# American Institute of Mathematical Sciences

March  2018, 17(2): 709-727. doi: 10.3934/cpaa.2018037

## On the nonlocal curvatures of surfaces with or without boundary

 1 DADU, University of Sassari, Palazzo del Pou Salit, Piazza Duomo 6,07041 Alghero (SS), Italy 2 Accademia Nazionale dei Lincei, Palazzo Corsini, Via della Lungara 10,00165 Roma, Italy 3 Department of Mathematics, University of Roma TorVergata, Via della Ricerca Scientifica 1,00133 Roma, Italy 4 Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60660, USA

* Corresponding author: Brian Seguin

Received  January 2017 Revised  September 2017 Published  March 2018

For surfaces without boundary, nonlocal notions of directional and mean curvatures have been recently given. Here, we develop alternative notions, special cases of which apply to surfaces with boundary. Our main tool is a new fractional or nonlocal area functional for compact surfaces.

Citation: Roberto Paroni, Podio-Guidugli Paolo, Brian Seguin. On the nonlocal curvatures of surfaces with or without boundary. Communications on Pure & Applied Analysis, 2018, 17 (2) : 709-727. doi: 10.3934/cpaa.2018037
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##### References:
The solid line depicts $\mathcal{S}$. The set $\mathcal{A}_i(z, 1)$ is shown in dark grey, the set $\mathcal{A}_e(z, 1)$ in light grey
The intersection of an open set $E$ and the half plane $\pi(z, {\pmb{e}})$
Several different ways a straight-line segment can intersect $\partial E$
How the mapping $\Phi$ in (11) associates $(z, {\pmb{u}}, \xi, \eta)$ with the pair of points $x$ and $y$ in ${\mathbb{R}}^n$
A depiction of $\mathcal{S}$, $\mathcal{S}_\varepsilon$, and $\mathcal{V}_\varepsilon$
Here, $\phi(z)>0$; $y_1, y_2 \in \mathcal{A}_e(z, \phi)$ and $y_3, y_4\in \mathcal{A}_i(z, \phi)$
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