The Poincaré Inequality for a Vector Field With Zero Tangential or Normal Component on the Boundary

Bishop, Nigel T.

doi:10.1080/16073606.1988.9631951

Cited by 3 publications

(7 citation statements)

References 4 publications

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“…. 1 We are able to prove that for smooth domains there always holds: κ(Ω) > √ 2. The proof of this fact will appear elsewhere.…”

Section: Introduction and The Main Resultsmentioning

confidence: 86%

“…Theorem 1.2. Assume that there exists a sequence {u k } ∞ k=1 , u k ∈ W 1,2 (Ω, R n ), u k • n = 0 on ∂Ω, with the following properties:…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…(i) u k converges to 0 weakly in W 1,2 (Ω, R n ), (ii) D(u k ) L 2 (Ω) = 1, (iii) lim k→∞ ∇u k L 2 (Ω) = κ(Ω).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…If κ(Ω) > √ 2 then the supremum in the definition (1.3) is attained. More precisely, for every A 0 ∈ L Ω there exists u ∈ W 1,2 (Ω, R n ), such that u • n = 0 on ∂Ω, D(u) ≡ 0, and:…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…It now suffices to check that every u ∈ W 1,2 (Q, R 2 ) with u · n = 0 on ∂Q, can be approximated by a sequence of C 2 (Q, R 2 ) vector fields satisfying the same boundary condition. To this end, define the extensionū 1…”

mentioning

confidence: 99%

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A note on the optimal constants in Korn\'s and geometric rigidity estimates in bounded and unbounded domains

Lewicka¹,

Müller²

2016

Indiana Univ. Math. J.

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We are concerned with the optimal constants: in the Korn inequality under tangential boundary conditions on bounded sets Ω ⊂ R n , and in the geometric rigidity estimate on the whole R 2 . We prove that the latter constant equals √ 2, and we discuss the relation of the former constants with the optimal Korn's constants under Dirichlet boundary conditions, and in the whole R n , which are well known to equal √ 2. We also discuss the attainability of these constants and the structure of deformations/displacement fields in the optimal sets.L Ω = {A ∈ so(n); ∃a ∈ R n ∀x ∈ ∂Ω (Ax + a) · n(x) = 0} .1991 Mathematics Subject Classification. 74B05.

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“…. 1 We are able to prove that for smooth domains there always holds: κ(Ω) > √ 2. The proof of this fact will appear elsewhere.…”

Section: Introduction and The Main Resultsmentioning

confidence: 86%

“…Theorem 1.2. Assume that there exists a sequence {u k } ∞ k=1 , u k ∈ W 1,2 (Ω, R n ), u k • n = 0 on ∂Ω, with the following properties:…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…(i) u k converges to 0 weakly in W 1,2 (Ω, R n ), (ii) D(u k ) L 2 (Ω) = 1, (iii) lim k→∞ ∇u k L 2 (Ω) = κ(Ω).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Section: Introduction and The Main Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

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A note on the optimal constants in Korn\'s and geometric rigidity estimates in bounded and unbounded domains

Lewicka¹,

Müller²

2016

Indiana Univ. Math. J.

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Bounds on heat transfer by incompressible flows between balanced sources and sinks

Song

Fantuzzi²,

Tobasco³

2023

Physica D: Nonlinear Phenomena

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Compressible Viscous Flows in a Symmetric Domain with Complete Slip Boundary

Liu¹

2016

Preprint

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This work is devoted to study the global behavior of viscous flows contained in a symmetric domain with complete slip boundary. In such scenario the boundary no longer provides friction and therefore the perturbation of angular velocity lacks decaying structure. In fact, we show the existence of uniformly rotating solutions as steady states for the compressible Navier-Stokes equations. By manipulating the conservation law of angular momentum, we establish a suitable Korn's type inequality to control the perturbation and show the asymptotic stability of the uniformly rotating solutions with small angular velocity. In particular, the initial perturbation which preserves the angular momentum would decay exponentially in time and the solution to the Navier-Stokes equations converges to the steady state as time grows up.

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The Poincaré Inequality for a Vector Field With Zero Tangential or Normal Component on the Boundary

Cited by 3 publications

References 4 publications

A note on the optimal constants in Korn\'s and geometric rigidity estimates in bounded and unbounded domains

A note on the optimal constants in Korn\'s and geometric rigidity estimates in bounded and unbounded domains

Bounds on heat transfer by incompressible flows between balanced sources and sinks

Compressible Viscous Flows in a Symmetric Domain with Complete Slip Boundary

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