On the relaxation of nonconvex superficial integral functionals (Q1593843)

The paper is concerned with an integral representation result for the relaxed functional of the integral \[ F: u\in{\mathcal D}({\mathbf R}^N;{\mathbf R}^m)\mapsto\int_{{\mathbf R}^N}W(\nabla u(x))d\mu \] where \({\mathcal D}({\mathbf R}^N;{\mathbf R}^m)\) is the set of the smooth functions with compact support in \({\mathbf R}^N\) with values in \({\mathbf R}^m\), \(W:{\mathbf R}^{Nm}\to[0,+\infty[\) is continuous, and \(\mu\) is the \(k\)-dimensional Hausdorff measure restricted to a fixed \(k\)-dimensional smooth submanifold \(M\) of \({\mathbf R}^N\), with \(0<k<N\). Let \(M\) and \(\mu\) be as above, and define \({\mathcal D}_\mu({\mathbf R}^N;{\mathbf R}^m)\) as the space of the equivalence classes of the elements of \({\mathcal D}({\mathbf R}^N;{\mathbf R}^m)\) that agree \(\mu\)-almost everywhere. Let now \(p\geq 1\), and define \(L^p_\mu({\mathbf R}^N;{\mathbf R}^m)\) as the completion of \({\mathcal D}_\mu({\mathbf R}^N;{\mathbf R}^m)\) with respect to the norm \[ \|u\|_{p,\mu}=\left(\int_{{\mathbf R}^N}|u|^pd\mu\right)^{1/p}. \] Then the author associates to \(F\) the functional \[ F_\mu: u\in L^p_\mu({\mathbf R}^N;{\mathbf R}^m)\mapsto\inf\{F(v) : v\in{\mathcal D}({\mathbf R}^N;{\mathbf R}^m),\;\overline v=u\}, \] where \(\overline v\) is the equivalence class of \(v\), and introduces the \(\mu\)-relaxed functional of \(F\) as \[ {\mathcal F}_\mu: u\in L^p_\mu({\mathbf R}^N;{\mathbf R}^m)\mapsto\inf\left\{\liminf_{n\to\infty}F_\mu(u_n) : \{u_n\}\subseteq{\mathcal D}({\mathbf R}^N;{\mathbf R}^m),\;u_n\to u\text{ in }L^p_\mu({\mathbf R}^N;{\mathbf R}^m)\right\}. \] The main representation result of the paper concerns \({\mathcal F}_\mu\), and states that, if \[ r|z|^p\leq W(z)\leq R(1+|z|^p) \] with \(R>0\) and \(p>1\), and if a suitable compatibility condition is fulfilled, then \({\mathcal F}_\mu\) can be represented on Sobolev spaces by means of an integral formula with a precisely described integrand. Similar results are also obtained in the framework of BV spaces under additional assumptions.