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
References:
[1]

N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35 (2014), 793-815.

[2]

G. Alberti, Distributional Jacobian and singularities of Sobolev maps, Ric. Mat., LIV (2006), 375-394.

[3] L. AmbrosioN Fusco and D Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000.
[4]

L. AmbrosioG. De Philippis and L. Martinazzi, Γ-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.

[5]

X. Cabré, M. M. Fall, J. Solá-Morales and T. Weth, Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delauney, Journal für die reine und angewandte Mathematik published on line 2016-04-16.

[6]

L. Caffarelli, Surfaces minimizing nonlocal energies, Rend. Lincei Mat. Appl., 20 (2009), 281-299.

[7]

L. Caffarelli, The mathematical idea of diffusion, Enrico Magenes Lecture, Pavia, March 2013.

[8]

L. CaffarelliJ.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.

[9]

L. Caffarelli and P. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Rational Mech. Anal., 195 (2010), 1-23.

[10]

L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limitingarguments, Adv. Math., 248 (2013), 843-871.

[11]

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.

[12]

A. ChambolleM. Morini and M. Ponsiglione, A nonlocal mean curvature flow and its semi-implicit time-discrete approximation, SIAM J. Math. Anal., 44 (2012), 4048-4077.

[13]

A. Chambolle, M. Morini and M. Ponsiglione, Minimizing movements and level set approaches to nonlocal variational geometric flows, Geometric partial differential equations, CRM Series, 15, Ed. Norm., Pisa, (2013), 93-104.

[14]

A. ChambolleM. Morini and M. Ponsiglione, Nonlocal Curvature Flows, Arch. Rational Mech. Anal., 218 (2015), 1263-1329.

[15]

S. Dipierro and E. Valdinoci, Nonlocal minimal surfaces: Interior regularity, quantitative estimates and boundary stickiness, preprint, arXiv: 1607.06872v2.

[16]

S. DipierroO. Savin and E. Valdinoci, Boundary behaviour of nonlocal minimal surfaces, J. Funct. Anal., 272 (2017), 1791-1851.

[17] M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, New Jersey, 1997.
[18]

A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math. published on line 2015-04-28.

[19] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, revised edition, CRC Press, 2015.
[20] D. Frenkel and B. Smit, Understanding Molecular Simulations, Academic Press, 2002.
[21] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Boston, 1984.
[22]

C. Imbert, Level set approach for fractional mean curvature flows, Interfaces Free Bound., 11 (2009), 153-176.

[23]

F. Maggi and E. Valdinoci, Capillarity problems with nonlocal surface tension energies, preprint, arXiv: 1606.08610.

[24]

B. Merriman, J. K. Bence and S. Osher, Diffusion Generated Motion by Mean Curvature, CAM report, Department of Mathematics, University of California, Los Angeles, 1992.

[25]

P. Podio-Guidugli, A notion of nonlocal Gaussian curvature, Rend. Lincei: Mat. e Appl., 27 (2016), 181-193.

[26]

O. Savin and E. Valdinoci, Γ-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500.

[27]

O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations, 48 (2013), 33-39.

[28]

H. Weyl, On the volume of tubes, Am. J. Math., 61 (1939), 461-472.

show all references

References:
[1]

N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35 (2014), 793-815.

[2]

G. Alberti, Distributional Jacobian and singularities of Sobolev maps, Ric. Mat., LIV (2006), 375-394.

[3] L. AmbrosioN Fusco and D Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000.
[4]

L. AmbrosioG. De Philippis and L. Martinazzi, Γ-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.

[5]

X. Cabré, M. M. Fall, J. Solá-Morales and T. Weth, Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delauney, Journal für die reine und angewandte Mathematik published on line 2016-04-16.

[6]

L. Caffarelli, Surfaces minimizing nonlocal energies, Rend. Lincei Mat. Appl., 20 (2009), 281-299.

[7]

L. Caffarelli, The mathematical idea of diffusion, Enrico Magenes Lecture, Pavia, March 2013.

[8]

L. CaffarelliJ.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.

[9]

L. Caffarelli and P. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Rational Mech. Anal., 195 (2010), 1-23.

[10]

L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limitingarguments, Adv. Math., 248 (2013), 843-871.

[11]

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.

[12]

A. ChambolleM. Morini and M. Ponsiglione, A nonlocal mean curvature flow and its semi-implicit time-discrete approximation, SIAM J. Math. Anal., 44 (2012), 4048-4077.

[13]

A. Chambolle, M. Morini and M. Ponsiglione, Minimizing movements and level set approaches to nonlocal variational geometric flows, Geometric partial differential equations, CRM Series, 15, Ed. Norm., Pisa, (2013), 93-104.

[14]

A. ChambolleM. Morini and M. Ponsiglione, Nonlocal Curvature Flows, Arch. Rational Mech. Anal., 218 (2015), 1263-1329.

[15]

S. Dipierro and E. Valdinoci, Nonlocal minimal surfaces: Interior regularity, quantitative estimates and boundary stickiness, preprint, arXiv: 1607.06872v2.

[16]

S. DipierroO. Savin and E. Valdinoci, Boundary behaviour of nonlocal minimal surfaces, J. Funct. Anal., 272 (2017), 1791-1851.

[17] M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, New Jersey, 1997.
[18]

A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math. published on line 2015-04-28.

[19] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, revised edition, CRC Press, 2015.
[20] D. Frenkel and B. Smit, Understanding Molecular Simulations, Academic Press, 2002.
[21] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Boston, 1984.
[22]

C. Imbert, Level set approach for fractional mean curvature flows, Interfaces Free Bound., 11 (2009), 153-176.

[23]

F. Maggi and E. Valdinoci, Capillarity problems with nonlocal surface tension energies, preprint, arXiv: 1606.08610.

[24]

B. Merriman, J. K. Bence and S. Osher, Diffusion Generated Motion by Mean Curvature, CAM report, Department of Mathematics, University of California, Los Angeles, 1992.

[25]

P. Podio-Guidugli, A notion of nonlocal Gaussian curvature, Rend. Lincei: Mat. e Appl., 27 (2016), 181-193.

[26]

O. Savin and E. Valdinoci, Γ-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500.

[27]

O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations, 48 (2013), 33-39.

[28]

H. Weyl, On the volume of tubes, Am. J. Math., 61 (1939), 461-472.

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
Figure 2.  The intersection of an open set $E$ and the half plane $\pi(z, {\pmb{e}})$
Figure 3.  Several different ways a straight-line segment can intersect $\partial E$
Figure 4.  How the mapping $\Phi$ in (11) associates $(z, {\pmb{u}}, \xi, \eta)$ with the pair of points $x$ and $y$ in ${\mathbb{R}}^n$
Figure 5.  A depiction of $\mathcal{S}$, $\mathcal{S}_\varepsilon $, and $\mathcal{V}_\varepsilon $
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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