Recent progress in elliptic equations and systems of arbitrary order with rough coefficients in Lipschitz domains

Maz′ya, Vladimir; Shaposhnikova, Tatyana

doi:10.1007/s13373-011-0003-6

Cited by 7 publications

(5 citation statements)

References 62 publications

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“…For a survey on regularity results for elliptic systems with rough coefficients, see e.g. [31]. For 1 ≤ j ≤ N let ∆ j be the weak Laplacian with form domain V j , cf.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

The Kato Square Root Problem for mixed boundary conditions

Egert

Haller-Dintelmann

Tolksdorf

2014

Journal of Functional Analysis

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Abstract. We consider the negative Laplacian subject to mixed boundary conditions on a bounded domain. We prove under very general geometric assumptions that slightly above the critical exponent 1 2 its fractional power domains still coincide with suitable Sobolev spaces of optimal regularity. In combination with a reduction theorem recently obtained by the authors, this solves the Kato Square Root Problem for elliptic second order operators and systems in divergence form under the same geometric assumptions. Thereby we answer a question posed by J. L. Lions in 1962 [29].

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“…For a survey on regularity results for elliptic systems with rough coefficients, see e.g. [31]. For 1 ≤ j ≤ N let ∆ j be the weak Laplacian with form domain V j , cf.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

The Kato Square Root Problem for mixed boundary conditions

Egert

Haller-Dintelmann

Tolksdorf

2014

Journal of Functional Analysis

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“…In this case, one has to study a Stokes system with mixed Dirichlet-Neumann boundary condition (or more precisely, an elliptic equation with mixed boundary condition). For the latter case, we refer to [3,17,18,26] and references therein for the latest developments.…”

Section: Existence Of Solutions Without Extensionmentioning

confidence: 99%

Interaction of a Vortex Induced by a Rotating Cylinder with a Plane

Han

Hou

Témam

2017

Commun. Comput. Phys.

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In this article,we study theoretically and numerically the interaction of a vortex induced by a rotating cylinder with a perpendicular plane. We show the existence of weak solutions to the swirling vortex models by using the Hopf extension method, and by an elegant contradiction argument, respectively. We demonstrate numerically that the model could produce phenomena of swirling vortex including boundary layer pumping and two-celled vortex that are observed in potential line vortex interacting with a plane and in a tornado.

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“…It has been extended by Gallouet and Monier in [9] to domains Ω with Lipschitz boundary. In their recent survey article [15], Maz'ya and Shaposhnikova give very general estimates working for Lipschitz domains Ω and systems of elliptic equations. Our Theorem 3.13 happens to be a very special case of Theorems 1 and 2 in [15].…”

Section: Prange / First-order Expansion For the Dirichlet Eigenvalmentioning

confidence: 99%

“…In their recent survey article [15], Maz'ya and Shaposhnikova give very general estimates working for Lipschitz domains Ω and systems of elliptic equations. Our Theorem 3.13 happens to be a very special case of Theorems 1 and 2 in [15]. To make the link obvious take m = 1, l = N , a = 0; then s = 1 − 1 p , W m,a p (Ω) = W 1,p (Ω) and V a p (Ω) = W 1,p 0 (Ω).…”

Section: Prange / First-order Expansion For the Dirichlet Eigenvalmentioning

confidence: 99%

First-order expansion for the Dirichlet eigenvalues of an elliptic system with oscillating coefficients

Prange

2013

Asymptotic Analysis

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This paper is concerned with the homogenization of the Dirichlet eigenvalue problem, posed in a bounded domain Ω ⊂ R 2 , for a vectorial elliptic operator −∇ · A ε (·)∇ with ε-periodic coefficients. We analyse the asymptotics of the eigenvalues λ ε,k when ε → 0, the mode k being fixed. A first-order asymptotic expansion is proved for λ ε,k in the case when Ω is either a smooth uniformly convex domain, or a convex polygonal domain with sides of slopes satisfying a small divisors assumption. Our results extend those of Moskow and Vogelius in restricted to scalar operators and convex polygonal domains with sides of rational slopes. We take advantage of the recent progress due to Gérard-Varet and Masmoudi [J.

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Recent progress in elliptic equations and systems of arbitrary order with rough coefficients in Lipschitz domains

Abstract: This is a survey of results mostly relating elliptic equations and systems of arbitrary even order with rough coefficients in Lipschitz graph domains. Asymptotic properties of solutions at a point of a Lipschitz boundary are also discussed. Keywords

Cited by 7 publications

References 62 publications

The Kato Square Root Problem for mixed boundary conditions

The Kato Square Root Problem for mixed boundary conditions

Interaction of a Vortex Induced by a Rotating Cylinder with a Plane

First-order expansion for the Dirichlet eigenvalues of an elliptic system with oscillating coefficients

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