Relaxation and optimal finiteness domain for degenerate quadratic functionals - one dimensional case

The aim of this paper is the study, in the one-dimensional case, of the relaxation of a quadratic functional admitting a very degenerate weight $w$, which may not satisfy both the doubling condition and the classical Poincar\'e inequality. The main result deals with the relaxation on the greatest ambient space $L^0(\Omega)$ of measurable functions endowed with the topology of convergence in measure $\tildew\,dx$. Here $\tildew$ is an auxiliary weight fitting the degenerations of the original weight $w$. Also the relaxation w.r.t. the $L^2(\Omega,\tildew)$-convergence is studied. The crucial tool of the proof is a Poincar\'e type inequality, involving the weights $w$ and $\tildew$, on the greatest finiteness domain $D_w$ of the relaxed functionals.