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
References:
[1]

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

[2]

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

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

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

[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. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[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.   Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

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

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

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

[23]

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

[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. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

show all references

References:
[1]

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

[2]

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

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

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

[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. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[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.   Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

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

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

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

[23]

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

[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. Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

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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