On Shape Optimization Theory with Fractional Laplacian

Malick Fall; Ibrahima Faye; Alassane Sy; Diaraf Seck

doi:doi:10.11648/j.acm.20211003.12

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On Shape Optimization Theory with Fractional Laplacian

Malick Fall, Ibrahima Faye, Alassane Sy, Diaraf Seck

Published in Applied and Computational Mathematics (Volume 10, Issue 3)

Received: 17 April 2021 Accepted: 13 May 2021 Published: 26 June 2021

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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_Ω) where u_Ω 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 D^{s, 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), 2024. Published by Science Publishing Group

Keywords

Shape Optimisation, Shape Derivative, Optimal Conditions, Fractional Laplacian

References

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[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.
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[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.
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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

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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

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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

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@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}
}

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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  -

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Author Information

Malick Fall

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

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

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

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

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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

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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

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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