Variational Problems for a Class of Functionals on Convex Domains

Let Ω be a bounded convex open subset of N, N 2, and let J be the integral functionalJ(u)ė∫Ω [f(Du(x))−u(x)] dx, where f: [0, +∞[→ {+∞} 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 W1, 10(Ω) functions that depend only on the distance from the boundary of Ω, provided that the ratio between the Lebesgue measure of Ω and the (N−1)-dimensional Hausdorff measure of ∂Ω is strictly less than a constant related to the growth of f at infinity.