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

Authors

  • 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
  • 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

DOI:

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

Keywords:

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

Abstract

In this note, we showcase some recent results obtained in [DSV19] concerning the stickiness properties of nonlocal minimal graphs in the plane. To start with, the nonlocal minimal graphs in the planeenjoy an enhanced boundary regularity, since boundary continuity with respect to the external datum is sufficient to ensure differentiability across the boundary of the domain. As a matter of fact, the Hoelder exponent of the derivative is in this situation sufficiently high to provide the validity of the Euler-Lagrange equation at boundary points as well. From this, using a sliding method, one also deduces that the stickiness phenomenon is generic for nonlocal minimal graphs in the plane, since an arbitrarily small perturbation of continuous nonlocal minimal graphs can produce boundary discontinuities (making the continuous case somehow ``exceptional'' in this framework.

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Published

2020-03-28

How to Cite

Dipierro, S., Dzhugan, A., Forcillo, N., & Valdinoci, E. (2020). Enhanced boundary regularity of planar nonlocal minimal graphs and a butterfly effect. Bruno Pini Mathematical Analysis Seminar, 11(1), 44–67. https://doi.org/10.6092/issn.2240-2829/10585