We present a variational framework for shape optimization
problems that hinges on devising energy decreasing flows based on
shape differential calculus followed by suitable space and time
discretizations (discrete gradient flows). A key ingredient is
the flexibility in choosing appropriate descent directions by
varying the scalar products, used for computation of normal
velocity, on the deformable domain boundary. We discuss
applications to image segmentation, optimal shape design for PDE,
and surface diffusion, along with several simulations exhibiting
large deformations as well as pinching and topological changes in
finite time. This work is joint with E. Baensch, G. Dogan, P.
Morin, and M. Verani.