On the Euler-Lagrange equation for a variational problem

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August 2007, 17(3): 449-480. doi: 10.3934/dcds.2007.17.449

On the Euler-Lagrange equation for a variational problem

1.	S.I.S.S.A. (I.S.A.S.), Via Beirut 2/4, 34013 Trieste

Received February 2006 Revised August 2006 Published December 2006

Abstract
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In this paper we prove the existence of a solution in L^∞_loc$(\Omega)$ to the Euler-Lagrange equation for the variational problem

inf_{$\bar u + W^{1,\infty}_0(\Omega)$}$ \int_{\Omega} (\I_D(\nabla u) + g(u)) dx,\ \ \ $ (0.1)

with $D$ convex closed subset of $\R^n$ with non empty interior. We next show that if D^* is strictly convex, then the Euler-Lagrange equation can be reduced to an ODE along characteristics, and we deduce that the solution to Euler-Lagrange is different from $0$ a.e. and satisfies a uniqueness property. Using these results, we prove a conjecture on the existence of variations on vector fields [6].

Keywords: Euler-Lagrange equation, Hamilton-Jacobi equation., Extended valued functions.

Mathematics Subject Classification: 49K2.

Citation: Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449

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