Absence of Lavrentiev's gap for anisotropic functionals (Q6591667)

The authors prove the absence of the Lavrentiev phenomenon for some functionals \N\[\N\mathcal{F} (u) = \int_{\Omega} F (x, u, \nabla u) \, dx,\N\]\Nwhere \(\Omega\) is a subset of \(\mathbf{R}^n\) and \(F: \Omega \times\mathbf{R}\times\mathbf{R}^n\) a measurable function, continuous with respect to the second and third variable satisfying\N\begin{gather*}\N\nu M (x , \beta \xi) \leqslant F (x, z, \xi ) \leqslant L \big( M (x , \beta \xi) + g (x) \big) , \\\N\nu, \beta \in (0,1), \ L > 1, \ g \geqslant 0, \ g \in L^1 (\Omega) \, .\N\end{gather*}\NHere, \(M\) is an anisotropic \(N\)-function. The authors prove that if there is \(u_o\) such that \(\mathcal{F} (u_o) < +\infty\) than the infimum of \(\mathcal{F}\) in \(u_o + C^{\infty}_ c (\Omega)\) coincides with the infimum of \(\mathcal{F}\) in \(u_o + X\), where \(X\) is the Sobolev type version of a certain Musielak-Orlicz space.