Definition
Theorems on the existence of minimizers for functionals defined on Banach spaces, and related approximation methods, applied to minimization problems arising in elasticity theory.
Introduction
Variational methods are a powerful tool in elasticity and in fact the only known approach able to guarantee sufficient generality in the treatment of problems arising in hyperelasticity (see the corresponding entry), in the asymptotic derivation of two-dimensional elastic models (Ciarlet, 1988, 1997), and in many other cases in continuum mechanics (Pedregal, 2000; Fonseca, 1987; dell’Isola and Placidi, 2011). Variational problems take usually the form of a minimization problem that can be described as follows: we search the minimum of a functional F(u) defined on a subset S of a Banach space B (i.e., a normed, complete vector space) and taking values in [−∞, +∞]. Direct method provides sufficient conditions on S, B, and F for the existence of a minimizer \(\tilde {u}\in S\). The main...
Notes
- 1.
We recall that compactness is a topological invariant.
- 2.
A set is said to be compact if each of its open covers admits a finite subcover.
- 3.
In French literature (e.g., in Ciarlet 1988), it is sometimes used the more logical name infimizing sequence.
- 4.
Here B∗ denotes the dual space of B, i.e., the vector space of continuous linear functionals on B, and 〈a, b∗〉 denotes the action of the continuous linear functional b∗∈ B∗ on a ∈ B. Note that in a Hilbert space H (e.g., the Sobolev spaces Hk := Wk, 2), 〈a, b∗〉 can be expressed by means of an inner product 〈a, b〉 with b ∈ H.
- 5.
By this we mean the convergence induced by the usual functional norm \(\vert \vert v^* \vert \vert _{B^*}:=\sup \langle u,v^*\rangle :\vert \vert u \vert \vert =1\).
- 6.
In fact, the same can be said for generic integral functionals.
- 7.
A real function f defined on a topological space X is called lower semicontinuous if the set {x ∈ X : f(x) > α} is an open real set for every \(\alpha \in \mathbb {R}\). A well-known example of a functional which is lower semicontinuous but not continuous is the arc-length functional, i.e., \(L(f):=\int _a^b \sqrt {1+(f'(x))^2} dx\). It is geometrically evident that, in the C0 norm, L is not continuous, as we can find functions arbitrarily close to a given rectifiable function whose graph has arbitrarily large length. On the other hand, it is easy to see that it is lower semicontinuous. This simple example shows how natural the concept of lower semicontinuity is. Clearly a similar result holds true for the existence of maxima of upper semicontinuous functions.
References
Alaoglu L (1940) Weak topologies of normed linear spaces. Ann Math 41:252–267
Axelsson O, Barker VA (2001) Finite element solution of boundary value problems: theory and computation. SIAM, Philadelphia
Ball JM (1976) Convexity conditions and existence theorems in nonlinear elasticity. Arch Ration Mech Anal 63(4):337–403
Ball JM, Mizel VJ (1987) One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. In: Analysis and thermomechanics. Springer, Berlin/Heidelberg, pp 285–348
Banach S (1932) Théorie des opérations linéaires
Belloni M (1995) Interpretation of lavrentiev phenomenon by relaxation: the higher order case. Trans Am Math Soc 347:2011–2023
Berkovitz LD (1974) Lower semicontinuity of integral functionals. Trans Am Math Soc 192:51–57
Bouchitté G, Braides A, Buttazzo G (1995) Relaxation results for some free discontinuity problems. Journal fur die Reine und Angewandte Mathematik 458:1–18
Ciarlet PG (1988) Mathematical elasticity. vol. I, volume 20 of studies in mathematics and its applications. North-Holland, Amsterdam
Ciarlet PG (1997) Mathematical elasticity, vol. II: theory of plates, volume 27 of studies in mathematics and its applications. North-Holland, Amsterdam
Dacorogna B (2007) Direct methods in the calculus of variations, vol 78. Springer, New York
Della Corte A, dell’Isola F, Esposito R, Pulvirenti M (2017) Equilibria of a clamped euler beam (elastica) with distributed load: large deformations. Math Models Methods Appl Sci 27(8):1391–1421
dell’Isola F, Placidi L (2011) Variational principles are a powerful tool also for formulating field theories. In: Variational models and methods in solid and fluid mechanics. Springer, Vienna, pp 1–15
Fonseca I (1987) Variational methods for elastic crystals. Arch Ration Mech Anal 97(3):189–220
Fonseca I, Leoni G (2007) Modern methods in the calculus of variations: Lˆ p spaces. Springer, New York
Galerkin BG (1915) Series solution of some problems of elastic equilibrium of rods and plates. Vestn Inzh Tekh 19:897–908
Gelfand IM, Fomin SV (1963) Richard A. Silverman (ed) Calculus of variations. Prentice-Hall, Englewood Cliffs
Kantorovich L, Krylov V (1962) Approximate methods of advanced analysis. Gos Izd Fiz-Math Lit, Leningrad 560
McShane E et al (1940) Necessary conditions in generalized-curve problems of the calculus of variations. Duke Math J 7(1):1–27
Mikhlin SG (1957) Variatsionnye metody v matematicheskoi fizike. Gostekhizdat
Morrey CB (1952) Quasi-convexity and the lower semicontinuity of multiple integrals. Pac J Math 2(1):25–53
Pedregal P (2000) Variational methods in nonlinear elasticity. SIAM, Philadelphia
Ritz W (1908) Uber eine neue methode zur losung gewisser variations-probleme der mathematischen physik. Journal fuur die reine und angewandte Mathematik 135(1):1–61
Roubicek T (1997) Relaxation in optimization theory and variational calculus, vol 4. Walter de Gruyter, Berlin
Rudin W (1991) Functional analysis. International series in pure and applied mathematics. McGraw-Hill, New York
Smirnov VI (2014) A course of higher mathematics, vol 62. Elsevier, Burlington
Sobolev SL (1950) Some applications of functional analysis in mathematical physics, vol 90. American Mathematical Society, Providence
Tonelli L (1911) Sui massimi e minimi assoluti del calcolo delle variazioni. Rendiconti del Circolo Matematico di Palermo (1884–1940) 32(1):297–337
Tonelli L (1920) La semicontinuita nel calcolo delle variazioni. Rendiconti del Circolo Matematico di Palermo (1884–1940) 44(1):167–249
Young LC (1937) Generalized curves and the existence of an attained absolute minimum in the calculus of variations. Comptes Rendus de la Société des Sci et des Lettres de Varsovie 30:212–234
Young LC (1938) Necessary conditions in the calculus of variations. Acta Mathematica 69(1):229–258
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2018 Springer-Verlag GmbH Germany
About this entry
Cite this entry
Corte, A.D., dell’Isola, F. (2018). Direct Method of Calculus of Variations in Elasticity. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_174-1
Download citation
DOI: https://doi.org/10.1007/978-3-662-53605-6_174-1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-53605-6
Online ISBN: 978-3-662-53605-6
eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering