SOLUTIONS OF THE DIVERGENCE AND ANALYSIS OF THE STOKES EQUATIONS IN PLANAR Hölder-Α DOMAINS

Abstract

Abstract.If Ω ⊂ R n is a bounded domain, the existence of solutions u ∈ H 1 0 (Ω) n of div u = f for f ∈ L 2 (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,, 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 (u, p) ∈ H 1 0 (Ω) n × L r (Ω) for some r < 2 depending on the power of the cusp.

Extension theorems for external cusps with minimal regularity

“…Indeed, the classical Korn inequality, fundamental to proving existence of solutions of the linearized elasticity equations, does not hold in domains with outer peaks, but a variant involving weights depending on the distance to the boundary or the distance to the tip of the cusp does hold. Similar results hold for the divergence operator for which a continuous right inverse can be defined on this kind of weighted spaces [Acosta et al 2006;2012;Durán and López García 2010a;2010b]. In this context it is clear that a version of Theorem A for weights of the type described can also be useful in applications.…”

“…Under extra assumptions on the boundary of D it is possible to find conditions under which weights of the type d(·, ∂ D) µ belong to A p . Indeed, that holds for −(n − m) < µ < (n − m)( p − 1), provided that ∂ D is a compact set contained in an m-regular set (see [Durán and López García 2010a]). In this case we can replicate Theorem 6.3 by using a weighted version of Proposition 6.1 and arguing along the lines given in Remark 6.2 (using Chua's results).…”

Muckenhoupt Ap-properties of Distance Functions and Applications to Hardy–Sobolev -type Inequalities

“…Corollary 3.8 is also closely related to [6,Lemma 3.3], which states that if a compact set ∅ = E ⊂ R n is a subset of an Ahlfors λ-regular set, for 0 ≤ λ < n, and if 1 < p < ∞ and α ∈ R are such that B(x, r)) ≤ Cr λ for each x ∈ F and all 0 < r ≤ diam(F ) (or for all r > 0 if F consists of a single point), and that then dim A (F ) = dim H (F ) = λ. In particular, if a closed set E ⊂ R n is a subset of an Ahlfors λ-regular set, then dim A (E) ≤ λ.…”

Weighted generalized Korn inequalities on John domains

“…Let Ω ⊂ R n be a bounded John domain, ≥ 0 and 1 < q < ∞. Then, the operator T defined in (17) is continuous from L q (Ω, −q ) to itself, where is the distance to the boundary of Ω. Moreover, its norm is bounded by…”