Interpretation of Lavrentiev phenomenon by relaxation: the higher order case

Abstract:

We consider integral functionals of the calculus of variations of the form \[ F(u) = \int \limits _0^1 {f(x,u,u’, \ldots ,{u^{(n)}})dx} \] defined for $u \in {W^{n,\infty }}(0,1)$, and we show that the relaxed functional $F$ with respect to the weak $W_{{\text {loc}}}^{n,1}(0,1)$ convergence can be written as \[ \overline F (u) = \int \limits _0^1 {f(x,u,u’, \ldots ,{u^{(n)}})dx + L(u),} \] where the additional term $L(u)$, the Lavrentiev Gap, is explicitly identified in terms of $F$.