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

Abstract

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

I(u)=\int_{\Omega}F(x, u(x), \nabla u(x))\,dx

where

\Omega

is a bounded open subset of

\mathbb{R}^n

and

F\colon \Omega\times\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}

is a given Carathéodory function; the admissible functions

u

coincide with a given Lipschitz function on

\partial\Omega

. We formulate some conditions under which a given function in

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

with

I(u)<+\infty

can be approximated by a sequence of functions

u_k\in\phi+W^{1,p}_0(\Omega)\cap L^{\infty}

converging to

u

in the norm of

W^{1,p}

, and such that

I(u_k)\rightarrow I(u)

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

Contact details are reproduced from the original publication and may be historical.

Dipartimento di Matematica, Università di Padova, 35121 Padova, Italy

Dipartimento di Matematica, Università di Padova, 35121 Padova, Italy

49N99; 49N60

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press.