On Lavrentiev’s phenomenon. (English) Zbl 0910.49020

The paper is a survey article, containing also many new results, on Lavrentiev’s phenomenon.
After a brief historical introduction, the author fixes his attention on some classes of integral functionals and discusses the presence of the Lavrentiev’s phenomenon for some associated variational problems. Sufficient conditions for the non-occurrence of Lavrentiev’s phenomenon are given for all the classes of functionals taken into account. Then some interesting examples are proposed that the above conditions are sharp, and thus describing cases of variational problems for the integral functionals of the type \[ \int_\Omega| Du|^{p_1} dx+ \int_\Omega a(x)| Du|^{p_2} dx\quad\text{with} \quad p_1<p_2,\quad\text{and} \quad \int_\Omega| Du|^{a(x)}dx, \] for which the Lavrentiev’s phenomenon occurs.
The validity of the Euler equation is also discussed in connection with Lavrentiev’s phenomenon, and finally the interpretation of Lavrentiev’s phenomenon in terms of relaxation is also discussed. Some sufficient conditions and counterexamples on the identity between an integral functional of the type \(\int_\Omega f(x,Du)dx\), for \(u\) smooth, and its relaxed functional are discussed.


49N60 Regularity of solutions in optimal control
49J45 Methods involving semicontinuity and convergence; relaxation
49J10 Existence theories for free problems in two or more independent variables