Two dimensional variational problems with linear growth (Q1396497)

Consider the variational functional with the integrand \(f \in C^{2}(\mathbb{R}^{nN}) \) of linear growth : \[ J(u)= \int _{\Omega} f( Du(x)) dx. \] and the generalized minimizers in the class \(u\in u_{0}+ W_{0}^{1,1}\), defined as the functions \(u\in \text{BV}\) which is the \(L^{1}\)-limit of some \(J\)-minimizing sequences \((u_{k}) \in u_{0}+ W_{0}^{1,1}\). Under some structure conditions, and \(f(z)=g(|z|)\) in the vectorial case \(N>1\), the author generalizes some previous results showing that when \(n=2\), the restriction \(u_{0}\in L^\infty \) is superfluous to obtain that the generalized minimizers are in \(C^{1,\alpha}\) .