  At the early, ejecta dominated, stage of supernova remnant (SNR) expansion a
fraction of the swept-up circumstellar magnetic field is dynamically compressed
to approximate equipartition at the contact discontinuity separating the SN
progenitor's wind (or the ISM) and the ejecta. We propose that the thin
non-thermal X-ray filaments observed by the Chandra satellite in several young
SNRs are associated with such ``pile-up'' of the magnetic field. We use a
one-dimensional diffusion-convection transport equation to describe the
propagation of non-thermal electrons near the contact discontinuity of a young
SNR and to calculate spatially resolved emission spectra in the X-ray and TeV
bands. The results suggest that the high-energy electrons are possibly
accelerated at the forward shock, and emitting efficiently only when they
diffuse into regions of high magnetic field near the contact discontinuity.
Much more likely, though, is that they are locally accelerated at the contact
discontinuity, in which case the acceleration cannot be related to Fermi-type
processes and should occur due to other plasma mechanisms. As a consequence,
the forward shock in young SNRs is inconspicuous and often unobservable,
similar to that in the Crab nebular.
