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

AbstractWe study the possible mechanisms of occurrence of the Lavrentiev phenomenon for the basic problem of the calculus of variations $$\mathcal{J}(x) = \int\limits_0^1 {\mathcal{L}(t,x(t),\dot x(t))dt \to \inf ,x(0) = x_0 ,x(1) = x1} $$ 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).A similar method is used to construct (Theorem 3.1) a class of autonomous C1-differentiable integrands $$\mathcal{L}(x,\dot x,\ddot x)$$ of the calculus of variations which are regular, i.e., convex, coercive w.r.t. $$\ddot x$$ , and exhibit the W2,1 – W2,∞ Lavrentiev gap, i.e., for some choice of boundary conditions of the variational problem $$\mathcal{J}(x( \cdot )) = \int\limits_0^1 {L(x(t),\dot x(t),\ddot x(t))dt \to \inf } $$ , the infimum of this problem over the space W2,1[0, 1] is strictly less than its infimum over the space W2,∞[0, 1]. This provides a negative answer to the question of whether functionals with regular autonomous second-order integrands should only have minimizers with essentially bounded second derivative.