Relaxation of convex variational problems with linear growth defined on classes of vector-valued functions

Abstract

For a bounded Lipschitz domain \Omega\subset\mathbb{R}^{n} and a function u_{0}\in W_{1}^{1}(\Omega;\mathbb{R}^{N}) we consider the minimization problem (\mathcal{P}) \int_{\Omega}f(\nabla u)dx\rightarrow\mbox{min in}\: u_{0}+\overset{\text{\textdegree}}{W_{1}^{1}}(\Omega;\mathbb{R}^{N}) where f:\mathbb{R}^{nN}\rightarrow[0,\infty) is a strictly convex integrand. Let \mathcal{M} denote the set of all L^{1}-cluster points of minimizing sequences of problem (\mathcal{P}) coincides with the relaxation based on the notation of the extended Lagrangian, moreover, we prove that the elements u of \mathcal{M}are in one-to-one correspondence with the solutions of the relaxed problems.