The paper provides a wide survey on the Lavrentiev phenomenon, i.e., the surprising features of some functionals of the calculus of variations to possess different infima if considered on the full class of admissible functions and on the smaller class of regular admissible functions. The phenomenon is strictly linked to partial regularity results for solutions of variational problems and can also be interpreted in terms of relaxation.
Such interpretation can be described by considering two topological spaces \(X\) and \(Y\), with \(Y\) dense in \(X\), a functional \(F: X\to ]- \infty, +\infty]\) lower semicontinuous in the topology of \(X\) and the relaxed functional \(\overline{F_{|Y}}\) of its restriction to \(Y\) defined by \(\overline{F_{|Y}}= \max\{G: X\to ]- \infty, +\infty]: G\) \(X\)-lower semicontinuous, \(G\leq F\) on \(Y\}\). Since \(F\leq \overline {F_{|Y}}\) on \(X\), it results \(\overline{F_{|Y}}= F+ L\), \(L\) being a nonnegative functional called Lavrentiev gap associated to \(X\) and \(Y\).
In the paper, the results existing in literature on \(L\) are reported starting from those in the one-dimensional case for integral functionals depending on functions possessing one derivative in which representation formulas are available and explicit computations in examples where \(L\) is not identically zero can be performed. The case of functionals depending on higher order derivatives is also reported, again representation formulas for \(L\) are described together with the study of the autonomous case. Sufficient conditions implying \(L\) to be identically equal to zero are also proposed.
In the \(n\)-dimensional case significant examples in which \(L\) is not identically zero are described, open problem are proposed and again sufficient conditions yielding \(L\) to be identically zero are reported.
A wide bibliography on the Lavrentiev phenomenon concludes the paper.
For the entire collection see [Zbl 0823.00006].