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.


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