AW-Convergence and Well-Posedness of Non Convex Functions

We consider the set of lower semicontinuous functions defined on a Banach space, equipped with AW-convergence. A function is called Tikhonov well-posed provided it has a unique minimizer to which every minimizing sequence converges. We show that well-posedness of f guarantees strong convergence of approximate minimizers of $\tau_{aw}$-approximating functions (under conditions of equiboundedness of sublevel sets), to the minimizer of f. Moreover we show that a lower semicontinuous function f which satisfies growth conditions at $\infty$ is well-posed iff its lower semicontinuous convex regularization is. Finally we investigate the link between AW-convergence of non convex integrands and that of the associated integral functionals.