homogenized matrix A * (x, t) is defined as in (4.5).
Let δ > 0 be arbitrary, then the domain Ω can be divided in to sub-domains Ω z , z = 1, 2..., m with Lipschitz boundaries and there exists a function u δ constant on every sub-domain Ω z such that for sufficiently small ǫ, we have This immediately implies |u(x) − u δ (x)| ≤ δ x ∈ Ω (4.14) Let u δ,ǫ ∈ H 1 (Ω) be a sequence of solutions to the linear PDEs Then we have the following identity which follows from (4.12) and (4.15).
This means that we have (4.20) where, u δ, * ∈ H 1 (Ω) is the solution to the linear PDE −div A * (x, u δ (x))∇u δ, * (x) = 0 in Ω u δ, * = f on ∂Ω. (4.21) Let w ∈ H 1 (Ω) be the solution to linear PDE −div (A * (x, u(x))∇w(x)) = 0 in Ω w = f on ∂Ω. Then similar to (4.16) we have the following identity which follows from the equations (4.21) and (4.22): Then together with (4.14) and the fact that A * (·, t) is uniformly Lipschitz in x we get the following estimate where C is independent of δ.
Combining (4.19), (4.20) and (4.23) and by writing where , , denotes the usual scalar product on H 1 (Ω) and v in H 1 (Ω) is arbitrary, we obtain where C is independent of ǫ, δ, v and the estimate holds for sufficiently small ǫ. From (4.24), since δ > 0 is arbitrary we can conclude that as ǫ → 0 and by uniqueness of the weak limit we have w = u. Hence we obtain the homogenized equation as −div (A * (x, u(x))∇u(x)) = 0 in Ω u = f on ∂Ω. (4.25) We note that as A * (·, t) is uniformly Lipschitz in x (By Lemma 4.2), so from Theorem A.1, it follows that the above problem (4.25) has the unique solution u ∈ H 1 (Ω).
Next we prove the the L 2 weak limit of the flux A ǫ (x, u ǫ )∇u ǫ is A * (x, u)∇u. Let us consider the following expression Then from the right hand side of (4.26) we get 27) and using the Lipschitz continuity of A ǫ (·, t) we get Thus from (4.26), (4.27) and (4.28) we have where C is independent of ǫ, δ and the estimates hold for sufficiently small ǫ.
In a similar manner, we can prove the estimates where C is independent of δ. Finally we consider here , denotes the usual inner product on L 2 (Ω) N and v ∈ L 2 (Ω) N is arbitrary. Now by using (4.20), (4.29), and (4.30) we conclude from (4.31) that where C is independent of ǫ, δ and v ∈ L 2 (Ω) N . This inequality holds for sufficiently small ǫ.
If we consider δ > 0 to be arbitrary, then (4.32) yields This completes the discussion of our proof.
Since the homogenization co-efficients A ǫ (x, t) = I in the ring Ω ′ \ Ω × R for any ǫ > 0, the conclusion of Theorem 4.1 still holds and passing through the limit in ǫ we obtain where, by localization principle, A * (x, t) = I in Ω ′ \ Ω × R and A * (x, t) in Ω × R is in (4.5).
Moreover, we also have

4.2.
Regular isotropic approximate cloak in R N . In this section we approximate the anisotropic approximate (or near) cloaks σ r A (x, t) as defined in (2.1) by isotropic conductivities, which then will themselves be approximate cloaks. We restrict our attention to the case when Ω = B 2 and E = B 1 needs to be cloaked.
Let σ(s 1 , t) denote the arithmetic mean of σ(s 1 , ·, t) in the second variable as: Then from (4.5) the homogenized conductivity, say σ kl (s 1 , t), turns out to be and can be written as where Π(x) : R N → R N is the projection on to the radial direction, defined by i.e., Π(x) is represented by the matrix |x| −2 xx t , cf. [20].
Next we give more explicit construction of the regular isotropic cloak.This construction is a generalization of the linear case presented in [20]. Let us consider functions φ : R → R and φ M : R → R given by and where a k (x, t), k = 1, 2 are chosen positive smooth functions such that σ(x, r ′ , t) satisfies the conditions immediately following (4.33) viz A1), A2) and A3). In particular, we choose a 1 and a 2 to satisfy conditions A1) and A2), for some possibly different choice of constants α, β, L. And for some positive integer M, we define ζ j : R → R to be 1−periodic functions, where φ M is as defined in (4.39).
To this end, we introduce a new parameter η > 0 and solve for each (x, t) the parameters a 1 (x, t), a 2 (x, t) from the following expressions of the equations for the harmonic and arithmetic averages for 1 < R < 2.
Now we first let ǫ → 0, then η → 0 and finally R → 1, the obtained homogenized conductivities approximate better and better the cloaking conductivity σ A (cf (3.5)). Thus we choose appropriate sequences R n → 1, η n → 0 and ε n → 0 and denote Let Ω ′ = B 3 . The above sequence σ n (x, t) is the desired regular isotropic sequence which approximates the cloaking for quasilinear problem in the following sense: Let is as defined in (3.5) in Ω and we assume σ A (x, t) = I on Ω ′ \ Ω × R.

Now if f ∈ H
Since A(x,v) ∈ L ∞ (Ω) we use Lax Milgram theorem and conclude that there exists a unique u that solves the linearized boundary value problem (A.1) with ||u|| H 1 (Ω) ≤ C||f || H 1/2 (∂Ω) where C depends only on the parameters mentioned in the statement of the theorem.
The existence of a solution to (1.2) will be proved if we show T has a fixed point in S. We will use Schauder fixed point theorem to to prove this. For that, we need to show T is a continuous operator and that S is closed and convex subset of L 2 .
Since f is fixed and ǫ is arbitrary, ||(A(x,v n )−A(x,v))∇u|| L 2 (Ω) → 0 as n → ∞. By (A.3), u n → u in L 2 (Ω) proving the continuity of T. (Actually this only shows that there is a subsequence of u n that converges in L 2 (Ω) to u. We run the entire argument with any arbitrary subsequence of u n and thereby get by the above argument that every subsequence of u n has a subsequence that converges in L 2 (Ω) to u. Hence we can say u n → u in L 2 (Ω).) Note that T is a compact operator in L 2 topology by Rellich compactness theorem.
Switch the roles of u 1 and u 2 to conclude that u 1 = u 2 a.e x ∈ Ω. where u is the unique solution to (1.2) and v is any H 1 (Ω) function with trace g. This is the natural generalization of the Dirichlet-to-Neumann map to a quasi-linear equation of divergence type.
We end this section by stating a result on higher global regularity of solutions to the quasilinear elliptic equation (1.2).
Proof. The existence and uniqueness result for the quasilinear boundary value problem considered in (A.8) can be found in [17] or [30].