SOLUTIONS OF THE DIVERGENCE AND ANALYSIS OF THE STOKES EQUATIONS IN PLANAR HÖLDER-α DOMAINS

Abstract. If Ω ⊂ Rn is a bounded domain, the existence of solutions u ∈ H10 (Ω)n of divu = f for f ∈ L2(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular it allows to show the existence of a solution (u, p) ∈ H10 (Ω) n × L2(Ω), where u is the velocity and p the pressure. It is known that the above mentioned result holds when Ω is a Lipschitz domain and that it is not valid for arbitrary Hölder-α domains. In this paper we prove that if Ω is a planar simply connected Hölder-α domain, there exist solutions of divu = f in appropriate weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal. For some particular domains with an external cusp we apply our results to show the well posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution (u, p) ∈ H10 (Ω)n × Lr(Ω) for some r < 2 depending on the power of the cusp. 1