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

Abstract

We give necessary and sufficient conditions for minimality of generalized minimizers for linear-growth functionals of the form \[ \mathcal F[u] := \int_Ωf(x,u(x)) \, \text{d}x, \qquad u:Ω\subset \mathbb R^N\to \mathbb R^d, \] where $u$ 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 $\mathcal{M}(\Omega ;{\mathbb{R}}^d)$, and the introduction of a set-valued pairing in $\mathcal{M}(\Omega ;{\mathbb{R}}^N)\times {\mathrm{L}}^{\infty }(\Omega ;{\mathbb{R}}^N)$. 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.

arXiv comment25 pages, several proofs have been shortened