On the minimum problem for non-quasiconvex vectorial functionals

We consider functionals of the form \begin{equation*} \mathcal{F}(u) = \int_{\Omega} f(x, u(x), D u(x))\,dx, \quad u\in u_0 + W_0^{1,r}(\Omega,\mathbb{R}^m), \end{equation*} \noindent where the integrand $f:\Omega\times \mathbb{R}^m\times \mathbb{M}^{m\times n} \to \mathbb{R}$ is assumed to be non-quasiconvex in the last variable and $u_0$ is an arbitrary boundary value. We study the minimum problem by the introduction of the lower quasiconvex envelope $\overline{f}$ of $f$ and of the relaxed functional \begin{equation*} \overline{\mathcal{F}}(u) = \int_{\Omega} \overline{f}(x, u(x), D u(x))\,dx, \quad u\in u_0 + W_0^{1,r}(\Omega,\mathbb{R}^m), \end{equation*} imposing standard differentiability and growth properties on $\overline{f}$. Then we assume a suitable structural condition on $\overline{f}$ and a special regularity on the minimizers of $\overline{\mathcal{F}}(u)$, showing that $\mathcal{F}(u)$ attains its infimum. In addition we treat a class of functionals with separate dependence on the gradients of competing maps by the use of integro-extremality method.