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A Maximization Problem Involving a Fractional Laplace Type Operator

Received: 25 May 2021     Accepted: 4 June 2021     Published: 16 June 2021
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Abstract

Fractional Laplacian is an important nonlocal operator which has many applications in different kinds of differential equations. Recently, optimization problems involving the fractional Laplacian have been studied a lot by many authors. However, most of these papers are focusing on the optimization problems related to the first eigenvalue of the equation. Optimization problems related to the energy functional of the equation have not been investigated well enough. In this paper, we are going to study a maximization problem related to the energy functional of an equation involving a fractional Laplace type operator. Firstly, by using suitable variational framework in a fractional Sobolev space, we can show that a fractional equation has a solution which is in fact the global minimum of the corresponding energy functional. Moreover, by using reduction to absurdity we can obtain the uniqueness of the solution of the fractional equation. Then, we focus on a maximization problem related to the equation which takes the energy functional as the objective functional. Finally, by carefully analysing the properties of an arbitrarily choosen minimizing sequence and the tools of the rearrangement theory, we can prove that the maximization problem is solvable.

Published in American Journal of Applied Mathematics (Volume 9, Issue 3)
DOI 10.11648/j.ajam.20210903.14
Page(s) 86-91
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

Maximization, Fractional Laplace, Rearrangement

References
[1] Burton, G. R. (1987) Rearrangements of functions, maximization of convex functionals and vortex rings. Mathematische Annalen, 276: 225-253.
[2] Burton, G. R. (1989) Variational problems on classes of rearrangements and multiple configurations for steady vortices. Ann. Inst. Henri Poincare., 6, 295-319.
[3] Cuccu, F., Emamizadeh, B., & Porru, G. (2006) Nonlinear elastic membrane involving the p-Laplacian operator. Electron. J. Differential Equations, 49, 1-10.
[4] Cuccu, F., Emamizadeh, B., & Porru, G. (2009) Optimization of the first eigenvalue in problems involving the p-Laplacian. Proc. Amer. Math. Soc., 137, 1677-1687.
[5] Cuccu, F., Porru, G., & Sakaguchi, S. (2011) Optimization problems on general classes of rearrangements. Nonlinear Anal., 74, 5554-5565.
[6] Emamizadeh, B., & Zivari-Rezapour, M. (2007) Rearrangement optimization for some elliptic equations. J. Optim. Theory Appl., 135, 367-379.
[7] Emamizadeh, B., & Prajapat, J. V. (2009) Symmetry in rearrangemet optimization problems. Electron. J. Differential Equations, 149, 1-10.
[8] Marras, M. (2010) Optimization in problems involving the p-Laplacian. Electron. J. Differential Equations, 2, 1-10.
[9] Qiu, C., Huang, Y. S., & Zhou, Y. Y. (2015) A class of rearrangement optimization problems involving the p-Laplacian. Nonlinear Anal., 112, 30-42.
[10] Qiu, C., Huang, Y. S., & Zhou, Y. Y. (2016) Optimization problems involving the fractional Laplacian. Electronic Journal of Differential Equations, 98, 1-15.
[11] Dalibard, A. L., & Gerard-Varet, D. (2013) On shape optimization problems involving the fractional Laplacian. ESAIM Control Optim. Calc. Var., 19 (4), 976-1013.
[12] Biswas, A., & Jarohs, S. (2020) On overdetermined problems for a general class of nonlocal operators. Journal of Differential Equations., 268, 2368-2393.
[13] Bonder, J. F., Ritorto, A., & Salort, A. M. (2018) Shape optimization problems for nonlocal operators. Advances in Calculus of Variations., 11 (4), 373-386.
[14] Bonder, J. F., Rossi, J. D., & Spedaletti, J. F. (2018) Optimal design problems for the first p-fractional eigenvalue with mixed boundary conditions. Advanced Nonlinear Studies., 18 (2), 323-335.
[15] Pezzo, L. D., Bonder, J. F., & Ríos, L. L.(2018) An optimization problem for the first eigenvalue of the p-fractional Laplacian. Mathematische Nachrichten., 291 (4), 632-651.
[16] Bonder, J. F., & Spedaletti, J. F. (2018) Some nonlocal optimal design problems. Journal of Mathematical Analysis And Applications., 459 (2), 906-931.
[17] Di Nezza, E., Palatucci, G., & Valdinoci, E. (2012) Hitchhiker’s guide to the fractional Sobolev spaces. Bulletin of Mathematical Sciences, 136, 521-573.
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  • APA Style

    Chong Qiu. (2021). A Maximization Problem Involving a Fractional Laplace Type Operator. American Journal of Applied Mathematics, 9(3), 86-91. https://doi.org/10.11648/j.ajam.20210903.14

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    ACS Style

    Chong Qiu. A Maximization Problem Involving a Fractional Laplace Type Operator. Am. J. Appl. Math. 2021, 9(3), 86-91. doi: 10.11648/j.ajam.20210903.14

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    AMA Style

    Chong Qiu. A Maximization Problem Involving a Fractional Laplace Type Operator. Am J Appl Math. 2021;9(3):86-91. doi: 10.11648/j.ajam.20210903.14

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  • @article{10.11648/j.ajam.20210903.14,
      author = {Chong Qiu},
      title = {A Maximization Problem Involving a Fractional Laplace Type Operator},
      journal = {American Journal of Applied Mathematics},
      volume = {9},
      number = {3},
      pages = {86-91},
      doi = {10.11648/j.ajam.20210903.14},
      url = {https://doi.org/10.11648/j.ajam.20210903.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20210903.14},
      abstract = {Fractional Laplacian is an important nonlocal operator which has many applications in different kinds of differential equations. Recently, optimization problems involving the fractional Laplacian have been studied a lot by many authors. However, most of these papers are focusing on the optimization problems related to the first eigenvalue of the equation. Optimization problems related to the energy functional of the equation have not been investigated well enough. In this paper, we are going to study a maximization problem related to the energy functional of an equation involving a fractional Laplace type operator. Firstly, by using suitable variational framework in a fractional Sobolev space, we can show that a fractional equation has a solution which is in fact the global minimum of the corresponding energy functional. Moreover, by using reduction to absurdity we can obtain the uniqueness of the solution of the fractional equation. Then, we focus on a maximization problem related to the equation which takes the energy functional as the objective functional. Finally, by carefully analysing the properties of an arbitrarily choosen minimizing sequence and the tools of the rearrangement theory, we can prove that the maximization problem is solvable.},
     year = {2021}
    }
    

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  • TY  - JOUR
    T1  - A Maximization Problem Involving a Fractional Laplace Type Operator
    AU  - Chong Qiu
    Y1  - 2021/06/16
    PY  - 2021
    N1  - https://doi.org/10.11648/j.ajam.20210903.14
    DO  - 10.11648/j.ajam.20210903.14
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    EP  - 91
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20210903.14
    AB  - Fractional Laplacian is an important nonlocal operator which has many applications in different kinds of differential equations. Recently, optimization problems involving the fractional Laplacian have been studied a lot by many authors. However, most of these papers are focusing on the optimization problems related to the first eigenvalue of the equation. Optimization problems related to the energy functional of the equation have not been investigated well enough. In this paper, we are going to study a maximization problem related to the energy functional of an equation involving a fractional Laplace type operator. Firstly, by using suitable variational framework in a fractional Sobolev space, we can show that a fractional equation has a solution which is in fact the global minimum of the corresponding energy functional. Moreover, by using reduction to absurdity we can obtain the uniqueness of the solution of the fractional equation. Then, we focus on a maximization problem related to the equation which takes the energy functional as the objective functional. Finally, by carefully analysing the properties of an arbitrarily choosen minimizing sequence and the tools of the rearrangement theory, we can prove that the maximization problem is solvable.
    VL  - 9
    IS  - 3
    ER  - 

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Author Information
  • Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian, China

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