  We examine the validity of the hypothesis of self-similarity in systems
coarsening under the driving force of interface energy reduction in which three
dimensional particles are intersected by a one or two dimensional diffusion
matrix. In both cases, solute fluxes onto the surface of the particles, assumed
spherical, depend on both particle radius and inter-particle distance. We argue
that overall mass conservation requires independent scalings for particle sizes
and inter-particle distances under magnification of the structure, and predict
power law growth for the average particle size in the case of a one dimensional
matrix (3D/1D), and a weak breakdown of self-similarity in the two dimensional
case (3D/2D). Numerical calculations confirm our predictions regarding
self-similarity and power law growth of average particle size with an exponent
1/7 for the 3D/1D case, and provide evidence for the existence of logarithmic
factors in the laws of boundary motion for the 3D/2D case. The latter indicate
a weak breakdown of self-similarity.
