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...