Local Lipschitz continuity of solutions to a problem in the calculus of variations (Q2464479)

The article deals with the problem of regularity of minimizers of the variational functional \[ I(u)= \int_{\Omega} F(Du) + G(x,u) \,dx \] over the function \(u \in W^{1,1}\) that assumme a given boundary values \(\phi\) on \(\partial\Omega\). It is well known that the minimizers are Lipschitz if the boundary data satisfies the so-called boundary slope condition. In the case \(G=0\) Clark introduced the lower (and also the upper) bounded slope condition, which is less restrictive. Under this condition and by means a new interesting methods the authors show the Lipschitz continuous regularity when \(G \neq 0\) is locally Lipschitz. Moreover the continuity of the solution at the boundary is discussed.