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 Poincaré inequality. The main result deals with the relaxation on the greatest ambient space L0(Ω) of measurable functions endowed with the topology of convergence in measure w dx. Here w is an auxiliary weight fitting the degenerations of the original weight w. Also the relaxation w.r.t. the L2(Ω, w˜)-convergence is studied. The crucial tool of the proof is a Poincaré type inequality, involving the weights w and w, on the greatest finiteness domain Dw of the relaxed functionals.