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

  • Piernicola Bettiol
  • Carlo MaricondaEmail author
Article
  • 13 Downloads

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 

Notes

Acknowledgements

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.

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© 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

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