Based on Fourier series, we adapt an approach discussed in a recent work on the Laplace operator to classical results obtained in the literature, describing the singularities of solutions to a fourth-order elliptic problem on a polygonal domain of the plane that may appear near a concave corner. We demonstrate how the Fourier series method provides explicit decomposition and precise description of the coefficients of singularities of the solution. As a main result, explicit and sharp estimates with respect to the opening angle parameter can be obtained via this method. We recall that such estimates can be useful for the asymptotic analysis of solutions near corners where the opening angle generates a jump in singularity in Sobolev's exponent.

Based on partial Fourier series analysis, we adapt on a model case a new approach to classical results obtained in the literature describing the singularities of a family a solutions of a second order elliptic problems on open non-convex planar sectors. The method allows the exhibition of singular and regular frequencies, explicit decomposition and description of coefficients of singularities of the solution. As a main result, explicit and sharp estimates with respect to the opening angle parameter are obtained via this method. They are not uniform near π where corners have opening angle generating a jump of singularity in Sobolev exponent, contrarily to the results obtained in A. Tami (2016),(2019),(2021) for harmonic and/or biharmonic problems on a family of convex planar sectors.