Partial Differential Equation Formulations from Variational Problems

Uchechukwu Opara

doi:doi:10.11648/j.pamj.20200901.11

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Partial Differential Equation Formulations from Variational Problems

Uchechukwu Opara

Published in Pure and Applied Mathematics Journal (Volume 9, Issue 1)

Received: 3 August 2019 Accepted: 29 August 2019 Published: 4 January 2020

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Abstract

The calculus of variations applied in multivariate problems can give rise to several classical Partial Differential Equations (PDE’s) of interest. To this end, it is acknowledged that a vast range of classical PDE’s were formulated initially from variational problems. In this paper, we aim to formulate such equations arising from the viewpoint of optimization of energy functionals on smooth Riemannian manifolds. These energy functionals are given as sufficiently regular integrals of other functionals defined on the manifolds. Relevant Banach domains which contain the optimal functional solutions are identified by preliminary analysis, and then necessary optimality conditions are discovered by differentiation in these Banach spaces. To determine specific optimal functionals in simple settings, smaller target domains are taken as appropriate subsets of the Banach (Sobolev) spaces. Briefings on analytical implications and approaches proffered are included for the aforementioned simple settings as well as more general case scenarios.

Published in	Pure and Applied Mathematics Journal (Volume 9, Issue 1)
DOI	10.11648/j.pamj.20200901.11
Page(s)	1-8
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

Riemannian Manifolds, Calculus of Variations, Energy Functionals, Ricci Flow, Potential Theory

References

[1]	Wolfgang Kuhnel – Differential Geometry: Curves – Surfaces – Manifolds, Copyright 2003 by the American Mathematical Society.
[2]	C. E. Chidume – Applicable Functional Analysis, Copyright 2014 by the Ibadan University Press Publishing House, University of Ibadan, Ibadan, Nigeria.
[3]	Haim Brezis – Functional Analysis, Sobolev Spaces and Partial Differential Equations, Copyright 2011 by Springer Science + Business Media.
[4]	H. D. Cao, B. Chow, S. C. Chu, S. T. Yau – Collected Papers on the Ricci Flow, Copyright 2003 by International Press.
[5]	Peter J. Olver – Applications of Lie Groups to Differential Equations (Second Edition), Copyright 1986, 1993 by Springer-Verlag New York, Inc.
[6]	Erich Miersemann – Partial Differential Equations, Copyright October 2012 by the Department of Mathematics, Leipzig University.
[7]	Bruce van Brunt – The Calculus of Variations, Copyright 2004 by Springer-Verlag New York, Inc.
[8]	Bennett Chow, Peng Lu, Lei Ni – Hamilton’s Ricci Flow, Copyright 2006 by the authors.
[9]	A. R. Forsyth – The Calculus of Variations, Copyright 1927 by the Cambridge University Press.
[10]	Jurgen Jost & Xianqing Li-Jost – Calculus of Variations, Copyright 1998 by the Cambridge University Press.
[11]	Peter Topping – Lectures on the Ricci Flow, Copyright 2006 by the author.
[12]	Peter J. Olver – Equivalence, Invariants and Symmetry, Copyright 1995 by the Cambridge University Press.

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APA Style

Uchechukwu Opara. (2020). Partial Differential Equation Formulations from Variational Problems. Pure and Applied Mathematics Journal, 9(1), 1-8. https://doi.org/10.11648/j.pamj.20200901.11

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ACS Style

Uchechukwu Opara. Partial Differential Equation Formulations from Variational Problems. Pure Appl. Math. J. 2020, 9(1), 1-8. doi: 10.11648/j.pamj.20200901.11

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AMA Style

Uchechukwu Opara. Partial Differential Equation Formulations from Variational Problems. Pure Appl Math J. 2020;9(1):1-8. doi: 10.11648/j.pamj.20200901.11

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@article{10.11648/j.pamj.20200901.11,
  author = {Uchechukwu Opara},
  title = {Partial Differential Equation Formulations from Variational Problems},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {1},
  pages = {1-8},
  doi = {10.11648/j.pamj.20200901.11},
  url = {https://doi.org/10.11648/j.pamj.20200901.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200901.11},
  abstract = {The calculus of variations applied in multivariate problems can give rise to several classical Partial Differential Equations (PDE’s) of interest. To this end, it is acknowledged that a vast range of classical PDE’s were formulated initially from variational problems. In this paper, we aim to formulate such equations arising from the viewpoint of optimization of energy functionals on smooth Riemannian manifolds. These energy functionals are given as sufficiently regular integrals of other functionals defined on the manifolds. Relevant Banach domains which contain the optimal functional solutions are identified by preliminary analysis, and then necessary optimality conditions are discovered by differentiation in these Banach spaces. To determine specific optimal functionals in simple settings, smaller target domains are taken as appropriate subsets of the Banach (Sobolev) spaces. Briefings on analytical implications and approaches proffered are included for the aforementioned simple settings as well as more general case scenarios.},
 year = {2020}
}

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TY  - JOUR
T1  - Partial Differential Equation Formulations from Variational Problems
AU  - Uchechukwu Opara
Y1  - 2020/01/04
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200901.11
DO  - 10.11648/j.pamj.20200901.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 1
EP  - 8
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200901.11
AB  - The calculus of variations applied in multivariate problems can give rise to several classical Partial Differential Equations (PDE’s) of interest. To this end, it is acknowledged that a vast range of classical PDE’s were formulated initially from variational problems. In this paper, we aim to formulate such equations arising from the viewpoint of optimization of energy functionals on smooth Riemannian manifolds. These energy functionals are given as sufficiently regular integrals of other functionals defined on the manifolds. Relevant Banach domains which contain the optimal functional solutions are identified by preliminary analysis, and then necessary optimality conditions are discovered by differentiation in these Banach spaces. To determine specific optimal functionals in simple settings, smaller target domains are taken as appropriate subsets of the Banach (Sobolev) spaces. Briefings on analytical implications and approaches proffered are included for the aforementioned simple settings as well as more general case scenarios.
VL  - 9
IS  - 1
ER  -

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Author Information

Uchechukwu Opara

Department of Mathematics & Statistics, Veritas University, Abuja, Nigeria

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Plain Text BibTeX RIS

APA Style

Uchechukwu Opara. (2020). Partial Differential Equation Formulations from Variational Problems. Pure and Applied Mathematics Journal, 9(1), 1-8. https://doi.org/10.11648/j.pamj.20200901.11

Copy | Download

ACS Style

Uchechukwu Opara. Partial Differential Equation Formulations from Variational Problems. Pure Appl. Math. J. 2020, 9(1), 1-8. doi: 10.11648/j.pamj.20200901.11

Copy | Download

AMA Style

Uchechukwu Opara. Partial Differential Equation Formulations from Variational Problems. Pure Appl Math J. 2020;9(1):1-8. doi: 10.11648/j.pamj.20200901.11

Copy | Download

@article{10.11648/j.pamj.20200901.11,
  author = {Uchechukwu Opara},
  title = {Partial Differential Equation Formulations from Variational Problems},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {1},
  pages = {1-8},
  doi = {10.11648/j.pamj.20200901.11},
  url = {https://doi.org/10.11648/j.pamj.20200901.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200901.11},
  abstract = {The calculus of variations applied in multivariate problems can give rise to several classical Partial Differential Equations (PDE’s) of interest. To this end, it is acknowledged that a vast range of classical PDE’s were formulated initially from variational problems. In this paper, we aim to formulate such equations arising from the viewpoint of optimization of energy functionals on smooth Riemannian manifolds. These energy functionals are given as sufficiently regular integrals of other functionals defined on the manifolds. Relevant Banach domains which contain the optimal functional solutions are identified by preliminary analysis, and then necessary optimality conditions are discovered by differentiation in these Banach spaces. To determine specific optimal functionals in simple settings, smaller target domains are taken as appropriate subsets of the Banach (Sobolev) spaces. Briefings on analytical implications and approaches proffered are included for the aforementioned simple settings as well as more general case scenarios.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Partial Differential Equation Formulations from Variational Problems
AU  - Uchechukwu Opara
Y1  - 2020/01/04
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200901.11
DO  - 10.11648/j.pamj.20200901.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 1
EP  - 8
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200901.11
AB  - The calculus of variations applied in multivariate problems can give rise to several classical Partial Differential Equations (PDE’s) of interest. To this end, it is acknowledged that a vast range of classical PDE’s were formulated initially from variational problems. In this paper, we aim to formulate such equations arising from the viewpoint of optimization of energy functionals on smooth Riemannian manifolds. These energy functionals are given as sufficiently regular integrals of other functionals defined on the manifolds. Relevant Banach domains which contain the optimal functional solutions are identified by preliminary analysis, and then necessary optimality conditions are discovered by differentiation in these Banach spaces. To determine specific optimal functionals in simple settings, smaller target domains are taken as appropriate subsets of the Banach (Sobolev) spaces. Briefings on analytical implications and approaches proffered are included for the aforementioned simple settings as well as more general case scenarios.
VL  - 9
IS  - 1
ER  -

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