Existence of minimizers for non-quasiconvex functionals by strict monotonicity

We consider functionals of the form $ \begin{equation*} \mathcal{F}(u) = \displaystyle{\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*} $ where the integrand $ f = f(x, p, \xi): \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) = \displaystyle{\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 the quasiaffinity of $ \overline{f} $ on the set in which $ f> \overline{f} $ and the strict monotonicity of the map $ \mathbb{R}\ni p^i \mapsto \overline{f}(x, p, \xi) $, where $ p^i $ is a single scalar component of the vector function variable $ p $, showing that any minimizer of $ \overline{\mathcal{F}} $ minimizes $ \mathcal{F} $ too.