Global integrability for minimizers of anisotropic functionals

We consider integral functionals in which the density has growth pi with respect to ∂u∂xi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{\partial u}{\partial x_i}}$$\end{document}, like in ∫Ω∂u∂x1(x)p1+∂u∂x2(x)p2+⋯+∂u∂xn(x)pndx.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int\limits_{\Omega}\left( \left| \frac{\partial u}{\partial x_1}(x) \right|^{p_1} + \left|\frac{\partial u}{\partial x_2}(x)\right|^{p_2} + \cdots + \left|\frac{\partial u}{\partial x_n}(x) \right|^{p_n} \right) dx.$$\end{document}We show that higher integrability of the boundary datum forces minimizer to be more integrable.