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One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. (English) Zbl 0585.49002

The present work is concerned with non smooth minimizers of \(I(u)=\int^{b}_{a}f(x,u(x),u'(x))dx\) in the set \({\mathcal A}\) of absolutely continuous functions u:[a,b]\(\to {\mathbb{R}}\) satisfying the end conditions \(u(a)=\alpha\) and \(u(b)=\beta\). The integrand f(x,u,p) is assumed to be smooth, nonnegative and to satisfy the regularity condition \(f_{pp}>0.\)
Firstly, an excellent review concerning results on regularity of minimizers and various forms of first order necessary conditions is given. Then a number of highly interesting examples is presented and carefully analyzed, where the minimizers \(u\in {\mathcal A}\) are not smooth and do not satisfy the Euler-Lagrange equation in integrated form. In particular, the examples are concerned with ”singular minimizers” as introduced by Tonelli, and with a phenomenon known as Lavrentiev gap respectively.
Reviewer: W.Velte

MSC:

49J05 Existence theories for free problems in one independent variable
49J15 Existence theories for optimal control problems involving ordinary differential equations
49K05 Optimality conditions for free problems in one independent variable
49K15 Optimality conditions for problems involving ordinary differential equations
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