By Fanghua Lin, New York University / NYU Shanghai
The formation and dynamics of interfaces are often dictated by an underlying energy functional and the medium they are located in. The variational energy functional can either be from the interface or the bulk or from a mixture of both.
In the former case, the geometric aspect of interfaces may often be related to "minimal surfaces", the latter two can vary in various situations. In this talk, I shall discuss the optimal partition problem for Dirichlet eigenvalues. Such problems appeared in classical optimal designs, and recently also in Data Searchings. It is a typical class of problem in which the interfaces are driven by the bulk energy. I shall discuss its existence, regularity and the problem of the large N asymptotics.