On Harnack inequality and Hölder continuity for non uniformly elliptic equations

We study Harnack inequality and a priori H & ouml;lder estimates for weak solutions to a new class of equations 1 partial derivative ziaij(z)partial derivative zju=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _{z_i} \left( a_{ij}(z)\partial _{z_j} u \right) =0 \end{aligned}$$\end{document}satisfying the non-uniform ellipticity condition C1 omega(x,t)|xi|2+|eta|2 <= aij(z)zeta i zeta j <= C2 omega(x,t)|xi|2+|eta|2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C_1 \left( \omega (x, t) \vert \xi \vert <^>2+\vert \eta \vert <^>2 \right) \le a_{ij}(z) \zeta _i \zeta _j \le C_2 \left( \omega (x, t) \vert \xi \vert <^>2+\vert \eta \vert <^>2 \right) \end{aligned}$$\end{document}where zeta=(xi,eta)is an element of RnxRm,zeta not equal 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \zeta =(\xi , \eta )\in \mathbb {R}<^>n\times \mathbb {R}<^>m, \, \zeta \ne 0 $$\end{document} where A=aij(z)i,j=1,...N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ A=\left\{ a_{ij}(z)\right\} _{i,j=1,...N} $$\end{document} is a positive matrix defined on a bounded domain Omega is an element of RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \in \mathbb {R}<^>N $$\end{document} of points z=(x,t),x is an element of Rn,t is an element of Rm,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=(x,t), \, x\in \mathbb {R}<^>n, \, t\in \mathbb {R}<^>m, $$\end{document}N=n+m;n,m >= 1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=n+m; \, n,m\ge 1.$$\end{document} The weight omega(x,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (x, t)$$\end{document} is a positive function satisfying also some additional conditions. We prove our results by using Sobolev and Poincare-type inequalities modeled on the non-uniformity of the gradient.