Diffuse Approximations for the Area and Willmore Functionals

In recent years, two different regularizations of the area functional have proven to be highly effective in minimal surface theory. These are the approximation via Allen--Cahn energies and the approximation based on the fractional perimeter. With both methods, it is possible to recover some cornerstone results regarding the existence and multiplicity of minimal hypersurfaces.The goal of this thesis is to investigate whether it is possible to approximate other geometric functionals related, in some ways, to the area functional in codimension one, starting from these approximations.More precisely, we are interested in exploring two different directions. On one hand, we want to consider the area functional in higher codimension. On the other hand, we aim to consider functionals that depend on the higher-order derivatives of the parameterizations of the set. Specifically, we will focus on the Willmore functional.Concerning the latter functional, in 1991, De Giorgi conjectured a possible approximation based on the first variations of the Allen--Cahn energies. The first result presented in this thesis provides a negative answer to this conjecture, and it first appeared in a joint work with G. Bellettini and N. Picenni.This result was rather unexpected; in fact, a modified version of the conjecture was proven in space dimension n=2,3 by Röger and Schätzle. In particular, this modification was originally proposed by Bellettini and Paolini in order to simplify the initial problem. In the same spirit, we will propose a nonlocal approximation of the Willmore functional using the first variations of the fractional Allen--Cahn energies and will prove the corresponding Gamma-limsup estimate in this context. This represents the second result presented in this thesis, which first appeared in a joint work with H. Chan and M. Inversi.As for the area functional in higher codimension, in the final part of this thesis, we will introduce, for every s in (0,1), a notion of s-fractional volume for boundaries of closed oriented hypersurfaces in the Euclidean space. Moreover, we will show that this quantity Gamma-converges, with respect to the flat distance, and pointwise converges to the (n-2)-dimensional area, in the limit as s goes to 1. In addition, we will discuss how this fractional volume can be seen as the codimension-two analogue of the well-established notion of fractional perimeter. This result first appeared in a joint work with M. Caselli and N. Picenni.