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. Building upon recent findings from [9, 12], our investigation focuses on establishing an explicit relaxation formula for a functional exhibiting p-growth for 1≤ p<+∞. For the case 1<p<∞, we adopt the approach developed in [12], where some assumptions like doubling or Muckenhoupt conditions are dropped. Moreover, for $p = 1$, we leverage novel concepts introduced in [9]. In both cases, our main tools consist of proving the validity of a weighted Poincar\'e inequality involving an auxiliary weight.