Convex functionals and the stratification of the singular set of their stationary points

We prove partial regularity of stationary solutions and minimizers u from a set Ω⊂Rn to a Riemannian manifold N, for the functional ∫ΩF(x,u,|∇u|2)dx. The integrand F is convex and satisfies some ellipticity and boundedness assumptions. We also develop a new monotonicity formula and an ϵ-regularity theorem for such stationary solutions with no restriction on their images. We then use the idea of quantitative stratification to show that the k-th strata of the singular set of such solutions are k-rectifiable.