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 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.

We consider the integral functional $\int f\left(x,Du\right)dx$ under non standard growth assumptions of $\left(p,q\right)$-type: namely, we assume that ${\left|z\right|}^{p\left(x\right)}\le f\left(x,z\right)\le L\left(1+{\left|z\right|}^{p\left(x\right)}\right)$, a relevant model case being the functional $\int {\left|Du\right|}^{p\left(x\right)}dx$. Under sharp assumptions on the continuous function $p\left(x\right)>1$ we prove regularity of minimizers both in the scalar and in the vectorial case, in which we allow for quasiconvex energy densities. Energies exhibiting this growth appear in several models from mathematical physics.

We prove a higher integrability result - similar to Gehring's lemma - for the metric space associated with a family of Lipschitz continuous vector fields by means of sub-unit curves. Applications are given to show the higher integrability of the gradient of minimizers of some noncoercive variational functionals.

Si prova la maggior sommabilità del gradiente dei minimi locali di funzionali integrali della forma $${\int }_{\mathrm{\Omega }}f\left(Du\right)dx,$$ dove $f$ soddisfa l'ipotesi di crescita $${\left|\xi \right|}^p-c_1\le f\left(\xi \right)\le c\left(1+{\left|\xi \right|}^q\right),$$ con $1<p<q\le 2$. L'integrando $f$ è $C^2$ e $DDf$ ha crescita $p-2$ dal basso e $q-2$ dall'alto.

We prove regularity results for real valued minimizers of the integral functional $\int f\left(x,u,Du\right)$ under non-standard growth conditions of $p\left(x\right)$-type, i.e. $$L^{-1}{\left|z\right|}^{p\left(x\right)}\le f\left(x,s,z\right)\le L\left(1+{\left|z\right|}^{p\left(x\right)}\right)$$ under sharp assumptions on the continuous function $p\left(x\right)>1$.

We give a different and probably more elementary proof of a good part of Jean Taylor’s regularity theorem for Almgren almost-minimal sets of dimension $2$ in ${ℝ}^3$. We use this opportunity to settle some details about almost-minimal sets, extend a part of Taylor’s result to almost-minimal sets of dimension $2$ in ${ℝ}^n$, and give the expected characterization of the closed sets $E$ of dimension $2$ in ${ℝ}^3$ that are minimal, in the sense that $H^2(E\setminus F)\le H^2(F\setminus E)$ for every closed set $F$ such that there is a bounded set $B$ so that $F=E$ out...

We prove that the higher integrability of the data $f,f_0$ improves on the integrability of minimizers $u$ of functionals $ℱ$, whose model is $${\int }_{\Omega }\left[{\left|Du\right|}^p+{(det\left(Du\right))}^2-\langle f,Du\rangle +\langle f_0,u\rangle \right]dx,$$ where $u:\Omega \subset {ℝ}^n\to {ℝ}^n$ and $p\ge 2$.