Properties of some weighted Sobolev spaces and application to spectral approximations

The weighted Sobolev spaces on a square, whose weight is a given power of the product of the distances to the edges, are introduced. Trace theorems related to these spaces are proved, then a regularity result for the Dirichlet problem for the Laplace operator is proved. Next several projection operators with polynomial values are considered for which approximation results in weighted norms are stated. Finally, a collocation spectral method for the Dirichlet problem for the Laplace operator with inhomogeneous boundary conditions is analyzed. This work includes new results on Chebyshev approximation.