We consider the classical functional of the Calculus of Variations of the form I(u) = ZΩ F(x, u(x), ∇u(x)) dx where Ω is a bounded open subset of Rn and F : Ω×R×Rn → R is a given Carathéodory function; the admissible functions u coincide with a given Lipschitz function on ∂Ω. We formulate some conditions under which a given function in ϕ + W01,p(Ω) with I(u) < +∞ can be approximated by a sequence of functions uk ∈ ϕ+W01,p(Ω)∩L∞ converging to u in the norm of W1,p, and such that I(uk) → I(u). The problem is strictly related with the non occurrence of the Lavrentiev gap.

Non-Occurrence of a Gap Between Bounded and Sobolev Functions for a Class of Nonconvex Lagrangians

Mariconda, Carlo
;
Treu, Giulia
2020

Abstract

We consider the classical functional of the Calculus of Variations of the form I(u) = ZΩ F(x, u(x), ∇u(x)) dx where Ω is a bounded open subset of Rn and F : Ω×R×Rn → R is a given Carathéodory function; the admissible functions u coincide with a given Lipschitz function on ∂Ω. We formulate some conditions under which a given function in ϕ + W01,p(Ω) with I(u) < +∞ can be approximated by a sequence of functions uk ∈ ϕ+W01,p(Ω)∩L∞ converging to u in the norm of W1,p, and such that I(uk) → I(u). The problem is strictly related with the non occurrence of the Lavrentiev gap.
2020
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/3324230
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