DOI: 10.68381/jca27043 ISSN: 0944-6532
A Varifold Perspective on the p-Elastic Energy of Planar Sets
Marco Pozzetta
Under suitable regularity assumptions, the
p
p
-elastic energy of a planar set
E\subset\mathbb{R}^2
E
⊂
R
2
is defined as
\mathcal{F}_p(E)=\int_{\partial E} 1 + |k_{\partial E}|^p \,\, d\mathcal{H}^1,
F
p
(
E
)
=
∫
∂
E
1
+
∣
k
∂
E
∣
p
d
H
1
,
where
k_{\partial E}
k
∂
E
is the curvature of the boundary
\partial E
∂
E
. In this work we use a varifold approach to investigate this energy, that can be well defined on varifolds with curvature. First we show new tools for the study of
1
1
-dimensional curvature varifolds, such as existence and uniform bounds on the density of varifolds with finite elastic energy. Then we characterize a new notion of
L^1
L
1
-relaxation of this energy by extending the definition of regular sets by an intrinsic varifold perspective, also comparing this relaxation with the classical one of G. Bellettini and L. Mugnai [Characterization and representation of the lower semicontinuous envelope of the elastica functional, Annales de l'Institut Henri Poincaré (C), Non Linear Analysis 21(6) (2004) 839–880; A varifolds representation of the relaxed elastica functional, J. Convex Analysis 14(3) (2007) 543–564]. Finally we discuss an application to the inpainting problem, examples and qualitative properties of sets with finite relaxed energy.