In this thesis we study the variational problem min u∈A ∫Ω F(Du)dx, where Ω ⊂ Rn is a bounded domain, A the set of functions in W1,2(Ω) with given boundary values, and F a smooth and strongly convex function. The aim is to show rigorously and in great detail that if we have a Lipschitz continuous minimizer, then it is, in fact, smooth. In order to prove the continuity of the first derivatives we use De Giorgi’s method, and for the higher derivatives the classical Schauder theory is applied. The question whether variational minimizers are smooth is a slightly weaker version of Hilbert’s 19th problem.
