Weighted Poincaré and Korn inequalities for Hölder <i>α</i> domains

Acosta, Gabriel; Durán, Ricardo G.; Lombardi, Ariel L.

doi:10.1002/mma.680

Cited by 29 publications

(41 citation statements)

References 15 publications

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“…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.…”

Section: Introductionmentioning

confidence: 54%

Extension theorems for external cusps with minimal regularity

Acosta¹,

Ojea²

2012

Pacific J. Math.

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Sobolev functions defined on certain simple domains with an isolated singular point (such as power type external cusps) can not be extended in standard, but in appropriate weighted spaces. In this article we show that this result holds for a large class of domains that generalizes external cusps, allowing minimal boundary regularity. The construction of our extension operator is based on a modification of reflection techniques originally developed for dealing with uniform domains. The weight involved in the extension appears as a consequence of the failure of the domain to comply with basic properties of uniform domains, and it turns out to be a quantification of that failure. We show that weighted, rather than standard spaces, can be treated with our approach for weights that are given by a monotonic function either of the distance to the boundary or of the distance to the tip of the cusp.

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Section: Introductionmentioning

confidence: 54%

Extension theorems for external cusps with minimal regularity

Acosta¹,

Ojea²

2012

Pacific J. Math.

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“…for all µ ∈ R. Indeed, this estimate was proved in [13] (see also Lema 3.1 in [1], and [31] for a different proof in the case p = 2 and µ = 0). Therefore, taking µ = β − α we obtain…”

Section: Preliminaries and Korn Type Inequalitiesmentioning

confidence: 77%

“…The statement given in the following theorem is slightly stronger than the result in Theorem 3.1 of [1]. Therefore, we include the proof for the sake of completeness although the arguments are essentially those given in that reference.…”

Section: Preliminaries and Korn Type Inequalitiesmentioning

confidence: 93%

“…Our approach use some of the ideas of the papers [1,24]. The existence of solutions of the divergence is derived from appropriate Korn type inequalities.…”

Section: Introductionmentioning

confidence: 99%

“…The existence of solutions of the divergence is derived from appropriate Korn type inequalities. The weighted Korn inequalities that we need are slight variants of those obtained in [1] but we include the proofs for the sake of completeness.…”

Section: Introductionmentioning

confidence: 99%

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

Durán

García

2010

Math. Models Methods Appl. Sci.

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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.

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