Phase field models with connectedness constraints
Patrick Dondl (Freiburg)
Abstract: Phase field models are useful tools for the approximation
of geometric variational problems, the classical example being the
Modica-Mortola-functional consisting of a gradient penalization
and a double-well potential. This functional, with its terms suitably
scaled, converges in the sense of Gamma-convergence to the perimeter,
implying that the zero level sets of its minimizers approximate minimal
surfaces. We consider the problem of including topological constraints
in phase field models, in particular the question whether it is possible
to constrain the zero level sets or sub/super-level sets to be connected
by adding a suitable term to the energy.
Our penalty term is based on a diffuse quantitative version of
path-connectedness. As a first application, we prove convergence of the
approximating penalized Modica-Mortola energies in the sense of
Gamma-convergence to a connected perimeter and present numerical
results and applications to image segmentation and Ohta-Kawasaki
functionals modeling charged droplets. Furthermore, we consider a
phase field model for Willmore's energy. The main problem in this
case is that, to enforce connectedness of surfaces in the limit,
a fairly strong convergence for sequences of bounded diffuse
Willmore energy on sets of codimension 2 is required. In two
space dimensions, uniform convergence (away from the limiting
surface) holds, for the three dimensional case we introduce
the notion of morally uniform convergence.