On removable sets for degenerated elliptic equations (Q317361)

Under consideration is the Dirichlet problem for an elliptic equation \[ -\sum_{i,j=1}^{n}\partial_{x_{j}} a_{ij}(x)\partial_{x_{i}} u =f, \;\;x\in G\subset {\mathbb R}^{n}. \eqno{(1)} \] The corresponding quadratic form is degenerate in the sense that there exists a positive constant \(\gamma>0\) such that \[ \gamma\sum_{i=1}|\xi_{i}|^{2}\lambda_{i}(x)\leq \sum_{i,j} a_{ij}\xi_{i}\xi_{j}\leq \gamma^{-1}\sum_{i=1}|\xi_{i}|^{2}\lambda_{i}(x)\text{ for almost all } x\in G, \;\forall \xi\in {\mathbb R}^{n}, \] where \(\lambda_{i}(x)=\Bigl(\sum_{i}^{n}2|x_{i}|/(2+\alpha_{i})\Bigr)^{\alpha_{i}}\) (\(\alpha_{i}\in (0,2/(n-1))\)). The coefficients \(a_{ij}\) are assumed to be measurable and bounded. The main results are necessary and sufficient conditions for removability of compact subsets in G relative to the Dirichlet problem for the equation (1). They are stated in terms of a special capacity.