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Journal of Convex Analysis 27 (2020), No. 4, 1247--1259
Copyright Heldermann Verlag 2020



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

Carlo Mariconda
Dipartimento di Matematica, UniversitÓ di Padova, 35121 Padova, Italy
carlo.mariconda@unipd.it

Giulia Treu
Dipartimento di Matematica, UniversitÓ di Padova, 35121 Padova, Italy
giulia.treu@unipd.it



[Abstract-pdf]

\def\R{\mathbb{R}} 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 $\R^n$ and $F\colon \Omega\times\R\times\R^n\to\R$ is a given Carath\'eodory 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.

Keywords: Lavrentiev, Lavrentieff, approximation, bounded functions, regularity.

MSC: 49N99; 49N60.

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