ENHANCEMENT OF NEAR-CLOAKING 3 region. We show that the cloaking problem becomes easier if only scattered waves at certain angles are visible. Finally, we extend our construction to the enhanced reshaping problem. We show how to make any target look like a disc with homogeneous physical parameters. 2. Enhancement of near cloaking in the quasi-static limit 2.1. Principles. To explain the principle of our new construction of cloaking structures, we review the results on the conductivity equation obtained in [4]. Let Ω be a domain in R2 containing 0 possibly with multiple components with Lipschitz boundary. For a given harmonic function H in R2, consider (2.1) ∇· σ0χ(R2 \ Ω) + σχ(Ω) ∇u = 0 in R2, u(x) H(x) = O(|x|−1) as |x| ∞, where σ0 and σ are conductivities (positive constants) of R2 and Ω, respectively. Here and throughout this paper, χ(Ω) (resp. χ(R2\Ω)) is the characteristic function of Ω (resp. χ(R2 \ Ω)). If the harmonic function H admits the expansion H(x) = H(0) + n=1 rn ( an(H) c cos + an(H) s sin nθ) ) with x = (r cos θ, r sin θ), then we have the following formula (u H)(x) = m=1 cos 2πmrm n=1 ( M cc mn ac n (H) + M cs mn as n (H) ) m=1 sin 2πmrm n=1 ( Mmnan(H) sc c + Mmnan(H) ss s ) as |x| ∞. (2.2) The coefficients M cc mn , M cs mn , M sc mn , and M sc mn are called the contracted generalized polarization tensors. In [4], we have constructed structures with vanishing contracted generalized polarization tensors for all |n|, |m| N. We call such structures GPT-vanishing structures of order N. For doing so, we use a disc with multiple coatings. Let Ω be a disc of radius r1. For a positive integer N, let 0 rN+1 rN . . . r1 and define (2.3) Aj := {rj+1 r = |x| rj}, j = 1, 2, . . . , N. Let A0 = R2 \ Ω and AN+1 = {r rN+1}. Set σj to be the conductivity of Aj for j = 1, 2, . . . , N + 1, and σ0 = 1. Let (2.4) σ = N+1 j=0 σjχ(Aj). Because of the symmetry of the disc, one can easily see that (2.5) Mmn[σ] cs = Mmn[σ] sc = 0 for all m, n, (2.6) Mmn[σ] cc = Mmn[σ] ss = 0 if m = n, and (2.7) Mnn[σ] cc = Mnn[σ] ss for all n.
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