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.