Show Summary Details
More options …

IMPACT FACTOR 2017: 1.029
5-year IMPACT FACTOR: 1.147

CiteScore 2017: 1.29

SCImago Journal Rank (SJR) 2017: 1.588
Source Normalized Impact per Paper (SNIP) 2017: 0.971

Mathematical Citation Quotient (MCQ) 2017: 1.03

Online
ISSN
2169-0375
See all formats and pricing
More options …
Volume 19, Issue 1

Fractional Perimeters from a Fractal Perspective

Luca Lombardini
• Corresponding author
• Dipartimento di Matematica, Università degli Studi di Milano, Via Cesare Saldini 50, 20100, Milano, Italy; and Département de Mathématiques, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens CEDEX 1, France
• Email
• Other articles by this author:
Published Online: 2018-06-13 | DOI: https://doi.org/10.1515/ans-2018-2016

Abstract

The purpose of this paper consists in a better understanding of the fractional nature of the nonlocal perimeters introduced in [L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 2010, 9, 1111–1144]. Following [A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math. 8 1991, 2, 175–201], we exploit these fractional perimeters to introduce a definition of fractal dimension for the measure theoretic boundary of a set. We calculate the fractal dimension of sets which can be defined in a recursive way, and we give some examples of this kind of sets, explaining how to construct them starting from well-known self-similar fractals. In particular, we show that in the case of the von Koch snowflake $S\subseteq {ℝ}^{2}$ this fractal dimension coincides with the Minkowski dimension. We also obtain an optimal result for the asymptotics as $s\to {1}^{-}$ of the fractional perimeter of a set having locally finite (classical) perimeter.

Keywords: 28A80; 35R11; 49Q05

References

• [1]

L. Ambrosio and N. Dancer, Calculus of Variations and Partial Differential Equations, Springer, Berlin, 2000. Google Scholar

• [2]

L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math. 134 (2011), no. 3–4, 377–403.

• [3]

G. Bellettini, Lecture Notes on Mean Curvature Flows, Barriers and Singular Perturbations, Appunti. Sc. Norm. Super. Pisa (N. S.) 12, Edizioni della Normale, Pisa, 2013. Google Scholar

• [4]

J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for ${W}^{s,p}$ when $s↑1$ and applications, J. Anal. Math. 87 (2002), 77–101. Google Scholar

• [5]

L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111–1144. Google Scholar

• [6]

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations 41 (2011), no. 1–2, 203–240.

• [7]

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

• [8]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573.

• [9]

S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as $s↘0$, Discrete Contin. Dyn. Syst. 33 (2013), no. 7, 2777–2790. Google Scholar

• [10]

S. Dipierro, O. Savin and E. Valdinoci, Graph properties for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Article ID 86.

• [11]

S. Dipierro and E. Valdinoci, Nonlocal minimal surfaces: Interior regularity, quantitative estimates and boundary stickiness, Recent Developments in Nonlocal Theory, De Gruyter, Berlin (2018), 165–209. Google Scholar

• [12]

P. Doktor, Approximation of domains with Lipschitzian boundary, Časopis Pěst. Mat. 101 (1976), no. 3, 237–255. Google Scholar

• [13]

K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 1990. Google Scholar

• [14]

D. Faraco and K. M. Rogers, The Sobolev norm of characteristic functions with applications to the Calderón inverse problem, Q. J. Math. 64 (2013), no. 1, 133–147.

• [15]

A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math. 729 (2017), 263–273.

• [16]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math. 80, Birkhäuser, Basel, 1984. Google Scholar

• [17]

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge Stud. Adv. Math. 135, Cambridge University Press, Cambridge, 2012. Google Scholar

• [18]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Stud. Adv. Math. 44, Cambridge University Press, Cambridge, 1995. Google Scholar

• [19]

A. C. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Differential Equations 19 (2004), no. 3, 229–255.

• [20]

O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations 48 (2013), no. 1–2, 33–39.

• [21]

W. Sickel, Pointwise multipliers of Lizorkin-Triebel spaces, The Maz’ya anniversary Collection. Vol. 2 (Rostock 1998), Oper. Theory Adv. Appl. 110, Birkhäuser, Basel (1999), 295–321. Google Scholar

• [22]

A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math. 8 (1991), no. 2, 175–201.

Revised: 2018-04-10

Accepted: 2018-04-12

Published Online: 2018-06-13

Published in Print: 2019-02-01

Citation Information: Advanced Nonlinear Studies, Volume 19, Issue 1, Pages 165–196, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.