• Willmore-type energies
• elastic energy
• geometric flows

We discuss variational problems concerning Willmore-type energies of curves
and surfaces. By Willmore-type energy of an immersed manifold we mean a functional depending
on the volume (length or area) of the manifold and on some L^p-norm of the mean curvature
of the manifold. We shall
consider problems of a variational nature both in the smooth setting of manifolds and in the
context of geometric measure theoretic objects. Necessary definitions and preliminaries are
collected in Chapter 1.
We then address the following problems.
• In Chapter 2 we consider a gradient flow of the p-elastic energy of immersed curves into
complete Riemannian manifolds. We investigate the smooth convergence of the flow to
critical points of the functional, proving that suitable hypotheses on the sub-convergence
of the flow imply the existence of the full limit of the evolving solution.
• In Chapter 3 we address the problem of finding a generalized weak definition of p-elastic
energy of subsets of the plane satisfying some meaningful variational requirement. We find
such a definition by characterizing a suitable relaxed functional, of which we then discuss
qualitative properties and applications.
• In Chapter 4 we study the minimization of the Willmore energy of surfaces with boundary
under different boundary conditions and constraints. We focus on the existence theory
for such minimization problems, proving both existence and non-existence theorems, and
some functional inequalities.