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

Serena Dipierro, Aleksandr Dzhugan, Nicolò Forcillo, Enrico Valdinoci

Research output: Contribution to journalArticle

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 plane enjoy 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 Hölder 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).

Original languageEnglish
Pages (from-to)44-67
Number of pages24
JournalBruno Pini Mathematical Analysis Seminar
Volume11
Issue number1
DOIs
Publication statusPublished - 1 Jan 2020

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