A multidirectional Dirichlet problem

Verchota, Gregory C.; Vogel, Andrew

doi:10.1007/bf02922056

Cited by 12 publications

(40 citation statements)

References 28 publications

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“…This follows because that term can be bounded by E N α (∇u) 2 ds with the measure of E controlled by 2 −m . This in turn follows in polyhedra by Lemma 5.3 of [VV03] and the Carleson lemma (see §4 of [VV03]). In addition W · ∇u on St(κ 0 ,K) is a tangential derivative.…”

Section: A Preliminary Argumentmentioning

confidence: 98%

“…where C is a constant depending only on κ 0 ∈ K. By Lemma 8.8 of [VV03] and the Poincaré inequality (2.3) may be replaced with…”

Section: A Preliminary Argumentmentioning

confidence: 99%

“…x ∈ R n , integration by parts (see §5 of [VV03] for polyhedra) yields the second equality in the identity…”

Section: A Preliminary Argumentmentioning

confidence: 99%

“…This paper and its companion [VV03] take the point of view that the approach to boundary value problems in Lipschitz domains that has been developed over the past few decades should be and can be extended to nongraph settings. It also indicates possible geometric and topological obstacles to such a project.…”

Section: Introductionmentioning

confidence: 99%

“…The uniformity is proved in Lemma 1 and the decomposition applied to get a priori estimates for both the Neumann and regularity (Dirichlet data in W 1,2 (∂ )) problems in 4-dimensional polyhedra (Lemma 2). That the domains of the decomposition are Lipschitz is shown in the Appendix ( §12) by a modification of an argument from [VV03].…”

Section: Introductionmentioning

confidence: 99%

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The multidirectional Neumann problem in ℝ4

Verchota

Vogel

2006

Math. Ann.

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The Neumann problem as formulated in Lipschitz domains with L p boundary data is solved for harmonic functions in any compact polyhedral domain of R 4 that has a connected 3-manifold boundary. Energy estimates on the boundary are derived from new polyhedral Rellich formulas together with a Whitney type decomposition of the polyhedron into similar Lipschitz domains. The classical layer potentials are thereby shown to be semi-Fredholm. To settle the onto question a method of continuity is devised that uses the classical 3-manifold theory of E. E. Moise in order to untwist the polyhedral boundary into a Lipschitz boundary. It is shown that this untwisting can be extended to include the interior of the domain in local neighborhoods of the boundary. In this way the flattening arguments of B. E. J. Dahlberg and C. E. Kenig for the H 1 at Neumann problem can be extended to polyhedral domains in R 4 . A compact polyhedral domain in R 6 of M. L. Curtis and E. C. Zeeman, based on a construction of M. H. A. Newman, shows that the untwisting and flattening techniques used here are unavailable in general for higher dimensional boundary value problems in polyhedra.

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Section: A Preliminary Argumentmentioning

confidence: 98%

“…where C is a constant depending only on κ 0 ∈ K. By Lemma 8.8 of [VV03] and the Poincaré inequality (2.3) may be replaced with…”

Section: A Preliminary Argumentmentioning

confidence: 99%

“…x ∈ R n , integration by parts (see §5 of [VV03] for polyhedra) yields the second equality in the identity…”

Section: A Preliminary Argumentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The multidirectional Neumann problem in ℝ4

Verchota

Vogel

2006

Math. Ann.

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The Stokes Equations

John

2016

Finite Element Methods for Incompressible Flow Problems

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Weighted Sobolev spaces and regularity for polyhedral domains

Ammann¹,

Nistor²

2007

Computer Methods in Applied Mechanics and Engineering

166

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Dedicated to Ivo Babuška on the occasion of his 80th birthday.Abstract. We prove a regularity result for the Poisson problem −∆u = f , u| ∂P = g on a polyhedral domain P ⊂ R 3 using the Babuška-Kondratiev spaces K m a (P). These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges [4,33]. In particular, we show that there is no loss of K m a -regularity for solutions of strongly elliptic systems with smooth coefficients. We also establish a "trace theorem" for the restriction to the boundary of the functions in K m a (P).

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A multidirectional Dirichlet problem

Cited by 12 publications

References 28 publications

The multidirectional Neumann problem in ℝ4

The multidirectional Neumann problem in ℝ4

The Stokes Equations

Weighted Sobolev spaces and regularity for polyhedral domains

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