Continuity of the temperature in a multi-phase transition problem.

Locally bounded, local weak solutions to a doubly nonlinear parabolic equation, which models the multi-phase transition of a material, is shown to be locally continuous. Moreover, an explicit modulus of continuity is given. The effect of the p-Laplacian type diffusion is also considered.

3D Multi-Branched SnO2 Semiconductor Nanostructures as Optical Waveguides.

Nanostructures with complex geometry have gathered interest recently due to some unusual and exotic properties associated with both their shape and material. 3D multi-branched SnO2 one-dimensional nanostructrures, characterized by a "node"-i.e., the location where two or more branches originate, are the ideal platform to distribute signals of different natures. In this work, we study how this particular geometrical configuration affects light propagation when a light source (i.e., laser) is focused onto it. Combining scanning electron microscopy (SEM) and optical analysis along with Raman and Rayleigh scattering upon illumination, we were able to understand, in more detail, the mechanism behind the light-coupling occurring at the node. Our experimental findings show that multi-branched semiconductor 1D structures have great potential as optically active nanostructures with waveguiding properties, thus paving the way for their application as novel building blocks for optical communication networks.