We prove Lipschitz continuity for local minimizers of integral functionals of the Calculus of Variations in the vectorial case, where the energy density depends explicitly on the space variables and has general growth with respect to the gradient. One of the models is$$F\left(u\right)={\int }_{\Omega }a\left(x\right){\left[h\left(\left|Du\right|\right)\right]}^{p\left(x\right)}\mathrm{d}x$$with $h$ a convex function with general growth (also exponential behaviour is allowed).

We prove Lipschitz continuity for local minimizers of integral functionals of the Calculus of Variations in the vectorial case, where the energy density depends explicitly on the space variables and has general growth with respect to the gradient. One of the models is $$F\left(u\right)={\int }_{\Omega }a\left(x\right){\left[h\left(\left|Du\right|\right)\right]}^{p\left(x\right)}\mathrm{d}x$$ with h a convex function with general growth (also exponential behaviour is allowed).

The criteria of extremality for classical variational integrals depending on several functions of one independent variable and their derivatives of arbitrary orders for constrained, isoperimetrical, degenerate, degenerate constrained, and so on, cases are investigated by means of adapted Poincare-Cartan forms. Without ambitions on a noble generalizing theory, the main part of the article consists of simple illustrative examples within a somewhat naive point of view in order to obtain results resembling...

Continuing the previous Part I, the degenerate first order variational integrals depending on two functions of one independent variable are investigated.

We show that local minimizers of functionals of the form ${\int }_{\Omega }\left[f\left(Du\right(x\left)\right)+g(x,u(x\left)\right)\right]\mathrm{d}x$,  $u\in u_0+W_0^{1,p}\left(\Omega \right)$, are locally Lipschitz continuous provided f is a convex function with $p-q$ growth satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous in u. As a consequence of this, we obtain an existence result for a related nonconvex functional.