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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 |
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].
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