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

TL;DR

This paper establishes necessary and sufficient conditions for the minimality of generalized solutions to linear-growth convex integral functionals under PDE constraints, using relaxation and duality methods.

Contribution

It introduces a novel relaxation approach into measure spaces and a set-valued pairing to characterize minimizers and dual maximizers in PDE-constrained variational problems.

Findings

Characterizes minimality conditions for generalized minimizers.

Establishes a duality relation between relaxed minimizers and maximizers.

Applies results to relaxation problems in BV and BD spaces.

Abstract

We give necessary and sufficient conditions for minimality of generalized minimizers for linear-growth functionals of the form \[ \mathcal F[u] := \int_\Omega f(x,u(x)) \, \text{d}x, \qquad u:\Omega \subset \mathbb R^N\to \mathbb R^d, \] 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 $\mathcal M(\Omega;\mathbb R^N) \times {\rm L}^\infty(\Omega;\mathbb R^N)$. 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.