One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation (Q1071247)

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.