First order approach to Lp estimates for the stokes operator on lipschitz domains

Alan McIntosh; Sylvie Monniaux

doi:10.1007/978-3-319-41945-9_3

First order approach to L^p estimates for the stokes operator on lipschitz domains

Alan McIntosh, Sylvie Monniaux^*

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

This paper concerns Hodge-Dirac operators D_H = d + δ acting in L^p(Ω,Λ) where Ω is a bounded open subset of ℝⁿ satisfying some kind of Lipschitz condition, Λ is the exterior algebra of ℝⁿ, d is the exterior derivative acting on the de Rham complex of differential forms on Ω, and δ is the interior derivative with tangential boundary conditions. In L²(Ω,Λ), d' = δ and D_H is self-adjoint, thus having bounded resolvent {(I + itD_H)}^{t^∈R} as well as a bounded functional calculus in L2(Ω,Λ). We investigate the range of values pH < p < p^H about p = 2 for which D_H has bounded resolvents and a bounded holomorphic functional calculus in L p(Ω,Λ).

Original language	English
Title of host publication	Mathematical Analysis, Probability and Applications – Plenary Lectures - ISAAC 2015
Editors	Tao Qian, Luigi G. Rodino
Publisher	Springer New York LLC
Pages	55-75
Number of pages	21
ISBN (Print)	9783319419435
DOIs	https://doi.org/10.1007/978-3-319-41945-9_3
Publication status	Published - 2016
Event	10th International Society of Analysis, its Applications and Computation, ISAAC 2015 - Macau, China Duration: 3 Aug 2015 → 8 Aug 2015

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	177
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	10th International Society of Analysis, its Applications and Computation, ISAAC 2015
Country/Territory	China
City	Macau
Period	3/08/15 → 8/08/15

Access to Document

10.1007/978-3-319-41945-9_3

Cite this

McIntosh, A., & Monniaux, S. (2016). First order approach to L^p estimates for the stokes operator on lipschitz domains. In T. Qian, & L. G. Rodino (Eds.), Mathematical Analysis, Probability and Applications – Plenary Lectures - ISAAC 2015 (pp. 55-75). (Springer Proceedings in Mathematics and Statistics; Vol. 177). Springer New York LLC. https://doi.org/10.1007/978-3-319-41945-9_3

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title = "First order approach to Lp estimates for the stokes operator on lipschitz domains",

abstract = "This paper concerns Hodge-Dirac operators DH = d + δ acting in Lp(Ω,Λ) where Ω is a bounded open subset of ℝn satisfying some kind of Lipschitz condition, Λ is the exterior algebra of ℝn, d is the exterior derivative acting on the de Rham complex of differential forms on Ω, and δ is the interior derivative with tangential boundary conditions. In L2(Ω,Λ), d' = δ and DH is self-adjoint, thus having bounded resolvent {(I + itDH)}{t∈R} as well as a bounded functional calculus in L2(Ω,Λ). We investigate the range of values pH < p < pH about p = 2 for which DH has bounded resolvents and a bounded holomorphic functional calculus in L p(Ω,Λ).",

keywords = "First order approach, Hodge boundary conditions, Hodge-Dirac operator, Lipschitz domains, Stokes operator",

author = "Alan McIntosh and Sylvie Monniaux",

note = "Publisher Copyright: {\textcopyright} Springer International Publishing Switzerland 2016.; 10th International Society of Analysis, its Applications and Computation, ISAAC 2015 ; Conference date: 03-08-2015 Through 08-08-2015",

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McIntosh, A & Monniaux, S 2016, First order approach to L^p estimates for the stokes operator on lipschitz domains. in T Qian & LG Rodino (eds), Mathematical Analysis, Probability and Applications – Plenary Lectures - ISAAC 2015. Springer Proceedings in Mathematics and Statistics, vol. 177, Springer New York LLC, pp. 55-75, 10th International Society of Analysis, its Applications and Computation, ISAAC 2015, Macau, China, 3/08/15. https://doi.org/10.1007/978-3-319-41945-9_3

First order approach to L^p estimates for the stokes operator on lipschitz domains. / McIntosh, Alan; Monniaux, Sylvie.
Mathematical Analysis, Probability and Applications – Plenary Lectures - ISAAC 2015. ed. / Tao Qian; Luigi G. Rodino. Springer New York LLC, 2016. p. 55-75 (Springer Proceedings in Mathematics and Statistics; Vol. 177).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - This paper concerns Hodge-Dirac operators DH = d + δ acting in Lp(Ω,Λ) where Ω is a bounded open subset of ℝn satisfying some kind of Lipschitz condition, Λ is the exterior algebra of ℝn, d is the exterior derivative acting on the de Rham complex of differential forms on Ω, and δ is the interior derivative with tangential boundary conditions. In L2(Ω,Λ), d' = δ and DH is self-adjoint, thus having bounded resolvent {(I + itDH)}{t∈R} as well as a bounded functional calculus in L2(Ω,Λ). We investigate the range of values pH < p < pH about p = 2 for which DH has bounded resolvents and a bounded holomorphic functional calculus in L p(Ω,Λ).

AB - This paper concerns Hodge-Dirac operators DH = d + δ acting in Lp(Ω,Λ) where Ω is a bounded open subset of ℝn satisfying some kind of Lipschitz condition, Λ is the exterior algebra of ℝn, d is the exterior derivative acting on the de Rham complex of differential forms on Ω, and δ is the interior derivative with tangential boundary conditions. In L2(Ω,Λ), d' = δ and DH is self-adjoint, thus having bounded resolvent {(I + itDH)}{t∈R} as well as a bounded functional calculus in L2(Ω,Λ). We investigate the range of values pH < p < pH about p = 2 for which DH has bounded resolvents and a bounded holomorphic functional calculus in L p(Ω,Λ).

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