On the nonlocal curvatures of surfaces with or without boundary

Roberto Paroni; Podio-Guidugli Paolo; Brian Seguin

doi:10.3934/cpaa.2018037

Article Contents

2018, Volume 17, Issue 2: 709-727. Doi: 10.3934/cpaa.2018037

This issue Previous Article Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature Next Article

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
* Corresponding author: Brian Seguin

Received: January 2017

Revised: September 2017

Published online: March 2018

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Abstract

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.

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Mathematics Subject Classification: Primary:35R11, 49Q05;Secondary:53A05.

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Figure 1. 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

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Figure 2. The intersection of an open set $E$ and the half plane $\pi(z, {\pmb{e}})$

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Figure 3. Several different ways a straight-line segment can intersect $\partial E$

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Figure 4. How the mapping $\Phi$ in (11) associates $(z, {\pmb{u}}, \xi, \eta)$ with the pair of points $x$ and $y$ in ${\mathbb{R}}^n$

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Figure 5. A depiction of $\mathcal{S}$, $\mathcal{S}_\varepsilon $, and $\mathcal{V}_\varepsilon $

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Figure 6. 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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