Worcester Polytechnic Institute
Worcester, Massachusetts, U. S. A.
Many problems in materials science, such as optimal design, phase transitions, liquid crystals, etc., involve free interfaces. A model problem is the optimal design of composites of two materials, with an energy penalty on the length of the interface separating the materials. Recently, Ambrosio-Buttazzo and Fanghua Lin independently studied such problems, showing existence of optimal designs, Holder continuity of the corresponding fields, and some regularity of the regions occupied by each material, including their essential openness. In this talk, we outline a proof that, in the plane, the components of these regions are a positive distance apart.