Creating desired potentials by embedding small inhomogeneities

Abstract

The governing equation is $[{\nabla }^2+k^2-q(x)]u=0$ in ${\mathbb{R}}^3$. It is shown that any desired potential $q(x)$, vanishing outside a bounded domain $D$, can be obtained if one embeds into D many small scatterers $q_m(x)$, vanishing outside balls $B_m:=\{x:|x-x_m|<a\}$, such that $q_m=A_m$ in $B_m$, $q_m=0$ outside $B_m$, $1\le m\le M$, $M=M(a)$. It is proved that if the number of small scatterers in any subdomain $\Delta$ is defined as $N(\Delta ):={\sum }_{x_m\in \Delta }1$ and is given by the formula $N(\Delta )=|V(a){|}^{-1}{\int }_{\Delta }n(x)dx[1+o(1)]$ as $a\to 0$, where $V(a)=4\pi a^3/3$, then the limit of the function $u_M(x)$, ${\lim }_{a\to 0}{U}_M=u_e(x)$ does exist and solves the equation $[{\nabla }^2+k^2-q(x)]u=0$ in ${\mathbb{R}}^3$, where $q(x)=n(x)A(x)$,and $A(x_m)=A_m$. The total number $M$ of small inhomogeneities is equal to $N(D)$ and is of the order $O(a^{-3})$ as $a\to 0$. A similar result is derived in the one-dimensional case.

References in corpus (4)

• Many-body wave scattering by small bodies and applications
• Wave scattering by many small particles embedded in a medium
• Electromagnetic wave scattering by small bodies
• Distribution of particles which produces a "smart" material

Cited by in corpus (4)

• Creating materials with a desired refraction coefficient
• Scattering by many small inhomogeneities and applications
• Many-body wave scattering problems in the case of small scatterers
• Scattering of scalar waves by many small particles