WEIGHT ESTIMATES FOR A SOLUTION OF AN ANISOTROPICALLY DEGENERATING EQUATION WITH NEUMANN BOUNDARY CONDITIONS IN THE POINTS OF DEGENERACY

摘要

The smoothness of a solution of a degenerating equation for Dirichlet boundary conditions in weight norms of the Sobolev spaces was considered in [1], [2]. Note that the behavior of a solution of a degenerating equation depends essentially on the type of boundary conditions given in the points of degeneracy of the coefficients of the differential operator. In [3], it was proved that, in the case of Dirichlet conditions, a solution has in a neighborhood of the points of degeneracy a boundary layer determined by the degree of degeneracy of the coefficients for any arbitrarily smooth right-hand side. In the same paper, it was established that the boundary layer can be factorized and represented as the product of a known singular function and a smooth function, which allows ones to construct, in spite of irregularity of the data, efficient finite-element schemes of numerical solution of the problem [4]. In the case of Neumann conditions on T, we have a qualitatively different picture. Namely, in the present paper, we show that in this case the solution in a neighborhood T is regular and its differential-integral properties are essentially determined by the corresponding properties of the right-hand side instead of the degeneracy of the coefficients of the differential operator.