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Advanced Nonlinear Studies

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Volume 19, Issue 1

Issues

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
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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 S2 this fractal dimension coincides with the Minkowski dimension. We also obtain an optimal result for the asymptotics as s1- of the fractional perimeter of a set having locally finite (classical) perimeter.

Keywords: 28A80; 35R11; 49Q05

MSC 2010: Nonlocal Minimal Surfaces; Fractal Dimensions; Fractional Operators

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. Web of ScienceCrossrefGoogle Scholar

  • [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 Ws,p when s1 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. CrossrefWeb of ScienceGoogle Scholar

  • [7]

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

  • [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. Web of ScienceCrossrefGoogle Scholar

  • [9]

    S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as s0, 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. Web of ScienceGoogle Scholar

  • [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. CrossrefGoogle Scholar

  • [15]

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

  • [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. CrossrefGoogle Scholar

  • [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. Web of ScienceCrossrefGoogle Scholar

  • [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. CrossrefGoogle Scholar

About the article


Received: 2018-01-06

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, DOI: https://doi.org/10.1515/ans-2018-2016.

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