Regularity of Lipschitz minima (Q2568709)

In the paper, applying directly the De Giorgi's techniques and ideas on perimeters, the following known result is proved: If \(F:R^n\longrightarrow R \) is a strictly convex function of class \(C^2\) and \(w \in \text{Lip}^{(n)}\) satisfies \(\int_{\Omega} F(Dw+D\psi)dx \geq \int_{\Omega} F(Dw)dx \) for all \(\psi \in\text{Lip}_c (\Omega),\) then \(w\) has Hölder-continuous first derivatives in \(\Omega.\) The main steps of the proof are the so-called Caccioppoli inequalities and Lemma 5 where a crucial inequality for the function \(u(x,t,\epsilon)= {\frac{w(x+t\epsilon)-w(x)}{t}}\) is established.