Inserted: 12 jan 2021
Last Updated: 1 apr 2021
In this thesis 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. The functionals we consider are the $p$-elastic energy of an immersed curve, defined as the sum of length and $L^p$-norm of the curvature vector, and the Willmore energy of an immersed surface, which is the $L^2$-norm of the mean curvature of the surface. We shall consider problems of a variational nature both in the smooth setting of manifolds and in the context of geometric measure theoretic objects. The necessary definitions and preliminaries are collected and discussed 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.