Under proper hypotheses, we rigorously derive the Plateau angle conditions at triple junctions of diffused interfaces in three dimensions, starting from the vector-valued Allen-Cahn equation with a triple-well potential. Our derivation is based on an application of the divergence theorem using the divergence-free form of the equation via an associated stress tensor.

Abstract. We rigorously derive a weak form of the Lifshitz-Slyozov-Wagner equation as the homogenization limit of a Stefan-type problem describing reaction-controlled coarsening of a large number of small spherical particles. Moreover, we deduce that the effective mean-field description holds true in the particular limit of vanishing surface-area density of particles.