On the validity of the Euler-Lagrange system

Menita Carozza; Jan Kristensen; Antonia Passarelli di Napoli

doi:10.3934/cpaa.2015.14.51

Article Contents

2015, Volume 14, Issue 1: 51-62. Doi: 10.3934/cpaa.2015.14.51

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On the validity of the Euler-Lagrange system

1.
Università del Sannio, Piazza Arechi II - 82100 Benevento
2.
Mathematical Institute, 24–29 St Giles’, University of Oxford, OX1 3LB Oxford
3.
Dipartimento di Matematica e Appl. “R. Caccioppoli”, Via Cintia- Monte S.Angelo, 80126 Napoli

Received: February 2014

Revised: March 2014

Published: January 2015

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Abstract

The minimizers of convex integral functionals of the form \begin{eqnarray} \mathfrak{F} (v, \Omega) = \int_{\Omega} F (Dv (x)) dx, \end{eqnarray} defined on Sobolev mappings $v$ in $W^{1,1}_{g}(\Omega R^N)$ are characterized as the energy solutions to the Euler--Lagrange system for $\mathfrak{F}$. We assume that the integrands $F: R^{N\times n} \to R$ are $C^1$, convex and super--linear at infinity, and the boundary datum $g \in W^{1,1}(\Omega, R^N)$ must satisfy $F(sDg) \in L^1(\Omega )$ for some number $s>1$.

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Mathematics Subject Classification: Primary: 49N15, 49N60; Secondary: 49N99.

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