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

We consider the classical functional of the Calculus of Variations of the form

where

\Omega Ω

is a bounded open subset of

\mathbb{R}^n R n

and

F\colon \Omega\times\mathbb{R}\times\mathbb{R}^n\to\mathbb{R} F  ⁣ : Ω × R × R n → R

is a given Carathéodory function; the admissible functions

u u

coincide with a given Lipschitz function on

\partial\Omega ∂ Ω

. We formulate some conditions under which a given function in

\phi+W^{1,p}_0(\Omega) ϕ + W 0 1 , p ( Ω )

with

I(u)<+\infty I ( u ) < + ∞

can be approximated by a sequence of functions

u_k\in\phi+W^{1,p}_0(\Omega)\cap L^{\infty} u k ∈ ϕ + W 0 1 , p ( Ω ) ∩ L ∞

converging to

u u

in the norm of

W^{1,p} W 1 , p

, and such that

I(u_k)\rightarrow I(u) I ( u k ) → I ( u )

. The problem is strictly related with the non occurrence of the Lavrentiev gap.