Partial Differential Equation Formulations from Variational Problems
Pure and Applied Mathematics Journal
Volume 9, Issue 1, February 2020, Pages: 1-8
Received: Aug. 3, 2019; Accepted: Aug. 29, 2019; Published: Jan. 4, 2020
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Author
Uchechukwu Opara, Department of Mathematics & Statistics, Veritas University, Abuja, Nigeria
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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.
Keywords
Riemannian Manifolds, Calculus of Variations, Energy Functionals, Ricci Flow, Potential Theory
To cite this article
Uchechukwu Opara, Partial Differential Equation Formulations from Variational Problems, Pure and Applied Mathematics Journal. Vol. 9, No. 1, 2020, pp. 1-8. doi: 10.11648/j.pamj.20200901.11
Copyright
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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