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

Article

## Abstract

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}
(P)
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.

## Keywords

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

## References

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)
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)
3. 3.
Bettiol, P., Mariconda, C.: A new variational inequality in the Calculus of Variations and Lipschitz regularity of minimizers. J. Differ. Equ. (2019).
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)
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)
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).
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)
10. 10.
Clarke, F.H.: Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, vol. 264. Springer, London (2013)
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)
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)
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)
14. 14.
Ferriero, A.: Relaxation and regularity in the calculus of variations. J. Differ. Equat. 249, 2548–2560 (2010)
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)
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)
17. 17.
Vinter, R.: Optimal Control. Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston (2000)Google Scholar