Relaxation results for higher order integrals below the natural growth exponent

Abstract

We consider higher-order variational integrals of the type $$ \mathcal{F}(u,\Omega )=\int_{\Omega }f(x,u,D^{[k]}u,D^{k+1}u)\ dx$$ and study relaxation and lower semicontinuity properties of such functionals. In particular, under a bound of the type $$ 0\leq f(x,u,z_{1},z_{2},\ldots,z_{k+1}) \leq L(1+| z_{k+1}|^{q}) $$ the following relaxed energies are studied: \begin{eqnarray*} \mathcal{F}^{q,p}(u,\Omega) & = & \inf_{\{u_n\}} \Big \{\liminf_n \int_{\Omega}f(x,u_n,D^{[k]}u_n ,D^{k+1}u_n)\ dx \ : \\ & & u_n\in W^{k+1 ,q}(\Omega ;\Bbb R^d),\ u_n \rightharpoonup u \mbox{ in }W^{k+1 ,p}(\Omega ;\Bbb R^d) \Big \} \\ \mathcal{F}^{q,p}_{{loc}}(u,\Omega) & = & \inf_{\{u_n\}} \Big \{\liminf_n \int_{\Omega}f(x,u_n,D^{[k]}u_n ,D^{k+1}u_n)\ dx \ : \\ & & u_n\in W^{k+1 ,q}_{{loc}}(\Omega ;\Bbb R^d),\ u_n \rightharpoonup u \mbox{ in }W^{k+1 ,p}(\Omega ;\Bbb R^d) \Big \} \end{eqnarray*} with $\frac{q}{p} < \frac{Nk}{Nk-1}.$