Preprint

Inserted: 27 mar 2024
Last Updated: 27 mar 2024

Year: 2024

Abstract:

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 (Comi, De Cicco, Scilla), (De Cicco, Serra Cassano) our investigation focuses on establishing an explicit relaxation formula for a functional exhibiting p-growth for $1\leq p<+\infty$. For the case $1<p<+\infty$, we adopt the approach developed in (De Cicco, Serra Cassano), where some assumptions like doubling or Muckenhoupt conditions are dropped. Moreover, for p=1, we leverage novel concepts introduced in (Comi, De Cicco, Scilla). In both cases, our main tools consist of proving the validity of a weighted Poincaré inequality involving an auxiliary weight.

Keywords: relaxation, degenerate variational integrals, Lower Semicontinuity, Poincaré inequalities, weight