On the Regularity of Non-scattering Anisotropic Inhomogeneities

Abstract

In this paper we examine the necessary conditions for an anisotropic inhomogeneous medium to be non-scattering at a single wave number and for a single incident field. These conditions are expressed in terms of the regularity of the boundary of the inhomogeneity. We assume that the coefficients, characterizing the constitutive material properties of the medium, are sufficiently smooth, and that the incident wave is appropriately non-degenerate. Our analysis utilizes the Hodograph transform as well as regularity results for nonlinear elliptic partial differential equations. Our approach requires that the boundary a-priori is of class $C^{1,\alpha }$ for some $0<\alpha <1$.

Appendix

Here we show that the first variation of the nonlinear partial differential equations (12)–(13) is uniformly strongly elliptic with a proper oblique derivative boundary condition; this allows us to apply Theorem 11.2 in [1]. The first variation

$$\begin{aligned} {\mathcal {L}}_z(y,\phi ,\nabla \phi )=0 \end{aligned}$$

of the set of nonlinear equations (12)–(13), in shorthand written as

$$\begin{aligned} {\mathcal {F}}(y,z,\nabla z)=0~, \end{aligned}$$

is defined by means of

$$\begin{aligned} {\mathcal {L}}_z={\mathcal {I}}'(0) \qquad \text{ where } \qquad {\mathcal {I}}(\tau )={\mathcal {F}}(y,z+\tau \phi ,\nabla (z+\tau \phi ))~; \end{aligned}$$

see [16, Chapter 8] and also [1, Page 684]. Assuming that z is a function of d variables, $\nabla z$ is regarded as d arguments, namely $\partial _{j}z$, $j=1,\ldots ,d$, of ${\mathcal {F}}$.

A straightforward calculation shows that the first variation of (12) has the following matrix coefficient ${\widetilde{A}}$ in the principal (divergence form second order) part of the operator at each point $y\in V^+$,

$$\begin{aligned} {\widetilde{A}} = -\frac{1}{\partial _{1} z} \begin{bmatrix} ({\widetilde{\nabla }}^{\top }z)\,A\,{\widetilde{\nabla }}z &{}\frac{1}{\partial _{1} z}(\textbf{a}'-A_{d-1}\nabla _{y'}z)^{\top }\\ \frac{1}{\partial _{1} z}(\textbf{a}'-A_{d-1}\nabla _{y'}z)&{} A_{d-1} \end{bmatrix}, \end{aligned}$$

where $\textbf{a}'^{\top }=(a_{12},a_{13},\ldots ,a_{1d})$ and $A_{d-1}=(a_{ij})_{i,j=2}^d$, that is,

$$\begin{aligned} A=\begin{bmatrix} a_{11}&{}\textbf{a}'^{\top }\\ \textbf{a}'&{}A_{d-1} \end{bmatrix}~. \end{aligned}$$

For simplicity of notations, we have again used $A=(a_{ij})$ to denote $A_H = A\circ H^{-1}=(a_{ij}\circ H^{-1})$. We calculate for $\xi =(\xi _1,\xi '^{\top })^{\top }\in {\mathbb {R}}^d$ that

$$\begin{aligned} -(\partial _{1} z)\, \xi ^{\top } {\widetilde{A}}\,\xi = \xi _1^2~({\widetilde{\nabla }}^{\top }z)\,A\,{\widetilde{\nabla }}z +2\,\xi _1\frac{1}{\partial _{1} z}(\textbf{a}'-A_{n-1}\nabla _{y'}z)^{\top }\xi ' +\xi '^{\top } {A}_{n-1}\xi ' = {\widetilde{\xi }}_z^{\top }A\,{\widetilde{\xi }}_z~, \end{aligned}$$

with

$$\begin{aligned} {\widetilde{\xi }}_z^{\top } = \left( \frac{1}{\partial _{1} z}\xi _1~,~ (\xi '-\frac{\nabla _{y'}z}{\partial _{1} z}\,\xi _1)^{\top }\right) =(0,\xi '^{\top })+\xi _1{\widetilde{\nabla }}^{\top }z~. \end{aligned}$$

It then follows by continuity (and compactness) that, for all $y\in V^+$,

$$\begin{aligned} (c_4)^{-1}|\xi |^2~\leqq ~ -\xi ^{\top } {\widetilde{A}}\,\xi ~\leqq ~ c_4 |\xi |^2, \end{aligned}$$

with some positive constants $c_4>0$ depending on $c_0$ in (2), $c_2$ in (10), and $\Vert {\widetilde{\nabla }}z\Vert _{V^+\cap \Sigma }=\Vert \nabla _x w\Vert _{{\overline{\Omega }}\cap B_r(0)}$. This verifies the uniform, strong ellipticity condition.

In regards to the boundary condition (13), the principal part of the first variation at each boundary point $y=(0,y')$ is given by $\sum _{j=1}^{d}b_j\partial _{j}$ with

$$\begin{aligned} b_1 = -\frac{1}{\partial _{1}z} \left( 2\,({\widetilde{\nabla }}^{\top }z)\,A\,{\widetilde{\nabla }}z + ({\widetilde{\nabla }}^{\top }z)\,(A-I)\,\nabla _x v\right) =-\frac{1}{\partial _{1}z}({\widetilde{\nabla }}^{\top }z)\,A\,{\widetilde{\nabla }}z~, \end{aligned}$$

and

$$\begin{aligned} b_j = -\frac{1}{\partial _{1}z} \left( 2A\,{\widetilde{\nabla }}z + (A-I)\,\nabla _x v\right) _j, \qquad j=2,\ldots ,d~, \end{aligned}$$

where again, by abuse of notation, we use A in place of $A\circ H^{-1}(0,y')$ and $\nabla _x v$ in place of $(\nabla _x v)\circ H^{-1}(0,y')$. From (2) and (14) we get that

$$\begin{aligned} 0< c_0^{-1}c_2^{-3}<c_0^{-1}c_2^{-1}|{\widetilde{\nabla }}z|^2< -b_1 \qquad \text{ for } \text{ all } y=(0,y') \,\,\text{ on }\,\,\Sigma ~. \end{aligned}$$

This ensures that the linearized boundary condition (on $\Sigma$) is a proper oblique derivative condition, and as a consequence it is “covering” for the linearized second order elliptic differential operator.