# American Institute of Mathematical Sciences

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

In this paper we prove the existence of a solution in Lloc$(\Omega)$ 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].

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