| Peer-Reviewed

On Shape Optimization Theory with Fractional Laplacian

Received: 17 April 2021     Accepted: 13 May 2021     Published: 26 June 2021
Views:       Downloads:
Abstract

The fractional Laplacian is a nonlocal operator that appears in biology, in physic, in fluids dynamic, in financial mathematics and probability. This paper deals with shape optimization problem associated to the fractional laplacian ∆s, 0 < s < 1. We focus on functional of the form J(Ω) = j(Ω,u) where u is solution to the fractional laplacian. A brief review of results related to fractional laplacian and fractional Sobolev spaces are first given. By a variational approach, we show the existence of a weak solution uΩbelonging to the fractional Sobolev spaces Ds, 2(Ω) of the boundary value problem considered. Then, we study the existence of an optimal shape of the functional J(Ω) on the class of admissible sets under constraints volume. Finally, shape derivative of the functional is established by using Hadamard formula’s and an optimality condition is also given.

Published in Applied and Computational Mathematics (Volume 10, Issue 3)
DOI 10.11648/j.acm.20211003.12
Page(s) 56-68
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Shape Optimisation, Shape Derivative, Optimal Conditions, Fractional Laplacian

References
[1] G. Allaire, Shape Optimization by Homogenization Method, Springer Verlag, 2001.
[2] G. Allaire and A. Henrot, On some recent Advances in shape optimization, C. R Acad. Sci. Paris, 7742 (01): 338-396, 2001.
[3] G. Allaire, E. Bonnetier, G. Francfort, F. Jouve, Shape optimization by homogenization method, Numerische Mathematik 76 (1997) 27-68.
[4] D. Bucur, A. Henrot, Minimization of the third eigenvalue of the Dirichlet Laplacian, Proc. R. Soc. London 456 (2000) 985-996.
[5] D. Bucur, P. Trebeschi, Shape optimization problem governed by nonlinear state equation. Proc. Roy. Soc. Edinburgh, 128 A (1998), 945-963.
[6] G. Buttazo, G. Dalmazo, An existence for a class of shape optimization problems. Arch. Rational Mech. Anal. 122 (1993) 183-195
[7] D. Bucur, J. P. Zolésio, Dimensional Shape Optimization under Capacity Constraints. J. Differential Equations, 123 (2) (1995), 504-522.
[8] G. Buttazzo, P. Trebeschi, The role of monotonicity in some shape optimization problems. Calculus of variations and differential equations (Haifa, 1998), 41-55, Chapman- Hall / CRC Res. Notes Math., 410, Chapman- Hall/CRD, Boca Raton, FL, 2000.
[9] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian. Comm. P. D. E. 32 (8), 1245- 1260 (2007).
[10] L. Caffarelli, L. Silvestre An Extension Problem Related to the Fractionnal Laplacian An Extension Problem Related to the Fractionnal Laplacian, Communications in Partial Differential Equation: 37-41, 2007.
[11] L. A. Cafarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, arXiv: 1409. 7721v2, 27 janvier 2015.
[12] A. L. Dalibard and D. Gerard-Varet,On Shape optimization problems involving the fractionnal laplacian, arXiv: 1202.4920 v1 [marh.Ap] 22 Feb 2012.
[13] G. Guy and S. Osher, Nonlocal operators with applications to image processing, multiscale model. Simul. 7 (2008), no.3, 1005-1028. MR 2480109.
[14] A. Henrot, M. Pierre, Shape Variation and Optimization, EMS Tracts in Mathematics Vol. 28, 2018.
[15] E. D. I. Nezza, G. Palatucci, and Enrico valdinoci, Hitchhiker’s guide to the fractionnal sobolev spaces, BULL. Sci. MATH. 2012; 136 : 521-573.
[16] A. Niang, Regularity of Solutions to Elliptic Partial Differential Equations with Mixed Boundary Conditions, Phd thesis, UCAD of Dakar, 2020.
[17] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer Verlag, New York, 1984.
[18] V. Schulz, M. Siebenborn and K. Welker, Structured inverse modeling in parabolic equations, arXiv: 1506.02244v4 [math. OC] 5 Jul. 2015.
[19] V. Schulz, Martin Siebenborn, and Kathrin Welker, Towards a lagrange-newton approach for pde contrained shape optimization, arxiv: 1405. 326v2 [math.NA] 27 nove 2014.
[20] L. Silvestre, On the diffferentiability of the solution to an equation with drift and fractional diffusion, Indiana University Mathematical Journal. 61 (2002), no. 2, 557- 584. 11, 16, 39.
[21] L. Silvestre, Regularity of the obstable problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), 67-112. 1, 9, 15.
[22] E. Val , D. Chamorro, Modelisation de l’operateur Laplacien Fractionnaire: A travers un probléme d’extension au demi-espace, 2014.
[23] M. Warmi, The fractionnal Relative Capacity and the Fractionnal laplacian with Neumann and Robin Boundary conditions on open sets, (X), 2014.
Cite This Article
  • APA Style

    Malick Fall, Ibrahima Faye, Alassane Sy, Diaraf Seck. (2021). On Shape Optimization Theory with Fractional Laplacian. Applied and Computational Mathematics, 10(3), 56-68. https://doi.org/10.11648/j.acm.20211003.12

    Copy | Download

    ACS Style

    Malick Fall; Ibrahima Faye; Alassane Sy; Diaraf Seck. On Shape Optimization Theory with Fractional Laplacian. Appl. Comput. Math. 2021, 10(3), 56-68. doi: 10.11648/j.acm.20211003.12

    Copy | Download

    AMA Style

    Malick Fall, Ibrahima Faye, Alassane Sy, Diaraf Seck. On Shape Optimization Theory with Fractional Laplacian. Appl Comput Math. 2021;10(3):56-68. doi: 10.11648/j.acm.20211003.12

    Copy | Download

  • @article{10.11648/j.acm.20211003.12,
      author = {Malick Fall and Ibrahima Faye and Alassane Sy and Diaraf Seck},
      title = {On Shape Optimization Theory with Fractional Laplacian},
      journal = {Applied and Computational Mathematics},
      volume = {10},
      number = {3},
      pages = {56-68},
      doi = {10.11648/j.acm.20211003.12},
      url = {https://doi.org/10.11648/j.acm.20211003.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211003.12},
      abstract = {The fractional Laplacian is a nonlocal operator that appears in biology, in physic, in fluids dynamic, in financial mathematics and probability. This paper deals with shape optimization problem associated to the fractional laplacian ∆s, 0 J(Ω) = j(Ω,uΩ) where uΩ is solution to the fractional laplacian. A brief review of results related to fractional laplacian and fractional Sobolev spaces are first given. By a variational approach, we show the existence of a weak solution uΩbelonging to the fractional Sobolev spaces Ds, 2(Ω) of the boundary value problem considered. Then, we study the existence of an optimal shape of the functional J(Ω) on the class of admissible sets  under constraints volume. Finally, shape derivative of the functional is established by using Hadamard formula’s and an optimality condition is also given.},
     year = {2021}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - On Shape Optimization Theory with Fractional Laplacian
    AU  - Malick Fall
    AU  - Ibrahima Faye
    AU  - Alassane Sy
    AU  - Diaraf Seck
    Y1  - 2021/06/26
    PY  - 2021
    N1  - https://doi.org/10.11648/j.acm.20211003.12
    DO  - 10.11648/j.acm.20211003.12
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
    SP  - 56
    EP  - 68
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20211003.12
    AB  - The fractional Laplacian is a nonlocal operator that appears in biology, in physic, in fluids dynamic, in financial mathematics and probability. This paper deals with shape optimization problem associated to the fractional laplacian ∆s, 0 J(Ω) = j(Ω,uΩ) where uΩ is solution to the fractional laplacian. A brief review of results related to fractional laplacian and fractional Sobolev spaces are first given. By a variational approach, we show the existence of a weak solution uΩbelonging to the fractional Sobolev spaces Ds, 2(Ω) of the boundary value problem considered. Then, we study the existence of an optimal shape of the functional J(Ω) on the class of admissible sets  under constraints volume. Finally, shape derivative of the functional is established by using Hadamard formula’s and an optimality condition is also given.
    VL  - 10
    IS  - 3
    ER  - 

    Copy | Download

Author Information
  • Department of Mathematics, Faculty of Applied Sciences and Information and Communication Technologies, Alioune Diop University, Bambey, Senegal

  • Department of Mathematics, Faculty of Applied Sciences and Information and Communication Technologies, Alioune Diop University, Bambey, Senegal

  • Department of Mathematics, Faculty of Applied Sciences and Information and Communication Technologies, Alioune Diop University, Bambey, Senegal

  • Department of Mathematics of Decision, Faculty of Economics and Management, Cheikh Anta Diop University, Dakar, Senegal

  • Sections