Let \(\Omega\) be a bounded open set in \(\mathbb R^d\) containing a small volume inclusion \(x_0+\varepsilon B\), where \(B\) is a bounded set with Lipschitz boundary and \(\varepsilon>0\) is as small as to satisfy \(\text{dist}(\partial \Omega,x_0+\varepsilon B)>d_0\). Related to \(\Omega\) the authors consider the Neumann problem for the equation \(\nabla\cdot (D^\varepsilon\nabla u^\varepsilon)=0\) with the additional condition \(\int_{\partial \Omega} u^\varepsilon\,d\sigma=0\), when the diffusion coefficient \(D^\varepsilon\) is assumed to have the properties: (i) \(D^\varepsilon(x)\geq C_0>0\) for a.e. \(x\in \Omega\); (ii) \(D^\varepsilon(x)=D_0(x)\) for all \(x\in {\overline \Omega}\setminus \overline{x_0+\varepsilon B}\); (iii) \(D^\varepsilon(x)=D_0(x)+D_1((x-x_0)/\varepsilon)\); (iv) \(D_0\in C^\infty({\overline \Omega})\) and \(D_1\in L^\infty(\Omega)\).
The authors provide an asymptotic expansion on the boundary \(\partial \Omega\) of the solution \(u^\varepsilon\), up to order \(\varepsilon^{2d}\), in terms of the small parameter \(\varepsilon\) and the polarizations tensors \(M_{i,j}\) and \(M^2_{i,j}\) first in the case when function \(D_1\) – taking account of the inclusion – is regular, i.e. \(D_1\in W^{1,\infty}(\Omega)\) and, then, when \(D_1\) is in the general case (iv).
The second part of the paper deals with the Neumann problem related to the Helmholtz equation in \(\Omega\), \(\Delta v^\varepsilon+\varepsilon^{-2+\eta}[q_0(x)+q_1((x-x_0)/\varepsilon)]v^\varepsilon=0\), with \(\eta\in [0,2]\), \(d\in [2,5]\) and \(q_0,q_1\in L^\infty(\Omega)\). When \(q_0\) is negative such a problem models the wave propagation in a medium perturbed by a small parameter \(\varepsilon\) with a refractive index of order \(\varepsilon^{-2+\eta}\).
Under suitable assumptions the authors prove an asymptotic expansion on the boundary \(\partial \Omega\) of the solution \(v^\varepsilon\), up to order \(\varepsilon^{2d}\), in terms of the small parameter \(\varepsilon\) and the polarization tensors \(Q_{i,j}\) (related to \(q_1\) only) and \(Q^\eta_{i,j}\).
Finally, when \(D_0(x)=D_0= \text{const.}\) and \(D_1\in C^2(\Omega)\), via the Liouville transformation, ensuring the equivalence between the diffusion equation \(\nabla\cdot (D^\varepsilon\nabla u^\varepsilon)=0\) and the Helmholtz equation \(\Delta\{[D^\varepsilon(x)]^{1/2}v^\varepsilon\} +[D^\varepsilon(x)]^{-1/2}\Delta\{[D^\varepsilon(x)]^{1/2}\}v^\varepsilon=0\), where \(D^\varepsilon(x)=D_0+D_1((x-x_0)/\varepsilon)\), the authors show the relation \(M_{i,j}=\widetilde M_{i,j}\) between the diffusion and the Helmholtz tensors.