Extending Sobolev functions with partially vanishing traces from locally<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:mi>ε</mml:mi><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-domains and applications to mixed boundary problems

Brewster, Kevin; Mitrea, Dorina; Mitrea, Irina; Mitrea, Marius

doi:10.1016/j.jfa.2014.02.001

Cited by 52 publications

(81 citation statements)

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“…4.3.1]). This interpolation result is closely related to those in , Ch. 6] and in , Ch.8] (in the latter proved via Calderon‐Zygmund decomposition and Hardy's inequality), but here, we avoid the general condition that Ω must allow locally around the boundary points of

\overset{\partial normalΩ ∖ D}{false}

the extendability of all W 1, p ‐functions, compare with Ch.…”

Section: Introductionsupporting

confidence: 86%

“…Then there exists a continuous, linear extension operator

frakturE : W_{D}^{1, p} (normalΩ) \to W_{D}^{1, p} ({double-struckR}^{d})

. Proof Both geometric configurations admit a continuous extension operator

W_{D}^{1, p} (normalΩ) \to W^{1, p} ({double-struckR}^{d})

, according to Theorem 4.5 – which is even uniform in p ∈]1, ∞ [. Thus, Proposition 4.10 applies. Remark Proposition 4.10, applied to the special case of Jones' extension operator, provides an alternative proof of , Theorem 1.3] in case of first‐ order Sobolev spaces, if D is a ( d − 1)‐set. In , this is achieved, even for arbitrary compact boundary parts D , by a deep analysis of the support properties of the functions obtained by Jones' extension operator.…”

Section: The Extension Operatormentioning

confidence: 93%

“…Thus, defining

{frakturF}_{p} (ψ) = ϕ + \sum_{j} ζ_{j} ϕ_{j}

, one gets a linear, continuous extension operator

{frakturF}_{p}

from

C_{F}^{\infty} (normalΛ)

into

W^{1, p} ({double-struckR}^{d})

. By density,

{frakturF}_{p}

extends uniquely to a linear, continuous operator

{frakturF}_{p} : W_{F}^{1, p} (normalΛ) \to W^{1, p} ({double-struckR}^{d}) .

□ Remark The theorem is essentially proved in for the case p = 2, see also , Lemma 3.2] and, in particular, , Thm. 3.7] for a far‐reaching result concerning even higher order Sobolev spaces.…”

Section: The Extension Operatormentioning

confidence: 95%

“…It allows to pass from the original domain Ω to another domain

{normalΩ}_{∙} \subseteq \overset{normalΩ}{false},

whose boundary is smaller and, in most cases, a more regular one. It is exactly this what enables also the treatment of geometric settings like in Figure , compare for a similar, but different approach – there even applied to higher order Sobolev spaces. Section 5 contains the aforementioned elliptic and parabolic regularity results.…”

Section: Introductionmentioning

confidence: 95%

“…Throughout, we stick to the condition that D is a ( d − 1)‐set, which in several instances can in fact be weakened, compare with . Since our motivation for this paper comes from the applications, our aim is to describe a very general but nevertheless easily accessible geometric constellation that allows to deduce our results.…”

Section: Introductionmentioning

confidence: 99%

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Elliptic and parabolic regularity for second‐ order divergence operators with mixed boundary conditions

Haller-Dintelmann

Jonsson

Knees

et al. 2015

Math Methods in App Sciences

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\overset{\partial normalΩ ∖ D}{false}

the extendability of all W 1, p ‐functions, compare with Ch.…”

Section: Introductionsupporting

confidence: 86%

“…Then there exists a continuous, linear extension operator

frakturE : W_{D}^{1, p} (normalΩ) \to W_{D}^{1, p} ({double-struckR}^{d})

. Proof Both geometric configurations admit a continuous extension operator

W_{D}^{1, p} (normalΩ) \to W^{1, p} ({double-struckR}^{d})

Section: The Extension Operatormentioning

confidence: 93%

“…Thus, defining

{frakturF}_{p} (ψ) = ϕ + \sum_{j} ζ_{j} ϕ_{j}

, one gets a linear, continuous extension operator

{frakturF}_{p}

from

C_{F}^{\infty} (normalΛ)

into

W^{1, p} ({double-struckR}^{d})

. By density,

{frakturF}_{p}

extends uniquely to a linear, continuous operator

{frakturF}_{p} : W_{F}^{1, p} (normalΛ) \to W^{1, p} ({double-struckR}^{d}) .

Section: The Extension Operatormentioning

confidence: 95%

“…It allows to pass from the original domain Ω to another domain

{normalΩ}_{∙} \subseteq \overset{normalΩ}{false},

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Elliptic and parabolic regularity for second‐ order divergence operators with mixed boundary conditions

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et al. 2015

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The Green function for the mixed problem for the linear Stokes system in domains in the plane

2014

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On the mixed problem for the semilinear Darcy‐Forchheimer‐Brinkman PDE system in Besov spaces on creased Lipschitz domains

Gutt

Kohr

Mikhailov

et al. 2017

Math Methods in App Sciences

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The purpose of this paper is to study the mixed Dirichlet-Neumann boundary value problem for the semilinear Darcy-Forchheimer-Brinkman system in L p -based Besov spaces on a bounded Lipschitz domain in R 3 , with p in a neighborhood of 2. This system is obtained by adding the semilinear term |u|u to the linear Brinkman equation. First, we provide some results about equivalence between the Gagliardo and nontangential traces, as well as between the weak canonical conormal derivatives and the nontangential conormal derivatives. Various mapping and invertibility properties of some integral operators of potential theory for the linear Brinkman system, and well-posedness results for the Dirichlet and Neumann problems in L p -based Besov spaces on bounded Lipschitz domains in R n (n ≥ 3) are also presented. Then, using integral potential operators, we show the well-posedness in L 2 -based Sobolev spaces for the mixed problem of Dirichlet-Neumann type for the linear Brinkman system on a bounded Lipschitz domain in R n (n ≥ 3). Further, by using some stability results of Fredholm and invertibility properties and exploring invertibility of the associated Neumann-to-Dirichlet operator, we extend the well-posedness property to some L p -based Sobolev spaces. Next, we use the well-posedness result in the linear case combined with a fixed point theorem to show the existence and uniqueness for a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy-Forchheimer-Brinkman system in L p -based Besov spaces, with p ∈ (2 − , 2 + ) and some parameter > 0. KEYWORDSSemilinear Darcy-Forchheimer-Brinkman system, mixed Dirichlet-Neumann problem, L p -based Besov spaces, layer potential operators, Neumann-to-Dirichlet operator, existence and uniqueness INTRODUCTIONBoundary integral methods are a powerful tool to investigate linear elliptic boundary value problems that appear in various areas of science and engineering (see, eg, previous studies 1-5 ). Among many valuable contributions in the field, we mention the well-posedness result of the Dirichlet problem for the Stokes system in Lipschitz domains in R n (n ≥ 3) with boundary data in L 2 -based Sobolev spaces, which have been obtained by Fabes, Kenig, and Verchota 6 by using a layer potential analysis. Also, Mitrea and Wright 7 obtained the well-posedness results for Dirichlet, 7780Neumann, and transmission problems for the Stokes system on arbitrary Lipschitz domains in R n (n ≥ 2), with data in Sobolev and Besov-Triebel-Lizorkin spaces. By using a boundary integral method, Mitrea and Taylor 5 obtained well-posedness results for the Dirichlet problem for the Stokes system on arbitrary Lipschitz domains on a compact Riemannian manifold, with boundary data in L 2 . Their results extended the results of Fabes, Kenig, and Verchota 6 from the Euclidean setting to the case of compact Riemannian manifolds. Continuing the study of Mitrea and Taylor 5 , Dindos and Mitrea 3 developed a layer potential analysis to obtain existence and uniqueness results for the Poiss...

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Extending Sobolev functions with partially vanishing traces from locally(ε,δ)-domains and applications to mixed boundary problems

Cited by 52 publications

References 50 publications

Elliptic and parabolic regularity for second‐ order divergence operators with mixed boundary conditions

Elliptic and parabolic regularity for second‐ order divergence operators with mixed boundary conditions

The Green function for the mixed problem for the linear Stokes system in domains in the plane

On the mixed problem for the semilinear Darcy‐Forchheimer‐Brinkman PDE system in Besov spaces on creased Lipschitz domains

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