A number of fundamental questions about the structure of matter can be formu lated as problems in energy-driven pattern formation. For example\, Gamow& #39\;s liquid drop model addresses the shape and stability properties of a tomic nuclei as arising from competition between long-range Coulomb repuls ion and short-range attraction modeled by surface tension. We discuss a re lated model for perpendicularly oriented dipoles in the plane\, and in whi ch perimeter (representing line tension) and regularized 3D dipolar repuls ion compete under a volume constraint. Examples of such situations are Lan gmuir monolayers and the patterns formed in ultrathin ferromagnetic films with perpendicular anisotropy.In contrast to previously studied similar pr oblems\, the nonlocal term contributes to the perimeter term to leading or der for small regularization cutoffs. For subcritical dipolar strengths we prove that the limiting functional is a renormalized perimeter and that f or small cutoff lengths all minimizers are disks. For critical dipolar str ength\, we identify the next-order ?-limit when sending the cutoff length to zero and prove that with a slight modification of the dipolar kernel th ere exist masses for which classical minimizers are not disks.

This talk is joint work with Cyrill B. Muratov

LOCATION:Big Seminar room Ground floor / Office Bldg West (I21.EG.101)\, IS T Austria ORGANIZER:jdeanton@ist.ac.at SUMMARY:A nonlocal isoperimetric problem with dipolar repulsion URL:https://talks-calendar.app.ist.ac.at/events/1701 END:VEVENT END:VCALENDAR