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Journal articles on the topic 'Calculus of variations'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

De Lellis, Camillo, Gerhard Huisken, and Robert Jerrard. "Calculus of Variations." Oberwolfach Reports 9, no. 3 (2012): 2205–68. http://dx.doi.org/10.4171/owr/2012/36.

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2

Brendle, Simon, Camillo De Lellis, and Robert Jerrard. "Calculus of Variations." Oberwolfach Reports 11, no. 3 (2014): 1801–60. http://dx.doi.org/10.4171/owr/2014/33.

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3

Brendle, Simon, Alessio Figalli, Robert Jerrard, and Neshan Wickramasekera. "Calculus of Variations." Oberwolfach Reports 13, no. 3 (2016): 1943–2008. http://dx.doi.org/10.4171/owr/2016/34.

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4

Figalli, Alessio, Robert Kohn, Tatiana Toro, and Neshan Wickramasekera. "Calculus of Variations." Oberwolfach Reports 15, no. 3 (2019): 2077–156. http://dx.doi.org/10.4171/owr/2018/35.

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5

Figalli, Alessio, Robert V. Kohn, Tatiana Toro, and Neshan Wickramasekera. "Calculus of Variations." Oberwolfach Reports 17, no. 2 (2021): 1139–96. http://dx.doi.org/10.4171/owr/2020/22.

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6

Bronsard, Lia, László Székelyhidi Jr., Yoshihiro Tonegawa, and Tatiana Toro. "Calculus of Variations." Oberwolfach Reports 19, no. 3 (2023): 2129–93. http://dx.doi.org/10.4171/owr/2022/37.

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7

Bronsard, Lia, Maria Colombo, László Székelyhidi Jr., and Yoshihiro Tonegawa. "Calculus of Variations." Oberwolfach Reports 21, no. 3 (2025): 2113–78. https://doi.org/10.4171/owr/2024/37.

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The Calculus of Variations is at the same time a classical subject, with long-standing open questions which have generated exciting discoveries in recent decades, and a modern subject in which new types of questions arise, driven by mathematical developments and emergent applications. It is also a subject with a very wide scope, touching on interrelated areas that include geometric variational problems, optimal transportation, geometric inequalities and domain optimization problems, elliptic regularity, geometric measure theory, harmonic analysis, physics, free boundary problems, etc. The work

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8

Dods, Victor. "Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus." Mathematics 10, no. 18 (2022): 3231. http://dx.doi.org/10.3390/math10183231.

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In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to “telescope” to accommodate a level of detail appropriate to a set of

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9

Chambers, Ll G., N. I. Akhiezer, Michael E. Alferieff, and B. Dacorogna. "The Calculus of Variations." Mathematical Gazette 74, no. 468 (1990): 191. http://dx.doi.org/10.2307/3619399.

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10

Gondran, M., and R. Hoblos. "Complex Calculus of Variations." IFAC Proceedings Volumes 34, no. 13 (2001): 727–30. http://dx.doi.org/10.1016/s1474-6670(17)39079-1.

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11

Neves, V. "Nonstandard Calculus of Variations." Journal of Mathematical Sciences 120, no. 1 (2004): 940–54. http://dx.doi.org/10.1023/b:joth.0000013557.48018.2a.

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12

Barron, E. N., and W. Liu. "Calculus of variations inL ∞." Applied Mathematics & Optimization 35, no. 3 (1997): 237–63. http://dx.doi.org/10.1007/bf02683330.

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13

Chrastina, Jan. "Examples from the calculus of variations. I. Nondegenerate problems." Mathematica Bohemica 125, no. 1 (2000): 55–76. http://dx.doi.org/10.21136/mb.2000.126263.

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14

Chrastina, Jan. "Examples from the calculus of variations. IV. Concluding review." Mathematica Bohemica 126, no. 4 (2001): 691–710. http://dx.doi.org/10.21136/mb.2001.134113.

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15

Torres, Delfim F. M. "On a Non-Newtonian Calculus of Variations." Axioms 10, no. 3 (2021): 171. http://dx.doi.org/10.3390/axioms10030171.

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The calculus of variations is a field of mathematical analysis born in 1687 with Newton’s problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler–Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton’s approach to calculus. As in many applications negative values of admissible functions are not physic

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16

Barron, E. N., and W. Liu. "Calculus of Variations in L ∞." Applied Mathematics and Optimization 35, no. 3 (1997): 237–63. http://dx.doi.org/10.1007/s002459900047.

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17

Hildebrandt, Stefan. "The calculus of variations today." Mathematical Intelligencer 11, no. 4 (1989): 50–60. http://dx.doi.org/10.1007/bf03025887.

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18

Zine, Houssine, and Delfim F. M. Torres. "A Stochastic Fractional Calculus with Applications to Variational Principles." Fractal and Fractional 4, no. 3 (2020): 38. http://dx.doi.org/10.3390/fractalfract4030038.

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We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation.

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19

Odzijewicz, Tatiana, Agnieszka B. Malinowska, and Delfim F. M. Torres. "Generalized fractional calculus with applications to the calculus of variations." Computers & Mathematics with Applications 64, no. 10 (2012): 3351–66. http://dx.doi.org/10.1016/j.camwa.2012.01.073.

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20

Massa, Enrico, Danilo Bruno, Gianvittorio Luria, and Enrico Pagani. "Geometric constrained variational calculus. II: The second variation (Part I)." International Journal of Geometric Methods in Modern Physics 13, no. 01 (2016): 1550132. http://dx.doi.org/10.1142/s0219887815501327.

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Within the geometrical framework developed in [Geometric constrained variational calculus. I: Piecewise smooth extremals, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550061], the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and reinterpreted in terms of Jacobi fields.

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21

Chrastina, Jan. "Solution of the inverse problem of the calculus of variations." Mathematica Bohemica 119, no. 2 (1994): 157–201. http://dx.doi.org/10.21136/mb.1994.126079.

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22

Chrastina, Jan. "Examples from the calculus of variations. II. A degenerate problem." Mathematica Bohemica 125, no. 2 (2000): 187–97. http://dx.doi.org/10.21136/mb.2000.125951.

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23

Bandyopadhyay, Saugata, Bernard Dacorogna, and Swarnendu Sil. "Calculus of variations with differential forms." Journal of the European Mathematical Society 17, no. 4 (2015): 1009–39. http://dx.doi.org/10.4171/jems/525.

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24

Nurbekyan, L. "Calculus of variations in Hilbert spaces." Journal of Contemporary Mathematical Analysis 47, no. 3 (2012): 148–60. http://dx.doi.org/10.3103/s1068362312030053.

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25

Chorlton, Frank, and U. Brechtken-Manderscheid. "Introduction to the Calculus of Variations." Mathematical Gazette 78, no. 481 (1994): 96. http://dx.doi.org/10.2307/3619470.

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26

Dickey, Leonid A. "Do Dogs Know Calculus of Variations?" College Mathematics Journal 37, no. 1 (2006): 20. http://dx.doi.org/10.2307/27646267.

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27

Dickey, Leonid A. "Do Dogs Know Calculus of Variations?" College Mathematics Journal 37, no. 1 (2006): 20–23. http://dx.doi.org/10.1080/07468342.2006.11922162.

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28

Carriero, Michele, Antonio Leaci, and Franco Tomarelli. "Calculus of variations and image segmentation." Journal of Physiology-Paris 97, no. 2-3 (2003): 343–53. http://dx.doi.org/10.1016/j.jphysparis.2003.09.008.

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29

Tan, H. H., and R. B. Potts. "A Discrete calculus of variations algorithm." Bulletin of the Australian Mathematical Society 38, no. 3 (1988): 365–71. http://dx.doi.org/10.1017/s0004972700027726.

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30

Maalaoui, Ali. "Bubbling phenomena in calculus of variations." Arabian Journal of Mathematics 6, no. 3 (2016): 213–37. http://dx.doi.org/10.1007/s40065-016-0157-x.

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31

Ramos, J. l. "Introduction to the calculus of variations." Applied Mathematical Modelling 17, no. 1 (1993): 53. http://dx.doi.org/10.1016/0307-904x(93)90134-3.

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32

Serfaty, Sylvia. "Lagrange and the calculus of variations." Lettera Matematica 2, no. 1-2 (2014): 39–46. http://dx.doi.org/10.1007/s40329-014-0049-x.

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33

Bailey, C. D. "Dynamics and the calculus of variations." Computer Methods in Applied Mechanics and Engineering 60, no. 3 (1987): 275–87. http://dx.doi.org/10.1016/0045-7825(87)90135-6.

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34

Gift, S. J. G. "Contributions to the calculus of variations." Journal of Optimization Theory and Applications 52, no. 1 (1987): 25–51. http://dx.doi.org/10.1007/bf00938463.

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35

Bögelein, Verena, Frank Duzaar, and Paolo Marcellini. "Existence of evolutionary variational solutions via the calculus of variations." Journal of Differential Equations 256, no. 12 (2014): 3912–42. http://dx.doi.org/10.1016/j.jde.2014.03.005.

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36

Chrastina, Jan. "Examples from the calculus of variations. III. Legendre and Jacobi conditions." Mathematica Bohemica 126, no. 1 (2001): 93–111. http://dx.doi.org/10.21136/mb.2001.133926.

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37

Matějík, M. "Calculus of variations and its application to division of forest land." Journal of Forest Science 50, No. 9 (2012): 439–46. http://dx.doi.org/10.17221/4639-jfs.

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The paper deals with an application of the least squares method (LSM) for the purposes of division and evaluation of land. This method can be used in all cases with redundant number of measurements, in this case of segments of plots. From the mathematical aspect, the minimisation condition of the LSM is a standardised condition &sum; pvv = min., minimising the Euclidean norm ||v||<sub>E</sub> of an n-dimensional vector of residues of plot segments at simultaneous satisfaction of the given conditions. The traditional procedure of calculus of variations with the use of Lagrangian

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38

Mwinken, Delphin. "The Interconnection Between Calculus of Variations, Partial Differential Equations and Differential Geometry." Selecciones Matemáticas 11, no. 02 (2024): 393–408. https://doi.org/10.17268/sel.mat.2024.02.11.

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Calculus of variations is a fundamental mathematical discipline focused on optimizing functionals, which map sets of functions to real numbers. This field is essential for numerous applications, including the formulation and solution of partial differential equations (PDEs) and the study of differential geometry. In PDEs, calculus of variations provides methods to find functions that minimize energy functionals, leading to solutions of various physical problems. In differential geometry, it helps understand the properties of curves and surfaces, such as geodesics, by minimizing arc-length func

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39

Kupershmidt, B. A. "An algebraic model of graded calculus of variations." Mathematical Proceedings of the Cambridge Philosophical Society 101, no. 1 (1987): 151–66. http://dx.doi.org/10.1017/s0305004100066494.

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The modern theory of integrable systems rests on two fundamental pillars: the classification of Lax [13] and zero-curvature equations [14, 1, 2]; and algebraic models of the classical calculus of variations [9,5] specialized to the residue calculus in modules of differential forms over rings of matrix pseudo-differential operators [9, 6]. Both these aspects of the theory are by now very well understood for integrable systems in one space dimension.

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40

Cellina, Arrigo. "Some Problems in the Calculus of Variations." Annales Mathematicae Silesianae 31, no. 1 (2017): 5–55. http://dx.doi.org/10.1515/amsil-2017-0005.

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41

PRUSINSKA, AGNIESZKA, EWA SZCZEPANIK, and ALEXEY A. TRETYAKOV. "High-order optimality conditions for degenerate variational problems." Carpathian Journal of Mathematics 30, no. 3 (2014): 387–94. http://dx.doi.org/10.37193/cjm.2014.03.03.

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The paper is devoted to the class of singular calculus of variations problems with constraints which are not regular mappings at the solution point. Methods of the p-regularity theory are used for investigation of isoperimetric and Lagrange singular problems. Necessary conditions for optimality in p-regular calculus of variations problem are presented.

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42

Ioffe, Alexander D., and Alexander J. Zaslavski. "Variational Principles and Well-Posedness in Optimization and Calculus of Variations." SIAM Journal on Control and Optimization 38, no. 2 (2000): 566–81. http://dx.doi.org/10.1137/s0363012998335632.

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43

Kohan, MahdiNakhaie, and Hamid Behnam. "Denoising medical images using calculus of variations." Journal of Medical Signals & Sensors 1, no. 3 (2011): 5. http://dx.doi.org/10.4103/2228-7477.95413.

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44

Giannetti, Flavia. "Elliptic complexes in the calculus of variations." Dissertationes Mathematicae 418 (2003): 1–63. http://dx.doi.org/10.4064/dm418-0-1.

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45

Bandyopadhyay, Saugata, and Swarnendu Sil. "Exterior convexity and classical calculus of variations." ESAIM: Control, Optimisation and Calculus of Variations 22, no. 2 (2016): 338–54. http://dx.doi.org/10.1051/cocv/2015007.

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46

Donaldson, Simon. "Karen Uhlenbeck and the Calculus of Variations." Notices of the American Mathematical Society 66, no. 03 (2019): 1. http://dx.doi.org/10.1090/noti1806.

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47

Sil, Swarnendu. "Calculus of variations: A differential form approach." Advances in Calculus of Variations 12, no. 1 (2019): 57–84. http://dx.doi.org/10.1515/acv-2016-0058.

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AbstractWe study integrals of the form {\int_{\Omega}f(d\omega_{1},\dots,d\omega_{m})}, where {m\geq 1} is a given integer, {1\leq k_{i}\leq n} are integers, {\omega_{i}} is a {(k_{i}-1)}-form for all {1\leq i\leq m} and {f:\prod_{i=1}^{m}\Lambda^{k_{i}}(\mathbb{R}^{n})\rightarrow\mathbb{R}} is a continuous function. We introduce the appropriate notions of convexity, namely vectorial ext. one convexity, vectorial ext. quasiconvexity and vectorial ext. polyconvexity. We prove weak lower semicontinuity theorems and weak continuity theorems and conclude with applications to minimization problems.

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48

Vyridís, Panayotis. "Bifurcation in calculus of variations with constraints." Acta Universitaria 23 (December 1, 2013): 27–31. http://dx.doi.org/10.15174/au.2013.583.

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We describe a variational problem on a domain of a plane under a constraint of geometrical character. We provide sufficient and necessary conditions for the existence of bifurcation points. The problem in 2 coordinate form, corresponds to a quasilinear elliptic boundary value problem. The problem provides a physical model for several applications referring to continuum media and membranes.

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49

Clarke, F. H., Yu S. Ledyaev, and J. B. Hiriart-Urruty. "Global optimality in the calculus of variations." Nonlinear Analysis: Theory, Methods & Applications 28, no. 7 (1997): 1187–92. http://dx.doi.org/10.1016/s0362-546x(97)82868-4.

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50

Khudaverdian, Hovhannes M., and Theodore Voronov. "On complexes related with calculus of variations." Journal of Geometry and Physics 44, no. 2-3 (2002): 221–50. http://dx.doi.org/10.1016/s0393-0440(02)00075-x.

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