A Wiener type criterion for weighted quasiminima (Q1175691)

Functionals of the form \(J(u,\Omega)=\int_ \Omega f(x,u,\nabla u)dx\) are considered, where \(\Omega\) is an open bounded connected subset of \(\mathbb{R}^ n\), \(n\geq 2\), \(f:\mathbb{R}^ n\times \mathbb{R}\times \mathbb{R}^ n\to\mathbb{R}\) is a Caratheodory function (measurable in \(x\) for every \((z,p)\) and continuous in \((z,p)\) for almost every \(x\in \Omega\)), \(u:\Omega\to \mathbb{R}\) is a scalar function belonging to the weighted Sobolev space \(H_{loc}^ 1(\Omega,w)\), the weight being in the \(A_ 2\) class of Muckenhoupt. A sufficient condition for continuity at the boundary for quasiminima of degenerate type is proved. The main tool in the proof is the Harnack inequality for functions in weighted De Giorgi classes.