This issuePrevious ArticleRecovering an obstacle using integral equationsNext ArticleA Newton method for reconstructing non star-shaped domains in
electrical impedance tomography
An estimate for the free Helmholtz equation that scales
Wavelength plays a distinguished role in classical
electromagnetic and acoustic scattering. Most significant
features of the far field patterns radiated by a
collection of sources or scatterers are related to their
sizes and relative distances, measured in
wavelengths. These significant features are reflected in
the invariance of the Helmholtz equation with respect to
translation, and its homogeneous scaling with respect
to dilations. The weighted norms that were first
developed to capture the correct decay properties of
waves in Rn do not scale homogeneously and are not
invariant with respect to translation. Lp
estimates scale homogeneously and commute with
translations and rotations. However, their scaling
properties give estimates with a weaker dependence
on wavenumber (for bounded sources and scatterers with
support that extends over many wavelengths). We
introduce some norms and estimates that commute with
translations and scale homogeneously under dilations,
while retaining the same sharp dependence on wavelength
for extended sources as that of the weighted
estimates.