The validity of the Euler-Lagrange equation

Giovanni Bonfanti; Arrigo Cellina

doi:10.3934/dcds.2010.28.511

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2010, Volume 28, Issue 2: 511-517. Doi: 10.3934/dcds.2010.28.511 open access

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The validity of the Euler-Lagrange equation

Giovanni Bonfanti^1, and
Arrigo Cellina^1,

1.
Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via R. Cozzi 53, I-20125 Milano

Received: November 30, 2009

Revised: January 31, 2010

Published: June 2010

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Abstract

We prove the validity of the Euler-Lagrange equationfor a solution $u$ to the problem of minimizing $\int_{\Omega}L(x,u(x),\nabla u(x))dx$,where $L$ is a Carathéodory function, convex in its last variable,without assuming differentiability with respect to this variable.

Keywords:

Euler-Lagrange equation.

Mathematics Subject Classification: Primary: 49K20.

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