Accepted Paper

Inserted: 11 may 2023
Last Updated: 14 mar 2024

Journal: ESAIM: Control, Optimisation and Calculus of Variations
Year: 2024

Abstract:

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 $L^0(\Omega)$ of measurable functions endowed with the topology of convergence in measure $\tilde w\,dx$. Here $\tilde w$ is an auxiliary weight fitting the degenerations of the original weight $w$. Also the relaxation w.r.t. the $L^2(\Omega,\tilde w)$-convergence is studied. The crucial tool of the proof is a Poincaré type inequality, involving the weights $w$ and $\tilde w$, on the greatest finiteness domain $D_w$ of the relaxed functionals.

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