The Lavrentiev phenomenon for invariant variational problems (Q1109324)

With the basic problem of the calculus of variations in mind, consider the function V: [1,\(\infty [\to {\mathbb{R}}\) defined by \[ V(p):=\inf \{\int^{1}_{0}L(t,x(t),\dot x(t))dt:\quad x\in AC[0,1],\quad \dot x\in L\quad p[0,1],\quad x(0)=\alpha,\quad x(1)=\beta \}. \] This function highlights the dependence of the problem on the class of arcs x(\(\cdot)\) over which minimization is performed. The authors present a detailed study of a class of smooth variational integrands L for which one has \(V(p)>V(1)\) for some \(p>1\) (``the Lavrentiev phenomenon''), and both infima defining V(1) and V(p) are actually attained. These integrands are essentially characterized by a certain homogeneity property. Noether's theorem and field theory play a crucial role in the proofs, which are also extended to cover certain free-endpoint problems. The robustness of the Lavrentiev phenomenon is then studied as the integrand L is subjected to additive and multiplicative perturbations.