  We study the minimizer u of a convex functional in the plane which is not
G\^ateaux-differentiable. Namely, we show that the set of critical points of
any C^1-smooth minimizer can not have isolated points. Also, by means of some
appropriate approximating scheme and viscosity solutions, we determine an
Euler-Lagrange equation that u must satisfy. By applying the same approximating
scheme, we can pair u with a function v which may be regarded as the stream
function of u in a suitable generalized sense.
