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<+\infty. We adopt the approach developed in the paper V. De Cicco, F. Serra Cassano: Relaxation and optimal finiteness domain for degenerate quadratic functionals - one dimensional case, ESAIM: Control, Optim. Calc. Var. 30 (2024), no. 31. 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.