CMSA Interdisciplinary Science Seminar: An isoperimetric problem with a competing nonlocal singular term
Yijing Wu - Department of Mathematics, University of Maryland, College Park
We are interested in the minimization problem of a functional in which the perimeter is competing with a nonlocal singular term comparable to a fractional perimeter, with volume constraint. We prove that minimizers exist and are radially symmetric for small mass, while minimizers cannot be radially symmetric for large mass. For large mass, we prove that the minimizing sequences either split into smaller sets that drift to infinity or develop fingers of a prescribed width. We connect these two alternatives to a related minimization problem for the optimal constant in a classical interpolation inequality.