Weighted estimates for a solution of an anisotropic degenerate equation with Neumann boundary conditions at points of degeneracy (Q881250)

The author deals with the model equation of elliptic type which degenerates on a part of the boundary of a cylindrical domain \(\Omega=\widetilde\Omega\times (0,1)\subset \mathbb R^m\) \[ -\partial_m(x^\alpha_m a_{mm}\partial_m u)- x^\beta_m \sum_{i,j< m} \partial_i(a_{ij}\partial_ju)= f,\tag{1} \] where \(\partial_i\) denotes the differentiation with respect to \(x_i\) with boundary conditions \[ x^\alpha_m a_{mm}\partial_mu= 0\quad\text{on }\Gamma,\quad u= 0\quad\text{on }\partial\Omega\setminus\Gamma.\tag{2} \] Here \(\widetilde\Omega\) is a domain in \(\mathbb R^{m-1}\) with sufficiently smooth boundary, the parameters \(\alpha\), \(\beta\) determine the degrees of singularity of the coefficients of the differential operator in the neighborhood \(\Gamma= \{x\in\overline\Omega\mid x_m= 0\}\). Under suitable assumptions on the data of (1)--(2) the author proves smoothness theorems and presents a priori estimate of solutions of (1)--(2) in the weighted Sobolev spaces.