Neumann problem for an elliptic operator degenerate on the boundary (Q1117401)

In the bounded domain \(\Omega\) of the Euclidean space \(R^ n\) the differential operator \[ Lu=\sum_{| k|,| \ell | \leq r} (-1)^{| \ell |}(a_{k\ell}(x)u^{(k)}(x))^{(l)} \] is considered. The domain \(\Omega\) is assumed to have (n-1)-dimensional boundary \(\partial \Omega\) of the class \(C^{\infty}\). Let \[ | a_{k\ell}^{(\lambda)}(x)| \leq C\rho (x)^{2\alpha -| \lambda |},\quad 0\leq \alpha <r, \] for all \(x\in \Omega\), all the multi- indices k and \(\ell\), not exceeding r in length, and all the derivatives up to the order \(| \lambda | \leq m\) and let \[ \sum_{| k|,| \ell | \leq r} a_{k\ell}(x)\xi_ k{\bar \xi}_{\ell}\geq \kappa \rho (x)^{2\alpha}\sum_{| k| =r} | \xi_ k|^ 2 \] for all \(x\in \Omega\) and all the sets of complex numbers \(\{\xi_ k\}_{| k| \leq r}.\) Here \(\rho\) (x) is the function of the class \(C^{\infty}({\bar \Omega})\), equivalent to the distance from \(x\in \Omega\) to \(\partial \Omega.\) Under certain assumptions the unique solvability of the variational generalized homogeneous Neumann problem is proved in weighted Sobolev classes \(W^{r+m}_{2,\alpha +m}(\Omega)\).