Local Boundedness for Vector Valued Minimizers of Anisotropic Functionals

For variational integrals F(u) = f(Omega) f (x, Du) dx defined on vector valued mappings u : Omega subset of R-N -> R-N, we establish some structure conditions on f that enable us to prove local boundedness for minimizers u is an element of W-1,W-1(Omega; R-N) of F. These structure conditions are satisfied in three remarkable examples: f (x, Du) = g(x, vertical bar Du vertical bar), f (x, Du) = Sigma(n)(j=1) gj(x,vertical bar u(xj)vertical bar) and f (x, Du) = a(x,vertical bar u(x1), ... , u(xn-1))vertical bar) + b(x, vertical bar ux(n)), for suitable convex functions t -> g(x,t), t -> g(j)(x, t), t -> a(x, t) and t -> b(x, t).