Some variational problems for a Föppl--von Kármán plate subject to general equilibrated loads are studied. The existence of global minimizers is proved under the assumption that the out-of-plane displacement fulfils homogeneous Dirichlet condition on the whole boundary while the in-plane displacement fulfils nonhomogeneous Neumann condition. If the Dirichlet condition is prescribed only on a subset of the boundary, then the energy may be unbounded from below over the set of admissible configurations, as shown by several explicit conterexamples: in these cases the analysis of critical points is addressed through an asymptotic development of the energy functional in a neighborhood of the flat configuration. By a $\Gamma$-convergence approach we show that critical points of the Föppl--von Kármán energy can be strongly approximated by uniform Palais--Smale sequences of suitable functionals: this property leads to identifying relevant features for critical points of approximating functionals, e.g., buckled configurations of the plate. The analysis for rescaled thickness is performed by assuming that the plate-like structure is initially prestressed, so that the energy functional depends only on the out-of-plane displacement and exhibits asymptotic oscillating minimizers as a mechanism to relax compressive states.


  1. Föppl--von Kármán
  2. calculus of variations
  3. elasticity
  4. nonlinear Neumann problems
  5. Monge--Ampère equation
  6. critical points
  7. $\Gamma$-convergence
  8. asymptotic analysis
  9. singular perturbations
  10. mechanical instabilities

MSC codes

  1. 49J45
  2. 74K30
  3. 74K35
  4. 74R10

Get full access to this article

View all available purchase options and get full access to this article.


G. Anzellotti, S. Baldo, and D. Percivale, Dimension reduction in variational problems, asymptotic development in $\Gamma$-convergence and thin structures in elasticity, Asymptot. Anal., 9 (1994), pp. 61--100.
M. Al-Gwaiz, V. Benci, and G. Gazzola, Bending and stretching energies in a rectangular plate modeling suspension bridges, Nonlinear Anal., 106 (2014), pp. 18--34.
B. Audoly and Y. Pomeau, Elasticity and Geometry, Oxford University Press, Oxford, 2010.
C. Baiocchi, G. Buttazzo, F. Gastaldi, and F. Tomarelli, General existence results for unilateral problems in continuum mechanics, Arch. Ration. Mech. Anal., 100 (1988), pp. 149--189.
J. Bedrossian and R. V. Kohn, Blister patterns and energy minimization in compressed thin films on compliant substrates, Comm. Pure Appl. Math., 68 (2015), pp. 472--510.
P. Bella and R. V. Kohn, Metric-induced wrinkling of a thin elastic sheet, J. Nonlinear Sci., 24 (2014), pp. 1147--1176.
P. Bella and R. V. Kohn, Wrinkles as the result of compressive stresses in an annular thin film, Comm. Pure Appl. Math., 67 (2014), pp. 693--747.
P. Bella and R. V. Kohn, Coarsening of folds in hanging drapes, Comm. Pure Appl. Math., 70 (2017), pp. 978--1021.
H. B. Belgacem, S. Conti, A. DeSimone, and S. Müller, Rigorous bounds for the Föppl--von Kármán theory of isotropically compressed plates, J. Nonlinear Sci., 106 (2000), pp. 661--683.
B. Belgacem, S. Conti, A. DeSimone, and S. Müller, Energy scaling of compressed elastic films, Arch. Ration. Mech. Anal., 164 (2002), pp. 1--37.
D. P. Bourne, S. Conti, and S. Müller, Energy bounds for a compressed elastic film on a substrate, J. Nonlinear Sci., 27 (2017), pp. 453--494.
J. Brandman, R. V. Kohn, and H.-M. Nguyen, Energy scaling laws for conically constrained thin elastic sheets, J. Elasticity, 113 (2013).
K. Bhattacharya, I. Fonseca, and G. Francfort, An asymptotic study of the debonding of thin films, Arch. Ration. Mech. Anal., 161 (2002), pp. 205--229.
G. Buttazzo and G. Dal Maso, Singular perturbation problems in the calculus of variations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 11 (1984), pp. 395--430.
G. Buttazzo and F. Tomarelli, Compatibility conditions for nonlinear Neumann problems, Adv. Math., 89 (1991), pp. 127--143.
M. Carriero, A. Leaci, and F. Tomarelli, Strong solution for an elastic-plastic plate, Calc. Var. Partial Differential Equations, 2 (1994), pp. 219--240.
E. Cerda and L. Mahadevan, Geometry and physics of wrinkling, Phys. Rev. Lett., 90 (2003), pp. 1--5.
P. G. Ciarlet, Mathematical Elasticity, Volume II: Theory of Plates, Elsevier, New York, 1997.
S. Conti, F. Maggi, and S. Müller, Rigorous derivation of Föppl's theory for clamped elastic membranes leads to relaxation, SIAM J. Math Anal., 38 (2006), pp. 657--680.
G. Dal Maso, M. Negri, and D. Percivale, Linearized elasticity as $\Gamma$-limit of finite elasticity, Set-Valued Anal., 10 (2002), pp. 165--183.
N. Damil, M. Potier-Ferry, and H. Hu, Membrane wrinkling revisited from a multi scale point of view, Adv. Model. Simulat. Eng. Science, 16 (2014).
B. Davidovitch, R. D. Schroll, D. Vella, M. Adda-Bedia, and E. A. Cerda, Prototypical model for tensional wrinkling in thin sheets, Proc. Natl. Acad. Sci. USA, 108 (2011), pp. 18227--18232.
G. Friesecke, R. D. James, and S. Müller, A hierarky of plate models from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal., 180 (2006), pp. 183--236.
G. Gioia and M. Ortiz, Delamination of compressed thin films, Adv. Appl. Mech., 33 (1997), pp. 119--192.
C. E. Guttiérrez, The Monge-Ampère Equation, Birkhäuser, Basel. Switzerland, 2001.
T. J. Healey, Q. Li, and R. B. Cheng, Wrinkling behavior of highly stretched rectangular elastic films via parametric global bifurcation, J. Nonlinear Sci., 23 (2013), pp. 777--805.
R. L. Jerrard and P. Sternberg, Critical points via $\Gamma$-convergence: General theory and applications, J. Eur. Math. Soc. (JEMS), 11 (2009), pp. 705--753.
R. V. Kohn and H.-M. Nguyen, Analysis of a compressed thin film bonded to a compliant substrate: The energy scaling law, J. Nonlinear Sci., 23 (2013), pp. 343--362.
M. Lecumberry and S. Müller, Stability of slender bodies under compression and validity of von Kármán theory, Arch. Ration. Mech. Anal., 193 (2009), pp. 255--310.
M. Leocmach, M. Nespoulous, S. Manneville, and T. Gibaud, Hierarchical wrinkling in a confined permeable biogel, Sci. Adv., 1 (2015), e1500608.
M. Lewicka, L. Mahadevan, and M. R. Pakzad, The Monge--Ampère constraint: Matching of isometries, density and regularity and elastic theories of shallow shells, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), pp. 45--67.
M. Lewicka and R. Pakzad, Prestrained elasticity: From shape formation to Monge--Ampère anomalies, Notices Amer. Math. Soc., 63 (2016), pp. 8--11.
M. Lewicka, P. Ochoa, and R. Pakzad, Variational models for prestrained plates with Monge--Ampère constraint, Differential Integral Equations, 28 (2015), pp. 861--898.
J. Maly, Absolutely continuous functions of several variables, J. Math. Anal. Appl., 231 (1999), pp. 492--508.
F. Maddalena and D. Percivale, Variational models for peeling problems, Interfaces Free Bound., 10 (2008), pp. 503--516.
F. Maddalena, D. Percivale, G. Puglisi, and L. Truskinowsky, Mechanics of reversible unzipping, Contin. Mech. Thermodyn., 21 (2009), pp. 251--268.
F. Maddalena, D. Percivale, and F. Tomarelli, Adhesive flexible material structures, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), pp. 553--574.
F. Maddalena, D. Percivale, and F. Tomarelli, Elastic Structures in Adhesion Interaction, Variational Analysis and Aerospace Engineering, A. Frediani and G. Buttazzo, eds., Ser. Springer Optim. Appl. 66, 2012, pp. 289--304.
F. Maddalena, D. Percivale, and F. Tomarelli, Local and nonlocal energies in adhesive interaction, IMA J. Appl. Math., 81 (2016), pp. 1051--1075.
H. Olbermann, Energy scaling law for a single disclination in a thin elastic sheet, Arch. Ration. Mech. Anal., 224 (2017), pp. 985--1019.
M. R. Pakzad, On the Sobolev space of isometric immersions, J. Differential Geom., 66 (2004), pp. 47--69.
D. Percivale and F. Tomarelli, From SBD to SBH: The elastic-plastic plate, Interfaces Free Bound., 4 (2002), pp. 137--165.
D. Percivale and F. Tomarelli, A variational principle for plastic hinges in a beam, Math. Models Methods Appl., 19 (2009), pp. 2263--2297.
D. Percivale and F. Tomarelli, Smooth and broken minimizers of some free discontinuity problems, to appear in Solvability, Regularity and Optimal Control of Boundary Value Problems for PDEs, P. Colli, A. Favini, E. Rocca, G. Schimperna, and J. Sprekels, eds., Springer INdAM Ser. 22, Springer, New York, 2017, pp. 431--468. https://doi.org/10.1007/978-3-319-64489-9_17.
E. Puntel, L. Deseri, and E. Fried, Wrinkling of a stretched thin sheet, J. Elasticity, 105 (2011), pp. 137--170.
J. Rauch and B. A. Taylor, The Dirichelet problem for the multidimensional Monge--Ampère equation, Rocky Mountain J. Math., 7 (1977), pp. 345--364.
S. K. Vodopyanov and V. M. Goldstein, Quasiconformal mappings and spaces with generalized first derivatives, Sib. Math. J., 17 (1976), pp. 399--411.

Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 251 - 282
ISSN (online): 1095-7154


Submitted: 8 February 2017
Accepted: 26 September 2017
Published online: 11 January 2018


  1. Föppl--von Kármán
  2. calculus of variations
  3. elasticity
  4. nonlinear Neumann problems
  5. Monge--Ampère equation
  6. critical points
  7. $\Gamma$-convergence
  8. asymptotic analysis
  9. singular perturbations
  10. mechanical instabilities

MSC codes

  1. 49J45
  2. 74K30
  3. 74K35
  4. 74R10



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options


View PDF







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.