Boundary Value Problems in Plane and Bihedral Angles

Abstract

In this chapter we consider the solvability in weighted Sobolev and Hölder spaces of the boundary value problems for the Stokes system in plane and bihedral angles. The proof of unique solvability in weighted Sobolev L ₂-spaces of these problems in a plane angle is simple. Rewriting the problem in polar coordinates and applying the Mellin transform, we obtain a boundary value problem for a system of ordinary differential equations with constant coefficients, which depends on the spectral parameter λ ∈ ℂ. This problem is solved explicitly. Then we determine the maximal strip in the complex plane ℂ, which contains the real axis and does not contain the spectrum of the corresponding homogeneous problem (it determines the maximal range for the weight exponent). Further, we prove an estimate for solutions to the nonhomogeneous problem depending on |λ|. Finally, applying the inverse Mellin transform and the Plancherel theorem, we deduce the required result.