Relaxation and optimal finiteness domain for degenerate quadratic functionals. One-dimensional case

¹ Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Univ. di Roma Via A. Scarpa 10, I-00185 Roma, Italy
² Dipartimento di Matematica, Via Sommarive 14, 38123 Povo, Trento, Italy

^* Corresponding author: virginia.decicco@uniroma1.it

Received: 13 May 2023
Accepted: 14 March 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 L⁰(Ω) of measurable functions endowed with the topology of convergence in measure w dx. Here w is an auxiliary weight fitting the degenerations of the original weight w. Also the relaxation w.r.t. the L²(Ω, w˜)-convergence is studied. The crucial tool of the proof is a Poincaré type inequality, involving the weights w and w, on the greatest finiteness domain D_w of the relaxed functionals.