`a`
Communications on Pure and Applied Analysis (CPAA)
 

On the nonlocal curvatures of surfaces with or without boundary
Pages: 709 - 727, Issue 2, March 2018

doi:10.3934/cpaa.2018037      Abstract        References        Full text (514.8K)           Related Articles

Roberto Paroni - DADU, University of Sassari, Palazzo del Pou Salit, Piazza Duomo 6, 07041 Alghero (SS), Italy (email)
Paolo Podio-Guidugli - Accademia Nazionale dei Lincei, Palazzo Corsini, Via della Lungara, 10 - 00165 Roma, Italy (email)
Brian Seguin - Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60660, United States (email)

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. Ambrosio, N Fusco and D Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000.       
4 L. Ambrosio, G. De Philippis and L. Martinazzi, $\Gamma$-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.       
5 X. Cabré, M.M. Fall, J. Solà-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. Caffarelli, J.-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. Chambolle, M. 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. Chambolle, M. 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. Dipierro, O. 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, 1976.       
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, $\Gamma$-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.       

Go to top