The stickiness property for antisymmetric nonlocal minimal graphs

Benjamin Baronowitz; Serena Dipierro; Enrico Valdinoci

doi:10.3934/dcds.2022103

Article Contents

2023, Volume 43, Issue 3&4: 1006-1025. Doi: 10.3934/dcds.2022103

This issue Previous Article Stability of a class of action functionals depending on convex functions Next Article Trace and boundary singularities of positive solutions of a class of quasilinear equations

The stickiness property for antisymmetric nonlocal minimal graphs

Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA 6009 Crawley, Australia

^*Corresponding author: Serena Dipierro
^*Corresponding author: Serena Dipierro

Received: December 2021

Revised: May 2022

Early access: August 2022

Published: March 2023

Abstract / Introduction Full Text(HTML) Figure(8) Related Papers Cited by

Abstract

We show that arbitrarily small antisymmetric perturbations of the zero function are sufficient to produce the stickiness phenomenon for planar nonlocal minimal graphs (with the same quantitative bounds obtained for the case of even symmetric perturbations, up to multiplicative constants).

In proving this result, one also establishes an odd symmetric version of the maximum principle for nonlocal minimal graphs, according to which the odd symmetric minimizer is positive in the direction of the positive bump and negative in the direction of the negative bump.

Keywords:

Mathematics Subject Classification: 35R11, 49Q05.

Citation:

Full Text(HTML)

Figure 1. The interactions contributing to the fractional perimeter

Download: Full-size image PowerPoint slide

Figure 2. The antisymmetric stickiness phenomenon, as given in Theorem 1.1

Download: Full-size image PowerPoint slide

Figure 3. The maximum principle for antisymmetric nonlocal minimal graphs, as given in Theorem 1.2

Download: Full-size image PowerPoint slide

Figure 4. The sets $ {\mathcal{A}} $ and $ {\mathcal{B}} $ in the proof of Theorem 1.1

Download: Full-size image PowerPoint slide

Figure 5. The set $ U_\delta $ used in the proof of Theorem 1.1

Download: Full-size image PowerPoint slide

Figure 6. The geometry involved in the proof of Lemma 3.1: detection of cancellations via isometric regions. In particular, the sets $ F_- $ and $ F_+ $ in (33) are given by the bold intervals along the horizontal axis

Download: Full-size image PowerPoint slide

Figure 7. The geometry involved in the proof of Theorem 1.2: detachment from the boundary

Download: Full-size image PowerPoint slide

Figure 8. Difficulties arising in a straightforward proof of the antisymmetric maximum principle

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	B. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 609-639.
[2]	J. P. Borthagaray, W. Li and R. H. Nochetto, Finite element discretizations of nonlocal minimal graphs: Convergence, Nonlinear Anal., 189 (2019), 111566, 31 pp. doi: 10.1016/j.na.2019.06.025.
[3]	J. P. Borthagaray, W. Li and R. H. Nochetto, Finite element algorithms for nonlocal minimal graphs, Math. Eng., 4 (2022), Paper No. 016, 29 pp. doi: 10.3934/mine.2022016.
[4]	C. Bucur, S. Dipierro, L. Lombardini and E. Valdinoci, Minimisers of a fractional seminorm and nonlocal minimal surfaces, Interfaces Free Bound., 22 (2020), 465-504. doi: 10.4171/IFB/447.
[5]	C. Bucur, L. Lombardini and E. Valdinoci, Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 655-703. doi: 10.1016/j.anihpc.2018.08.003.
[6]	X. Cabré, E. Cinti and J. Serra, Stable $s$-minimal cones in $\Bbb{R}^3$ are flat for $s \sim 1$, J. Reine Angew. Math., 764 (2020), 157-180. doi: 10.1515/crelle-2019-0005.
[7]	X. Cabré and M. Cozzi, A gradient estimate for nonlocal minimal graphs, Duke Math. J., 168 (2019), 775-848. doi: 10.1215/00127094-2018-0052.
[8]	X. Cabré, M. M. Fall, J. Solà-Morales and T. Weth, Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay, J. Reine Angew. Math., 745 (2018), 253-280. doi: 10.1515/crelle-2015-0117.
[9]	X. Cabré, M. M. Fall and T. Weth, Delaunay hypersurfaces with constant nonlocal mean curvature, J. Math. Pures Appl. (9), 110 (2018), 32-70. doi: 10.1016/j.matpur.2017.07.005.
[10]	L. Caffarelli, D. De Silva and O. Savin, Obstacle-type problems for minimal surfaces, Comm. Partial Differential Equations, 41 (2016), 1303-1323. doi: 10.1080/03605302.2016.1192646.
[11]	L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144. doi: 10.1002/cpa.20331.
[12]	L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013), 843-871. doi: 10.1016/j.aim.2013.08.007.
[13]	L. A. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal., 195 (2010), 1-23. doi: 10.1007/s00205-008-0181-x.
[14]	A. Chambolle, M. Morini and M. Ponsiglione, Minimizing movements and level set approaches to nonlocal variational geometric flows, Geometric Partial Differential Equations, 93–104, CRM Series, 15, Ed. Norm., Pisa, 2013. doi: 10.1007/978-88-7642-473-1_4.
[15]	A. Chambolle, M. Novaga and B. Ruffini, Some results on anisotropic fractional mean curvature flows, Interfaces Free Bound., 19 (2017), 393-415. doi: 10.4171/IFB/387.
[16]	E. Cinti, J. Serra and E. Valdinoci, Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces, J. Differential Geom., 112 (2019), 447-504. doi: 10.4310/jdg/1563242471.
[17]	E. Cinti, C. Sinestrari and E. Valdinoci, Neckpinch singularities in fractional mean curvature flows, Proc. Amer. Math. Soc., 146 (2018), 2637-2646. doi: 10.1090/proc/14002.
[18]	G. Ciraolo, A. Figalli, F. Maggi and M. Novaga, Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature, J. Reine Angew. Math., 741 (2018), 275-294. doi: 10.1515/crelle-2015-0088.
[19]	M. Cozzi, J. Dávila and M. del Pino, Long-time asymptotics for evolutionary crystal dislocation models, Adv. Math., 371 (2020), 107242,109 pp. doi: 10.1016/j.aim.2020.107242.
[20]	M. Cozzi and L. Lombardini, On nonlocal minimal graphs, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 136, 72 pp. doi: 10.1007/s00526-021-02002-9.
[21]	J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Nonlocal Delaunay surfaces, Nonlinear Anal., 137 (2016), 357-380. doi: 10.1016/j.na.2015.10.009.
[22]	J. Dávila, M. del Pino and J. Wei, Nonlocal $s$-minimal surfaces and Lawson cones, J. Differential Geom., 109 (2018), 111-175. doi: 10.4310/jdg/1525399218.
[23]	A. Di Castro, M. Novaga, B. Ruffini and E. Valdinoci, Nonlocal quantitative isoperimetric inequalities, Calc. Var. Partial Differential Equations, 54 (2015), 2421-2464. doi: 10.1007/s00526-015-0870-x.
[24]	S. Dipierro, A. Dzhugan, N. Forcillo and E. Valdinoci, Enhanced boundary regularity of planar nonlocal minimal graphs and a butterfly effect, Something About Nonlinear Problems, 44–67, Bruno Pini Math. Anal. Semin., 11, Univ. Bologna, Alma Mater Stud., Bologna, 2020.
[25]	S. Dipierro, F. Onoue and E. Valdinoci, (Dis)connectedness of nonlocal minimal surfaces in a cylinder and a stickiness property, Proc. Amer. Math. Soc., 150 (2022), 2223-2237. doi: 10.1090/proc/15796.
[26]	S. Dipierro, O. Savin and E. Valdinoci, Graph properties for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 55 (2016), Art. 86, 25 pp. doi: 10.1007/s00526-016-1020-9.
[27]	S. Dipierro, O. Savin and E. Valdinoci, Boundary behavior of nonlocal minimal surfaces, J. Funct. Anal., 272 (2017), 1791-1851. doi: 10.1016/j.jfa.2016.11.016.
[28]	S. Dipierro, O. Savin and E. Valdinoci, Boundary properties of fractional objects: Flexibility of linear equations and rigidity of minimal graphs, J. Reine Angew. Math., 769 (2020), 121-164. doi: 10.1515/crelle-2019-0045.
[29]	S. Dipierro, O. Savin and E. Valdinoci, Nonlocal minimal graphs in the plane are generically sticky, Comm. Math. Phys., 376 (2020), 2005-2063. doi: 10.1007/s00220-020-03771-8.
[30]	S. Dipierro, J. Thompson and E. Valdinoci, On the Harnack inequality for antisymmetric $s$-harmonic functions, arXiv e-prints.
[31]	A. Farina and E. Valdinoci, Flatness results for nonlocal minimal cones and subgraphs, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19 (2019), 1281-1301.
[32]	J.-C. Felipe-Navarro, Uniqueness for linear integro-differential equations in the real line and applications, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 220, 25 pp. doi: 10.1007/s00526-021-02084-5.
[33]	J.-C. Felipe-Navarro and T. Sanz-Perela, Semilinear integro-differential equations, i: Odd solutions with respect to the Simons cone, J. Funct. Anal., 278 (2020), 108309, 48 pp. doi: 10.1016/j.jfa.2019.108309.
[34]	A. Figalli, N. Fusco, F. Maggi, V. Millot and M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies, Comm. Math. Phys., 336 (2015), 441-507. doi: 10.1007/s00220-014-2244-1.
[35]	A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math., 729 (2017), 263-273. doi: 10.1515/crelle-2015-0006.
[36]	N. Fusco, V. Millot and M. Morini, A quantitative isoperimetric inequality for fractional perimeters, J. Funct. Anal., 261 (2011), 697-715. doi: 10.1016/j.jfa.2011.02.012.
[37]	C. Imbert, Level set approach for fractional mean curvature flows, Interfaces Free Bound., 11 (2009), 153-176. doi: 10.4171/IFB/207.
[38]	S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pura Appl. (4), 195 (2016), 273-291. doi: 10.1007/s10231-014-0462-y.
[39]	M. Sáez and E. Valdinoci, On the evolution by fractional mean curvature, Comm. Anal. Geom., 27 (2019), 211-249. doi: 10.4310/CAG.2019.v27.n1.a6.
[40]	O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations, 48 (2013), 33-39. doi: 10.1007/s00526-012-0539-7.
[41]	O. Savin and E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm, J. Math. Pures Appl. (9), 101 (2014), 1-26. doi: 10.1016/j.matpur.2013.05.001.