Enhanced boundary regularity of planar nonlocal minimal graphs and a butterfly effect

Serena Dipierro; Aleksandr Dzhugan; Nicolò Forcillo; Enrico Valdinoci

doi:10.6092/issn.2240-2829/10585

Autori

Serena Dipierro Serena Dipierro, Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley WA 6009 http://orcid.org/0000-0003-4386-4485
Serena Dipierro took her PhD in Mathematical Analysis at the International School for Advanced Studies (SISSA, Trieste) in 2012. After PostDoc positions at the Universidad de Chile and University of Edinburgh, and a Humboldt Fellowship, she held permanent positions at the University of Melbourne and the Università di Milano. Since August 2018, she is Associate Professor at the University of Western Australia.
Aleksandr Dzhugan Aleksandr Dzhugan, Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna
Nicolò Forcillo Nicolò Forcillo, Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna
Enrico Valdinoci Enrico Valdinoci, Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley WA 6009 http://orcid.org/0000-0001-6222-2272

Enrico Valdinoci received his PhD in Mathematics at the University of Texas at Austin in 2002 and later obtained a PostDoc Position at the Scuola Normale Superiore in Pisa. He was appointed Assistant/Associate Professor at the University of Rome Tor Vergata, Full Professor at the University of Milan, European Research Council Group Leader at the Weierstrass Institute in Berlin, and Professor at the University of Melbourne.
Since August 2019, he is Professor at the University of Western Australia.

DOI:

https://doi.org/10.6092/issn.2240-2829/10585

Parole chiave:

Nonlocal minimal surfaces, fractional equations, stickiness phenomena, regularity and maximum principles

Abstract

In questa nota, presentiamo alcuni risultati recenti ottenuti in [DSV19] relativi alla proprietà di ``appiccicosità'' dei grafici minimi nonlocali nel piano. I grafici minimi non locali nel piano godono di una regolarità ``accresciuta'' al bordo, in quanto la continuità al bordo rispetto al dato esterno è sufficiente a garantire la differenziabilità attraverso il bordo del dominio. Inoltre, l'esponente di Hoelder della derivata è sufficientemente grande da garantire la validità dell'equazione di Eulero-Lagrange anche ai punti di bordo del dominio. Da ciò, usando un metodo di scivolamento, si ottiene anche cheil fenomeno di appiccicosità è generico per grafici minimi non locali nel piano, nel senso che una perturbazione arbitrariamente piccola di i grafici minimi nonlocali continui produce discontinuità al bordo (rendendo quindi il caso continuo in qualche modo ``eccezionale'').

Riferimenti bibliografici

Nicola Abatangelo and Enrico Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim. 35 (2014), no. 7-9, 793–815, DOI 10.1080/01630563.2014.901837. MR3230079

Luigi Ambrosio, Guido De Philippis, and Luca Martinazzi, Gamma-convergence of non- local perimeter functionals, Manuscripta Math. 134 (2011), no. 3-4, 377–403, DOI

1007/s00229-010-0399-4. MR2765717

Begon ̃a Barrios, Alessio Figalli, and Enrico Valdinoci, Bootstrap regularity for integro-

differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), no. 3, 609–639. MR3331523

Kh. Brezis, How to recognize constant functions. A connection with Sobolev spaces, Uspekhi Mat. Nauk 57 (2002), no. 4(346), 59–74, DOI 10.1070/RM2002v057n04ABEH000533 (Russian, with Russian summary); English transl., Russian Math. Surveys 57 (2002), no. 4, 693–708. MR1942116

Jean Bourgain, Haïm Brezis, and Petru Mironescu, Limiting embedding theorems for Ws,p when s ↑ 1 and applications, J. Anal. Math. 87 (2002), 77–101, DOI 10.1007/BF02868470. Dedicated to the memory of Thomas H. Wolff. MR1945278

Claudia Bucur and Luca Lombardini, Asymptotics as s ↘ 0 of the nonlocal nonparametric Plateau problem with obstacles, In preparation.

Claudia Bucur, Luca Lombardini, and Enrico Valdinoci, Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter, Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 3, 655–703, DOI 10.1016/j.anihpc.2018.08.003. MR3926519

Xavier Cabré, Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory, arXiv e-prints (2019), arXiv:1905.10790, available at 1905.10790.

Xavier Cabré and Matteo Cozzi, A gradient estimate for nonlocal minimal graphs, Duke Math. J. 168 (2019), no. 5, 775–848, DOI 10.1215/00127094-2018-0052. MR3934589

Xavier Cabré, Mouhamed Moustapha Fall, Joan Solà-Morales, and Tobias 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. MR3881478

Xavier Cabré, Mouhamed Moustapha Fall, and Tobias 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 (English, with English and French summaries). MR3744919

------, Near-sphere lattices with constant nonlocal mean curvature, Math. Ann. 370 (2018), no. 3-4, 1513–1569, DOI 10.1007/s00208-017-1559-6. MR3770173

L. Caffarelli, D. De Silva, and O. Savin, Obstacle-type problems for minimal surfaces, Comm. Partial Differential Equations 41 (2016), no. 8, 1303–1323, DOI 10.1080/03605302.2016.1192646. MR3532394

L. Caffarelli, J.-M. Roquejoffre, and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111–1144, DOI 10.1002/cpa.20331. MR2675483

Luis A. Caffarelli and Panagiotis E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal. 195 (2010), no. 1, 1–23, DOI 10.1007/s00205-008-0181-x. MR2564467

Luis Caffarelli and Enrico Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations 41 (2011), no. 1-2, 203–240, DOI 10.1007/s00526-010-0359-6. MR2782803

------, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math. 248 (2013), 843–871, DOI 10.1016/j.aim.2013.08.007. MR3107529

Annalisa Cesaroni, Serena Dipierro, Matteo Novaga, and Enrico Valdinoci, Fattening and nonfattening phenomena for planar nonlocal curvature flows, Math. Ann. 375 (2019), no. 1- 2, 687–736, DOI 10.1007/s00208-018-1793-6. MR4000255

Annalisa Cesaroni and Matteo Novaga, Volume constrained minimizers of the fractional perimeter with a potential energy, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), no. 4, 715–727, DOI 10.3934/dcdss.2017036. MR3640534

------, The isoperimetric problem for nonlocal perimeters, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 3, 425–440, DOI 10.3934/dcdss.2018023. MR3732175

Antonin Chambolle, Massimiliano Morini, and Marcello Ponsiglione, A nonlocal mean curvature flow and its semi-implicit time-discrete approximation, SIAM J. Math. Anal. 44 (2012), no. 6, 4048–4077, DOI 10.1137/120863587. MR3023439

------, Minimizing movements and level set approaches to nonlocal variational geometric flows, Geometric partial differential equations, CRM Series, vol. 15, Ed. Norm., Pisa, 2013, pp. 93–104, DOI 10.1007/978-88-7642-473-1 4. MR3156889

------, Nonlocal curvature flows, Arch. Ration. Mech. Anal. 218 (2015), no. 3, 1263–1329, DOI 10.1007/s00205-015-0880-z. MR3401008

Antonin Chambolle, Matteo Novaga, and Berardo Ruffini, Some results on anisotropic fractional mean curvature flows, Interfaces Free Bound. 19 (2017), no. 3, 393–415, DOI 10.4171/IFB/387. MR3713894

Eleonora Cinti, Joaquim Serra, and Enrico Valdinoci, Quantitative flatness results and BV - estimates for stable nonlocal minimal surfaces, J. Differential Geom. 112 (2019), no. 3, 447–504, DOI 10.4310/jdg/1563242471. MR3981295

Eleonora Cinti, Carlo Sinestrari, and Enrico Valdinoci, Neckpinch singularities in fractional mean curvature flows, Proc. Amer. Math. Soc. 146 (2018), no. 6, 2637–2646, DOI 10.1090/proc/14002. MR3778164

Giulio Ciraolo, Alessio Figalli, Francesco Maggi, and Matteo 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. MR3836150

Matteo Cozzi and Luca Lombardini, On nonlocal minimal graphs, In preparation.

J. Dávila, On an open question about functions of bounded variation, Calc. Var. Partial Differential Equations 15 (2002), no. 4, 519–527, DOI 10.1007/s005260100135. MR1942130

Juan Dávila, Manuel del Pino, Serena Dipierro, and Enrico Valdinoci, Nonlocal Delaunay surfaces, Nonlinear Anal. 137 (2016), 357–380, DOI 10.1016/j.na.2015.10.009. MR3485130

Agnese Di Castro, Matteo Novaga, Berardo Ruffini, and Enrico Valdinoci, Nonlocal quantitative isoperimetric inequalities, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2421–2464, DOI 10.1007/s00526-015-0870-x. MR3412379

Serena Dipierro, Alessio Figalli, Giampiero Palatucci, and Enrico Valdinoci, Asymptotics of the s-perimeter as s ↘ 0, Discrete Contin. Dyn. Syst. 33 (2013), no. 7, 2777–2790, DOI 10.3934/dcds.2013.33.2777. MR3007726

Serena Dipierro, Francesco Maggi, and Enrico Valdinoci, Asymptotic expansions of the contact angle in nonlocal capillarity problems, J. Nonlinear Sci. 27 (2017), no. 5, 1531–1550, DOI 10.1007/s00332-017-9378-1. MR3707346

Serena Dipierro, Ovidiu Savin, and Enrico Valdinoci, Graph properties for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Art. 86, 25, DOI 10.1007/s00526-016-1020-9. MR3516886

------, Boundary behavior of nonlocal minimal surfaces, J. Funct. Anal. 272 (2017), no. 5, 1791–1851, DOI 10.1016/j.jfa.2016.11.016. MR3596708

[DSV19]------, Nonlocal minimal graphs in the plane are generically sticky, arXiv e-prints (2019), arXiv:1904.05393, available at 1904.05393.

------, Boundary properties of fractional objects: flexibility of linear equations and rigidity of minimal graphs, J. Reine Angew. Math.

Alberto Farina and Enrico Valdinoci, Flatness results for nonlocal minimal cones and sub- graphs, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5).

Fausto Ferrari, Michele Miranda Jr., Diego Pallara, Andrea Pinamonti, and Yannick Sire, Fractional Laplacians, perimeters and heat semigroups in Carnot groups, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 3, 477–491, DOI 10.3934/dcdss.2018026. MR3732178

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), no. 1, 441–507, DOI 10.1007/s00220-014-2244-1. MR3322379

Alessio Figalli and Enrico Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math. 729 (2017), 263–273, DOI 10.1515/crelle-2015-0006. MR3680376

Rupert L. Frank, Elliott H. Lieb, and Robert Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc. 21 (2008), no. 4, 925–950, DOI 10.1090/S0894-0347-07-00582-6. MR2425175

Rupert L. Frank and Robert Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal. 255 (2008), no. 12, 3407–3430, DOI 10.1016/j.jfa.2008.05.015. MR2469027

Nicola Fusco, Vincent Millot, and Massimiliano Morini, A quantitative isoperimetric inequality for fractional perimeters, J. Funct. Anal. 261 (2011), no. 3, 697–715, DOI 10.1016/j.jfa.2011.02.012. MR2799577

Cyril Imbert, Level set approach for fractional mean curvature flows, Interfaces Free Bound. 11 (2009), no. 1, 153–176, DOI 10.4171/IFB/207. MR2487027

Vesa Julin and Domenico La Manna, Short time existence of the classical solution to the fractional mean curvature flow, arXiv e-prints (2019), arXiv:1906.10990, available at 1906. 10990.

Luca Lombardini, Approximation of sets of finite fractional perimeter by smooth sets and comparison of local and global s-minimal surfaces, Interfaces and Free Boundaries 20 (2018), no. 2, 261-296, DOI 10.4171/IFB/402. MR3827804

Francesco Maggi and Enrico Valdinoci, Capillarity problems with nonlocal surface ten- sion energies, Comm. Partial Differential Equations 42 (2017), no. 9, 1403–1446, DOI 10.1080/03605302.2017.1358277. MR3717439

Jos é M. Maz ́on, Julio D. Rossi, and Julián Toledo, Nonlocal perimeter, curvature and minimal surfaces for measurable sets, J. Anal. Math. 138 (2019), no. 1, 235–279, DOI 10.1007/s11854-019-0027-5. MR3996039

V. Maz′ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal. 195 (2002), no. 2, 230–238, DOI 10.1006/jfan.2002.3955. MR1940355

Valerio Pagliari, Halfspaces minimise nonlocal perimeter: a proof via calibrations, arXiv e-prints (2019), arXiv:1905.00623, available at 1905.00623.

Roberto Paroni, Paolo Podio-Guidugli, and Brian Seguin, On the nonlocal curvatures of surfaces with or without boundary, Commun. Pure Appl. Anal. 17 (2018), no. 2, 709–727, DOI 10.3934/cpaa.2018037. MR3733825

Xavier Ros-Oton and Joaquim Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl. (9) 101 (2014), no. 3, 275–302, DOI 10.1016/j.matpur.2013.06.003 (English, with English and French summaries). MR3168912

Ovidiu Savin and Enrico Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations 48 (2013), no. 1-2, 33–39, DOI 10.1007/s00526- 012-0539-7. MR3090533

Mariel Sáez and Enrico Valdinoci, On the evolution by fractional mean curvature, Comm. Anal. Geom. 27 (2019), no. 1, 211–249, DOI 10.4310/CAG.2019.v27.n1.a6. MR3951024