Integrability for solutions to some anisotropic obstacle problems

This paper deals with Kψ,θ(pi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{K}}_{\psi, \theta}^{(p_i)}}$$\end{document}-obstacle problems of some anisotropic elliptic equations of the type ∑i=1nDi(ai(x,Du(x)))=∑i=1nDifi(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum_{i=1}^{n} D_i (a_i(x,Du(x)))=\sum_{i=1}^{n} D_i f^i(x)$$\end{document}under some suitable coercivity and controllable growth conditions on the vector a(x,z)=(a1(x,z),a2(x,z),…,an(x,z))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a(x,z)=(a_1(x,z),a_2(x,z), \ldots, a_n(x,z))}$$\end{document}.Assumptions on ai(x, z) are suggested by the Euler equation of the anisotropic functional ∫Ωh+∑j=1n|Dju|pjp1-2p1|D1u|2+⋯+h+∑j=1n|Dju|pjpn-2pn|Dnu|2dx.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int_{\Omega} \left(\left(h+ \sum_{j=1}^n |D_ju|^{p_j}\right)^{\frac{p_1-2}{p_1}}|D_1u|^2+ \cdots + \left(h+\sum_{j=1}^n |D_ju|^{p_j}\right)^{\frac {p_n-2}{p_n}}|D_nu|^2 \right) dx.$$\end{document}We show that, higher integrability of the datum θ∗=max{ψ,θ}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta_*=\max\{\psi, \theta\}}$$\end{document} forces solutions u to have higher integrability as well.