Variational problems for a class of functionals on convex domains

Let Omega be a bounded convex open subset of R-N, N greater than or equal to 2, and let J be the integral functional J(u) = integral(Omega)[f(\Du(x)\) - u(x)]dx, where f:[0, +infinity[ --> R boolean OR {+infinity} is a lower semicontinuous function (possibly nonconvex and with linear growth). We prove that the functional J admits a unique minimizer in the space of W-0(1,1)(Omega) functions that depend only on the distance from the boundary of Omega, provided that the ratio between the Lebesgue measure of Omega and the (N -1)-dimensional Hausdorff measure of partial derivativeOmega is strictly less than a constant related to the growth of f at infinity