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

Abstract

If Ω ⊂ ℝnis a bounded domain, the existence of solutions [Formula: see text] of div u = 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 [Formula: see text], 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 div u = 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 [Formula: see text] for some r < 2 depending on the power of the cusp.