First- and Second-Order Integral Functionals of the Calculus of Variations which Exhibit the Lavrentiev Phenomenon

We study the possible mechanisms of occurrence of the Lavrentiev phenomenon for the basic problem of the calculus of variations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{J}(x) = \int\limits_0^1 {\mathcal{L}(t,x(t),\dot x(t))dt \to \inf ,x(0) = x_0 ,x(1) = x1} $$ \end{document} when the infimum of the problem in the class of absolutely continuous functions W1,1[0, 1] is strictly less than the infimum of the same problem in the class of Lipshitzian functions W1,∞[0,1]. We suggest an approach to constructing new classes of integrands which exhibit the Lavrentiev phenomenon (Theorem 2.1).