ABSTRACT
The Lavrentiev phenomenon is associated with the sensitivity of the infimum of a variational problem to the smoothness of the class of admissible functions. Since the determination by Tonelli that the class of absolutely continuous functions is an appropriate class of admissible functions with which to obtain existence of minimizers to variational problems on a real interval by direct methods, it was shown that many of the classical problems yield to this approach. Moreover, in many of the classical problems the minimizers were actually Lipschitz or better, so that the problems were insensitive to the particular subclass of absolutely continuous admissible functions chosen for applying the direct method for existence. Therefore it was quite surprising when in 1926 M. Lavrentiev published, in response to a challenge issued by Tonelli during a lecture in Moscow, an example [L] of a functional of the form
,6 J[y] = I f(x,y(x),y'(x))dx with y-K b\
subject to certain constraints and smoothness conditions, in which the infimum of J over the class of all absolutely continuous functions subject to certain boundary conditions at a and b is strictly smaller than its infimum over the class of all Cl functions meeting the same boundary conditions. Thereafter in 1934 B. Mania published an example [Ma] involving a much simpler (polynomial) integrand. The presentation of an example involving a functional ,7 possessing a strictly elliptic integrand occurred only in 1985 [B&M] during the course of an investigation stimulated by the possible relevance of such questions to the theory of (multidimensional) hyperelastic materials.
