Relaxation and Minimization of Convex Integral Energies with Linear Growth under a General PDE Constraint
Adolfo Arroyo-Rabasa · arXiv (Cornell University) · 2016
We deal with lower semi-continuity and minimization properties of functionals of the form \[ F(\tau) = \int_\Omega f(\tau) \, dx, \qquad \tau:\Omega \subset \mathbb R^N \to \mathbb R^d \] under the PDE constraint \[ \mathcal A \tau = 0, \] where $\Omega \subset \mathbb R^N$ is an open, bounded, and strictly star shaped domain, $\mathcal A : D(A) \subset L^1(\Omega)^d \to L^1(\Omega)^m$ is a linear differential operator, and $f : \mathbb R^d \to \mathbb R$ is a convex integrand with linear growth at infinity. We provide an explicit formula for the relaxed energy of $F$ in a space of Radon measures for which we do not assume Murat's constant rank condition on $\mathcal A$. We also study the dual energy and give a point-wise characterization for generalized minimizers of $\bar F$. Our technique does not rely on any special topological assumption on $\Omega$ or a any constant rank condition on $\mathcal A$ that may facilitate the existence of recovery sequences.