Quasiconvexification of geometric integrals (Q2505604)

Given the integral functional \(F(u)=\int_{\Omega} f(\nabla u(x))\,d\mu (x)\) where \(\Omega\) is an open bounded set in \(\mathbb R^N\), \(f:M^{m\times N}\to [0, +\infty)\) is a continuous (not necessarily convex) function and \(\mu\) is a positive Radon measure on \(\mathbb R^N\), the author finds an integral representation formula for the relaxed functional: \[ \inf\left\{\liminf_n \int_{\Omega} f(\nabla u_n)d\mu \mid u_n \in C^{\infty} (\overline{\Omega}; R^m), u_n \to_{L^p_{\mu}} u\right\}. \] Under suitable condition on the so-called \`\` low-dimensional smooth points\'\'\ of spt\(\mu\), the author proves the integral representation formula where the integrand is given by the \(\mu_x\)-quasicon\-vexification of \(f\) introduced in the paper. The equivalence with other related results on the \(\mu_x\)-quasiconvexification is also shown.