A Du Bois-Reymond Convex Inclusion for Nonautonomous Problems of the Calculus of Variations and Regularity of Minimizers

  • Piernicola Bettiol
  • Carlo MaricondaEmail author


We consider a local minimizer, in the sense of the \(W^{1,m}\) norm (\(m\ge 1\)), of the classical problem of the calculus of variations
$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathrm{Minimize}}\quad &{}\displaystyle I(x):=\int _a^b\varLambda (t,x(t), x'(t))\,dt+\varPsi (x(a), x(b))\\ \text {subject to:} &{}x\in W^{1,m}([a,b];\mathbb {R}^n),\\ &{}x'(t)\in C\,\text { a.e., } \,x(t)\in \varSigma \quad \forall t\in [a,b].\\ \end{array}\right. } \end{aligned}$$
where \(\varLambda :[a,b]\times \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}\cup \{+\infty \}\) is just Borel measurable, C is a cone, \(\varSigma \) is a nonempty subset of \(\mathbb {R}^n\) and \(\varPsi \) is an arbitrary possibly extended valued function. When \(\varLambda \) is real valued, we merely assume a local Lipschitz condition on \(\varLambda \) with respect to t, allowing \(\varLambda (t,x,\xi )\) to be discontinuous and nonconvex in x or \(\xi \). In the case of an extended valued Lagrangian, we impose the lower semicontinuity of \(\varLambda (\cdot ,x,\cdot )\), and a condition on the effective domain of \(\varLambda (t,x,\cdot )\). We use a recent variational Weierstrass type inequality to show that the minimizers satisfy a relaxation result and an Erdmann – Du Bois-Reymond convex inclusion which, remarkably, holds whenever \(\varLambda (x,\xi )\) is autonomous and just Borel. Under a growth condition, weaker than superlinearity, we infer the Lipschitz continuity of minimizers.


Weierstrass, directional Nonautonomous Lagrangian Tonelli–Morrey Proximal Maximum principle Calculus of variations Du Bois-Reymond Erdmann Regularity Lipschitz Growth Slow growth 

Mathematics Subject Classification

49N60 49K05 90C25 



We thank Richard Vinter for pointing out the lack of a regularity results for problems concerning nonautonomous Lagrangians with state constraints. C. M. wishes to thank the University of Brest and P. B. for the hospitality during the preparation of the paper.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Ambrosio, L., Ascenzi, O., Buttazzo, G.: Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands. J. Math. Anal. Appl. 142, 301–316 (1989)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ball, J.M., Mizel, V.J.: One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. Arch. Ration. Mech. Anal. 90, 325–388 (1985)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bettiol, P., Mariconda, C.: A new variational inequality in the Calculus of Variations and Lipschitz regularity of minimizers. J. Differ. Equ. (2019). CrossRefGoogle Scholar
  4. 4.
    Bettiol, P., Mariconda, C.: On a new necessary condition in the Calculus of Variations for highly discontinuous Lagrangians in the state and velocity. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30, 649–663 (2019)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Buttazzo, G., Giaquinta, M., Hildebrandt, S.: One-dimensional variational problems. An introduction., Oxford Lecture Series in Mathematics and its Applications, vol. 15. The Clarendon Press, Oxford University Press, New York (1998)Google Scholar
  6. 6.
    Cellina, A.: The classical problem of the calculus of variations in the autonomous case: relaxation and Lipschitzianity of solutions. Trans. Am. Math. Soc. 356, 415–426 (2004). (electronic)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cellina, A., Treu, G., Zagatti, S.: On the minimum problem for a class of non-coercive functionals. J. Differ. Equat. 127(1), 225–262 (1996). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cesari, L.: Optimization–theory and applications, Applications of Mathematics (New York), vol. 17. Springer, New York (1983). Problems with ordinary differential equationsGoogle Scholar
  9. 9.
    Clarke, F.H.: An indirect method in the calculus of variations. Trans. Am. Math. Soc. 336, 655–673 (1993)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Clarke, F.H.: Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, vol. 264. Springer, London (2013)CrossRefGoogle Scholar
  11. 11.
    Clarke, F.H., Vinter, R.B.: Regularity properties of solutions to the basic problem in the calculus of variations. Trans. Am. Math. Soc. 289, 73–98 (1985)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cupini, G., Guidorzi, M., Marcelli, C.: Necessary conditions and non-existence results for autonomous nonconvex variational problems. J. Differ. Equat. 243, 329–348 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dal Maso, G., Frankowska, H.: Autonomous integral functionals with discontinuous nonconvex integrands: Lipschitz regularity of minimizers, DuBois-Reymond necessary conditions, and Hamilton-Jacobi equations. Appl. Math. Optim. 48, 39–66 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ferriero, A.: Relaxation and regularity in the calculus of variations. J. Differ. Equat. 249, 2548–2560 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mariconda, C., Treu, G.: Lipschitz regularity of the minimizers of autonomous integral functionals with discontinuous non-convex integrands of slow growth. Calc. Var. Partial Differ. Equat. 29, 99–117 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Quincampoix, M., Zlateva, N.: On Lipschitz regularity of minimizers of a calculus of variations problem with non locally bounded Lagrangians. C. R. Math. Acad. Sci. Paris 343(1), 69–74 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Vinter, R.: Optimal Control. Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston (2000)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques de Bretagne AtlantiqueUniv Brest, UMR CNRS 6205F-BrestFrance
  2. 2.Dipartimento di Matematica “Tullio Levi-Civita”Università degli Studi di PadovaPadovaItaly

Personalised recommendations