## Relaxation results for higher order integrals below the natural growth exponent.(English)Zbl 1030.49013

Let $$k\in{\mathbb N}$$, $$q>1$$, and let $$f\colon{\mathbb R}^N\times{\mathbb R}^d\times{\mathbb R}^{Nd}\times\dots\times{\mathbb R}^{dN^{k+1}}\to[0,+\infty[$$ be a Carathéodory function satisfying $0\leq f(x,u,z_1,z_2,\dots,z_{k+1})\leq L\bigg(1+\sum_{h=1}^k|z_h|^{r_h}+|z_{k+1}|^q\bigg).$ For $$1<p<q$$, every $$u\in W^{k+1,q}(\Omega;{\mathbb R}^d)$$, and every bounded open subset $$\Omega$$ of $${\mathbb R}^N$$, in the paper the following relaxed functionals $\begin{split} {\mathcal F}^{q,p}(u,\Omega)= \inf\bigg\{\liminf_n\int_\Omega f(x,u_n,Du_n,D^2u_n,\dots,D^{k+1}u_n)dx: \\ \{u_n\}\subseteq W^{k+1,q}(\Omega;{\mathbb R}^d),\;u_n\rightharpoonup u\text{ in }W^{k+1,q}(\Omega;{\mathbb R}^d)\bigg\},\end{split}$ and $\begin{split} {\mathcal F}^{q,p}_{\text{ loc}}(u,\Omega)=\inf\bigg\{\liminf_n\int_\Omega f(x,u_n,Du_n,D^2u_n,\dots,D^{k+1}u_n)dx :\\ \{u_n\}\subseteq W^{k+1,q}_{\text{ loc}}(\Omega;{\mathbb R}^d),\;u_n\rightharpoonup u\text{ in }W^{k+1,q}(\Omega;{\mathbb R}^d)\bigg\}\end{split}$ are studied.
Starting from the results of I. Fonseca and J. Malý [Ann. Inst. Henri Poincaré, Analyse Non Linéaire 14, 309-338 (1997; Zbl 0868.49011)] who treated the case $$k=0$$ by showing that the functionals $${\mathcal F}^{q,p}(u,\cdot)$$ and $${\mathcal F}^{q,p}_{\text{ loc}}(u,\cdot)$$ enjoy suitable measure theoretic properties, the authors prove the same properties when $$k>1$$ provided $$q/p<Nk/(Nk-1)$$, and similar restrictions for the lower-order exponents $$r_h$$ hold.
A sequential lower semicontinuity result in the weak-$$W^{k+1,q}(\Omega;{\mathbb R}^d)$$ topology is also proved for the functional ${\mathcal F}(u,\Omega)=\int_\Omega f(D^{k+1}u)dx,$ where $$f\colon{\mathbb R}^{dN^{k+1}}\to[0,+\infty[$$ is quasiconvex and satisfies $0\leq f(z_{k+1})\leq L(1+|z_{k+1}|^q),$ and $$q/p<Nk/(Nk-1)$$.
The central technical point for the achievement of the above results is, again following Fonseca and Malý, the construction of a linear continuous operator from $$W^{k+1,q}(\Omega;{\mathbb R}^d)$$ to $$W^{k+1,q}(\Omega;{\mathbb R}^d)$$ improving the integrability of functions on certain thin layers.

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation 49J10 Existence theories for free problems in two or more independent variables 35J20 Variational methods for second-order elliptic equations 35G20 Nonlinear higher-order PDEs

Zbl 0868.49011