On the Euler-lagrange Equation for a Variational Problem: the General Case
نویسنده
چکیده
In this paper we study the existence of a solution in Lloc(Ω) to the Euler-Lagrange equation for the variational problem (0.1) inf ū+W 1,∞ 0 (Ω) Z Ω (1ID(∇u) + g(u))dx, with D convex closed subset of Rn with non empty interior. By means of a disintegration theorem, we next show that the Euler-Lagrange equation can be reduced to an ODE along characteristics, and we deduce that there exists a solution to Euler-Lagrange is different from 0 a.e. and satisfies a uniqueness property. These results prove a conjecture on the existence of variations on vector fields stated in [3].
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