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. |
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Copyright © The Author(s), 2021. Published by Science Publishing Group |
Maximization, Fractional Laplace, Rearrangement
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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
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
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
@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} }
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 SP - 86 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 -