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Axioms
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Principles of Variational Methods in Mathematical Physics
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Principles of Variational Methods in Mathematical Physics

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 31 December 2025 | Viewed by 11252

Special Issue Editor

Prof. Dr. Irina Meghea

E-Mail Website
Guest Editor
Faculty of Applied Sciences, Department of Mathematical Methods and Models, University POLITEHNICA of Bucharest, 313 Splaiul Independentei, RO-060042 Bucharest, Romania
Interests: variational methods; mathematical analysis; applied statistics

Special Issue Information

Dear Colleagues,

This Special Issue is devoted to the fundamental principles of variational methods, theoretical aspects related to main theorems and the multitude of variants for the mentioned results, together with the various problems in mathematical physics that are solved in such a way. Review articles, original papers and short communications are welcome. Both theoretical approaches and applications are expected.

The aim of this Special Issue is to encourage scientists to publish their experimental and theoretical results in as much detail as possible; there is no restriction on the length of the papers. The full experimental details must be provided so that the results can be reproduced.

The main topics of this Special Issue:

  • Fundamental variational principles—variants, related results, and applications;
  • Minimax, mountain pass and saddle-point-type theorems and their applications;
  • Main mathematical physics problems solved with the above statements;
  • Numerical methods to achieve the passage from mentioned theory towards the design of the solutions for mathematical physics problems evolved from modeling real phenomena.

Interdisciplinary and/or multidisciplinary papers are welcome.

Drawing this general line from the most abstract frame toward the design of a concrete solution of a real-world model highlights the importance of such a collection of works, as this is the key to scientific knowledge. The novelty is the expression of such a vision, and I consider that this Special Issue can supplement  the existing literature.

Prof. Dr. Irina Meghea
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Ekeland variational principle
  • perturbed variational principle
  • minimax-type theorems
  • mountain-pass-type theorems
  • saddle-point-type theorems
  • critical points
  • Dirichlet problem
  • von Neumann problem
  • fractional calculus
  • numerical methods
  • real-world modeling

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (7 papers)

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Research

Jump to: Review

21 pages, 322 KiB  
Article
A Quasi-Explicit Method Applied to Missing Boundary Data Reconstruction for the Stokes System
by Abdulaziz H. Alharbi, Fadhel Jday, Abdulrahman B. Albidah and Ali M. Alhartomi
Axioms 2025, 14(3), 177; https://doi.org/10.3390/axioms14030177 - 27 Feb 2025
Viewed by 307
Abstract
In this paper, we study the data completion problem for the Cauchy–Stokes equation in a cylindrical domain, Ω. Neumann and Dirichlet boundary conditions are prescribed on part of the overdetermined boundary, Γ0, and the goal is to complete the data [...] Read more.
In this paper, we study the data completion problem for the Cauchy–Stokes equation in a cylindrical domain, Ω. Neumann and Dirichlet boundary conditions are prescribed on part of the overdetermined boundary, Γ0, and the goal is to complete the data on the other part of the boundary, Γa. Here, Γ0 and Γa represent the side faces of the cylinder Ω. This problem is known to be ill-posed and is formulated as an optimal control problem with a regularized cost function. To directly approximate the missing data on Γa, we employ the method of factorization of elliptic boundary value problems. This technique allows the factorization of a boundary value problem into a product of parabolic problems. It is successfully applied to the optimality system in this work, yielding new and significant results. Full article
(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)
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15 pages, 459 KiB  
Article
Elasticity Problem with a Cusp between Thin Inclusion and Boundary
by Alexander Khludnev
Axioms 2023, 12(12), 1081; https://doi.org/10.3390/axioms12121081 - 27 Nov 2023
Cited by 1 | Viewed by 1130
Abstract
This paper concerns an equilibrium problem for an an elastic body with a thin rigid inclusion crossing an external boundary of the body at zero angle. The inclusion is assumed to be exfoliated from the surrounding elastic material that provides an interfacial crack. [...] Read more.
This paper concerns an equilibrium problem for an an elastic body with a thin rigid inclusion crossing an external boundary of the body at zero angle. The inclusion is assumed to be exfoliated from the surrounding elastic material that provides an interfacial crack. To avoid nonphysical interpenetration of the opposite crack faces, we impose inequality type constraints. Moreover, boundary conditions at the crack faces depend on a positive parameter describing a cohesion. A solution existence of the problem with different conditions on the external boundary is proved. Passages to the limit are analyzed as the damage parameter tends to infinity and to zero. Finally, an optimal control problem with a suitable cost functional is investigated. In this case, a part of the rigid inclusion is located outside of the elastic body, and a control function is a shape of the inclusion. Full article
(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)
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17 pages, 376 KiB  
Article
Solutions for Some Specific Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 2
by Irina Meghea
Axioms 2023, 12(8), 726; https://doi.org/10.3390/axioms12080726 - 26 Jul 2023
Cited by 1 | Viewed by 1217
Abstract
This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using the widely debated results of surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or [...] Read more.
This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using the widely debated results of surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important operators. The relevance of these operators and the possibility to be involved in the modeling of an important class of real phenomena is once again revealed by their applications. The use of certain variational methods facilitates the complete solution of the problem using appropriate numerical methods and computational algorithms. Some theoretical results are involved to complete the solutions for a sequence of models issued from real phenomena drawing. Full article
(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)
33 pages, 653 KiB  
Article
A Systematic Approach to Standard Dissipative Continua
by Sebastian Stark
Axioms 2023, 12(3), 267; https://doi.org/10.3390/axioms12030267 - 4 Mar 2023
Cited by 4 | Viewed by 1566
Abstract
Many isothermal dissipative continuum problems can be formulated in a variational setting using the concept of “standard dissipative continua”. A major advantage of this approach is that complex problems can be cast into a compact, thermodynamically consistent formulation based on a single space–time [...] Read more.
Many isothermal dissipative continuum problems can be formulated in a variational setting using the concept of “standard dissipative continua”. A major advantage of this approach is that complex problems can be cast into a compact, thermodynamically consistent formulation based on a single space–time continuous functional together with a corresponding variational principle. Formulating the problem in terms of a functional provides an immediate avenue for performing spatial and temporal discretization, which are the prerequisites for a numerical solution. Within the present contribution, a novel systematic approach to standard dissipative formulations is proposed, with the main goal being the development and implementation of generic procedures and algorithms for the formulation as well as the computational solution of a subset of isothermal dissipative continuum problems. In order to demonstrate the capabilities of the approach, its application to example problems is discussed. Full article
(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)
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15 pages, 302 KiB  
Article
Ekeland Variational Principle and Some of Its Equivalents on a Weighted Graph, Completeness and the OSC Property
by Basit Ali, Ştefan Cobzaş and Mokhwetha Daniel Mabula
Axioms 2023, 12(3), 247; https://doi.org/10.3390/axioms12030247 - 28 Feb 2023
Cited by 1 | Viewed by 2182
Abstract
We prove a version of the Ekeland Variational Principle (EkVP) in a weighted graph G and its equivalence to Caristi fixed point theorem and to the Takahashi minimization principle. The usual completeness and topological notions are replaced with some weaker versions expressed in [...] Read more.
We prove a version of the Ekeland Variational Principle (EkVP) in a weighted graph G and its equivalence to Caristi fixed point theorem and to the Takahashi minimization principle. The usual completeness and topological notions are replaced with some weaker versions expressed in terms of the graph G. The main tool used in the proof is the OSC property for sequences in a graph. Converse results, meaning the completeness of weighted graphs for which one of these principles holds, are also considered. Full article
(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)
7 pages, 235 KiB  
Article
Unfolding a Hidden Lagrangian Structure of a Class of Evolution Equations
by Philip Rosenau
Axioms 2023, 12(1), 2; https://doi.org/10.3390/axioms12010002 - 20 Dec 2022
Cited by 1 | Viewed by 1402
Abstract
It is shown that a simple modification of the standard Lagrangian underlying the dynamics of Newtonian lattices enables one to infer the hidden Lagrangian structure of certain classes of first order in time evolution equations which lack the conventional Lagrangian structure. Implication to [...] Read more.
It is shown that a simple modification of the standard Lagrangian underlying the dynamics of Newtonian lattices enables one to infer the hidden Lagrangian structure of certain classes of first order in time evolution equations which lack the conventional Lagrangian structure. Implication to other setups is outlined and exemplified. Full article
(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)

Review

Jump to: Research

66 pages, 6454 KiB  
Review
Solutions for Some Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 1
by Irina Meghea
Axioms 2023, 12(6), 532; https://doi.org/10.3390/axioms12060532 - 29 May 2023
Cited by 2 | Viewed by 1534
Abstract
This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important [...] Read more.
This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important operators. This article details three ways to use surjectivity results for a special type of operator involving the duality mapping and a Nemytskii operator, three methods starting from Ekeland’s variational principle and, lastly, one with a generalized variational principle to solve or describe the above-mentioned solutions. The relevance of these operators and the possibility of their involvement in the modeling of an important class of real phenomena determined the author to group these seven procedures together, presented in detail, followed by many applications, accompanied by a general overview of specialty domains. The use of certain variational methods facilitates the complete solution of the problem via appropriate numerical methods and computational algorithms. The exposure of the sequence of theoretical results, together with their demonstration in as much detail as possible has been fulfilled as an opportunity for the complete development of these topics. Full article
(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)
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