Growth conditions and regularity, an optimal local boundedness result

@article{Hirsch2019GrowthCA,
  title={Growth conditions and regularity, an optimal local boundedness result},
  author={Jonas Hirsch and Mathias Schaffner},
  journal={arXiv: Analysis of PDEs},
  year={2019}
}
We prove local boundedness of local minimizers of scalar integral functionals $\int_\Omega f(x,\nabla u(x))\,dx$, $\Omega\subset\mathbb R^n$ where the integrand satisfies $(p,q)$-growth of the form \begin{equation*} |z|^p\lesssim f(x,z)\lesssim |z|^q+1 \end{equation*} under the optimal relation $\frac1p-\frac1q\leq \frac1{n-1}$. 
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