Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang


IMPACT FACTOR 2018: 2.316

CiteScore 2018: 1.77

SCImago Journal Rank (SJR) 2018: 2.350
Source Normalized Impact per Paper (SNIP) 2018: 1.465

Mathematical Citation Quotient (MCQ) 2018: 1.44

Online
ISSN
1864-8266
See all formats and pricing
More options …
Volume 11, Issue 4

Issues

A class of shape optimization problems for some nonlocal operators

Julián Fernández BonderORCID iD: http://orcid.org/0000-0003-1097-4776 / Antonella Ritorto / Ariel Martin Salort
Published Online: 2017-05-11 | DOI: https://doi.org/10.1515/acv-2016-0065

Abstract

In this work we study a family of shape optimization problem where the state equation is given in terms of a nonlocal operator. Examples of the problems considered are monotone combinations of fractional eigenvalues. Moreover, we also analyze the transition from nonlocal to local state equations.

Keywords: Fractional partial differential equations; shape optimization

MSC 2010: 35R11; 49Q10

References

  • [1]

    V. Akgiray and G. G. Booth, The siable-law model of stock returns, J. Biopharm. Statist. 6 (1988), no. 1, 51–57. Google Scholar

  • [2]

    G. Allaire, Shape Optimization by the Homogenization Method, Appl. Math. Sci. 146, Springer, New York, 2002. Google Scholar

  • [3]

    J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, Optimal Control and Partial Differential Equations (Paris 2000), IOS Press, Amsterdam (2001), 439–455. Google Scholar

  • [4]

    L. Brasco and E. Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var. 9 (2016), no. 4, 323–355. Google Scholar

  • [5]

    L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional p-Laplacian, Discrete Contin. Dyn. Syst. 36 (2016), no. 4, 1813–1845. Web of ScienceGoogle Scholar

  • [6]

    H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. Google Scholar

  • [7]

    D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progr. Nonlinear Differential Equations Appl. 65, Birkhäuser, Boston, 2005. Google Scholar

  • [8]

    A. Burchard, R. Choksi and I. Topaloglu, Nonlocal shape optimization via interactions of attractive and repulsive potentials, preprint (2015), https://arxiv.org/abs/1512.07282.

  • [9]

    G. Buttazzo and G. Dal Maso, An existence result for a class of shape optimization problems, Arch. Ration. Mech. Anal. 122 (1993), no. 2, 183–195. CrossrefGoogle Scholar

  • [10]

    P. Constantin, Euler equations, Navier–Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows, Lecture Notes in Math. 1871, Springer, Berlin (2006), 1–43. Google Scholar

  • [11]

    A.-L. Dalibard and D. Gérard-Varet, On shape optimization problems involving the fractional Laplacian, ESAIM Control Optim. Calc. Var. 19 (2013), no. 4, 976–1013. CrossrefWeb of ScienceGoogle Scholar

  • [12]

    G. Dal Maso, An Introduction to Γ-Convergence, Progr. Nonlinear Differential Equations Appl. 8, Birkhäuser, Boston, 1993. Google Scholar

  • [13]

    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

  • [14]

    Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev. 54 (2012), no. 4, 667–696. Web of ScienceCrossrefGoogle Scholar

  • [15]

    A. C. Eringen, Nonlocal Continuum Field Theories, Springer, New York, 2002. Google Scholar

  • [16]

    J. Fernandez Bonder and J. Spedaletti, Some nonlocal optimal design problems, preprint (2016), https://arxiv.org/abs/1601.03700.

  • [17]

    G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits, J. Stat. Phys. 87 (1997), no. 1–2, 37–61. CrossrefGoogle Scholar

  • [18]

    G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul. 7 (2008), no. 3, 1005–1028. Web of ScienceGoogle Scholar

  • [19]

    A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Front. Math., Birkhäuser, Basel, 2006. Google Scholar

  • [20]

    N. Humphries, Environmental context explains Lévy and Brownian movement patterns of marine predators, Nature 465 (2010), 1066–1069. Web of ScienceCrossrefGoogle Scholar

  • [21]

    H. Knüpfer and C. B. Muratov, On an isoperimetric problem with a competing nonlocal term. I: The planar case, Comm. Pure Appl. Math. 66 (2013), no. 7, 1129–1162. CrossrefGoogle Scholar

  • [22]

    H. Knüpfer and C. B. Muratov, On an isoperimetric problem with a competing nonlocal term. II: The general case, Comm. Pure Appl. Math. 67 (2014), no. 12, 1974–1994. CrossrefGoogle Scholar

  • [23]

    N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), no. 4–6, 298–305. CrossrefGoogle Scholar

  • [24]

    S. Levendorski, Pricing of the american put under Lévy processes, Int. J. Theor. Appl. Finance 7 (2004), no. 3, 303–335. CrossrefGoogle Scholar

  • [25]

    A. Massaccesi and E. Valdinoci, Is a nonlocal diffusion strategy convenient for biological populations in competition?, preprint (2015), https://arxiv.org/abs/1503.01629.

  • [26]

    R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, Article ID 77. Google Scholar

  • [27]

    O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer Ser. Comput. Phys., Springer, New York, 1984. Google Scholar

  • [28]

    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

  • [29]

    C. Qiu, Y. Huang and Y. Zhou, Optimization problems involving the fractional Laplacian, Electron. J. Differential Equations 2016 (2016), Paper No. 98. Google Scholar

  • [30]

    A. M. Reynolds and C. J. Rhodes, The Lévy flight paradigm: Random search patterns and mechanisms, Ecology 90 (2009), no. 4, 877–887. Web of ScienceCrossrefGoogle Scholar

  • [31]

    W. Schoutens, Lévy Processes in Finance: Pricing Financial Derivatives, Wiley Ser. Probab. Stat., Wiley, New York, 2003. Google Scholar

  • [32]

    S. Shi and J. Xiao, On fractional capacities relative to bounded open Lipschitz sets, Potential Anal. 45 (2016), no. 2, 261–298. Web of ScienceCrossrefGoogle Scholar

  • [33]

    Y. Sire, J. L. Vazquez and B. Volzone, Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application, preprint (2015), https://arxiv.org/abs/1506.07199.

  • [34]

    J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Ser. Comput. Math. 16, Springer, Berlin, 1992. Google Scholar

  • [35]

    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University Press, Princeton, 1970. Google Scholar

  • [36]

    K. Zhou and Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions, SIAM J. Numer. Anal. 48 (2010), no. 5, 1759–1780. CrossrefWeb of ScienceGoogle Scholar

About the article


Received: 2016-12-27

Revised: 2017-03-29

Accepted: 2017-04-27

Published Online: 2017-05-11

Published in Print: 2018-10-01


Funding Source: Universidad de Buenos Aires

Award identifier / Grant number: 20020130100283BA

Funding Source: Consejo Nacional de Investigaciones Científicas y Técnicas

Award identifier / Grant number: PIP 11220150100032CO

Funding Source: Agencia Nacional de Promoción Científica y Tecnológica

Award identifier / Grant number: PICT 2012-0153

The present paper was partially supported by grants UBACyT 20020130100283BA, CONICET PIP 11220150100032CO and ANPCyT PICT 2012-0153. Julián Fernández Bonder and Ariel M. Salort are members of CONICET and Antonella Ritorto is a doctoral fellow of CONICET.


Citation Information: Advances in Calculus of Variations, Volume 11, Issue 4, Pages 373–386, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2016-0065.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Comments (0)

Please log in or register to comment.
Log in