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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 prove some optimal regularity results for minimizers of the integral functional $\int f(x,u,Du)\mathrm{d}x$ belonging to the class $K:=\{u\in W^{1,p}\left(\Omega \right)u\ge \psi \}$, where $\psi$ is a fixed function, under standard growth conditions of $p$-type, i.e. $$L^{-1}{\left|z\right|}^p\le f(x,s,z)\le {L(1+|z|}^p).$$