Relaxation for degenerate nonlinear functionals in the onedimensional case

In this study, we approach the analysis of a degenerate nonlinear functional in one dimension, accommodating a degenerate weight $w$. Our investigation focuses on establishing an explicit relaxation formula for a functional exhibiting $p$-growth for $1<p<+\mathrm{\infty }$. We adopt the approach developed in [6], where some assumptions like doubling or Muckenhoupt conditions are dropped. Our main tools consist of proving the validity of a weighted Poincaré inequality involving an auxiliary weight.