Comportement des coefficients de Fourier de la solution d’un problème de Dirichlet à domaine polygonal. (Behavior of the Fourier coefficients of the solution of a Dirichlet problem in a polygonal domain).(French)Zbl 0635.35019

Near each vertice of a polygonal domain, the solution of the Dirichlet problem (for the Laplacian) is represented by a Fourier series. The formal description of the Fourier coefficients is given for analytic data. It is shown that the ideal spaces for studying the behaviour of the Fourier coefficients are the weighted Sobolev spaces. In these spaces, every Fourier coefficient is split into a regular part and a singular part which is proportional to at most one intrinsic singular function.

MSC:

 35J25 Boundary value problems for second-order elliptic equations 35C10 Series solutions to PDEs 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 42B05 Fourier series and coefficients in several variables