Relaxation and optimization for linear-growth convex integral functionals under PDE constraints

We give necessary and sufficient conditions for the minimality of generalized minimizers of linear-growth integral functionals of the form F[u]=∫Ωf(x,u(x))dx,u:Ω⊂Rd→RN, where f:Ω×RN→R is a convex integrand and u is an integrable function satisfying a general PDE constraint. Our analysis is based on two ideas: a relaxation argument into a subspace of the space of bounded vector-valued Radon measures M(Ω;RN), and the introduction of a set-valued pairing on M(Ω;RN)×L∞(Ω;RN). By these means we are able to show an intrinsic relation between minimizers of the relaxed problem and maximizers of its dual formulation also known as the saddle-point conditions. In particular, our results can be applied to relaxation and minimization problems in BV, BD and divergence-free spaces.