Potential Theory on Lipschitz Domains in Riemannian Manifolds: Sobolev–Besov Space Results and the Poisson Problem

Mitrea, Marius; Taylor, Michael E.

doi:10.1006/jfan.2000.3619

Cited by 121 publications

(160 citation statements)

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“…This is a slight generalization of a lemma in [41] which, in turn, extends some Euclidean estimates from [14]. Two key observations which allow us to use this result in the present context are as follows.…”

Section: Theorem 34mentioning

confidence: 64%

“…As for C 0 , the following result from [41] applies. Let q(D, x) ∈ ØPC 0 S −1 cl have an odd principal symbol.…”

Section: Theorem 32mentioning

confidence: 96%

“…Throughout the proof, Comp will stand for generic compact operators from H 1,p 0 (Ω, E). With D * D replaced by ∆, the Laplace-Beltrami operator associated with the Riemannian metric g on M (and assuming that E is trivial) such an estimate has been proved in [41]. Our strategy is to utilize this in conjuction with the identity…”

Section: Dirac Operators On Manifolds With Rough Boundariesmentioning

confidence: 99%

“…D * D has scalar principal symbol), then (1.8) is shown to hold for the larger range 3/2 − < p < 3 + . A key ingredient in the latter result is the recent solution of the Poisson problem for the Laplace-Beltrami operator in Lipschitz domains from [41].…”

Section: Introductionmentioning

confidence: 99%

“…See [25] and the references cited there for a survey of the state of the art in this field up to early 1990's. A more recent line of research, initiated in [39], [34] (and further developed in [40], [41], [42]), is the use of layer potentials in order to solve boundary problems for general second-order, variable coefficient, elliptic systems in non-smooth manifolds. The present paper, dealing with variable first-order elliptic systems, is a natural continuation of this work.…”

Section: Introductionmentioning

confidence: 99%

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Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

Mitrea

2001

The Journal of Fourier Analysis and Applications

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We develop a function theory associated with Dirac type operators on Lipschitz subdomains of Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems, and our aim is to identify the geometric and analytic assumptions guaranteeing the validity of basic results from complex function theory in this general setting. For example, we study PlemeljCalderón-Seeley-Bojarski type splittings of Cauchy boundary data into traces of 'inner' and 'outer' monogenics and show that this problem has finite index. We also consider Szegö projections and the corresponding L p -decompositions. Our approach relies on an extension of the classical Calderón-Zygmund theory of singular integral operators which allows one to consider Cauchy type operators with variable kernels on Lipschitz graphs. In the second part, where we explore connections with Maxwell's equations, the main novelty is the treatment of the corresponding electro-magnetic boundary value problem by recasting it as a 'half' Dirichlet problem for a suitable Dirac operator.

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Section: Theorem 34mentioning

confidence: 64%

“…As for C 0 , the following result from [41] applies. Let q(D, x) ∈ ØPC 0 S −1 cl have an odd principal symbol.…”

Section: Theorem 32mentioning

confidence: 96%

Section: Dirac Operators On Manifolds With Rough Boundariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

Mitrea

2001

The Journal of Fourier Analysis and Applications

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Differential operators and boundary value problems on hypersurfaces

Duduchava

Mitrea

2006

Mathematische Nachrichten

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We explore the extent to which basic differential operators (such as Laplace-Beltrami, Lamé, Navier-Stokes, etc.) and boundary value problems on a hypersurface S in R n can be expressed globally, in terms of the standard spatial coordinates in R n . The approach we develop also provides, in some important cases, useful simplifications as well as new interpretations of classical operators and equations.Boundary value problems (BVP's) for partial differential equations (PDE's) on surfaces arise in a variety of situations and have many practical applications. See, for example, [15, §72] for the heat conduction by surfaces, [3, §10] for the equations of surface flow, [7], [8] and [13] for shell problems in elasticity, [2] for the vacuum Einstein equations describing gravitational fields, [29] for the Navier-Stokes equations on spherical domains, as well as the references therein. Furthermore, while studying the asymptotic behavior of solutions to elliptic boundary value problems in the neighborhood of a conical point one is led to considering a one-parameter family of boundary value problems in a subdomain S of S n−1 , the unit sphere in R n , naturally associated (via the Mellin transform) with the original elliptic problem. A classical reference in this regard is [16]. Finally, PDE's on surfaces also turn up naturally in the limit case, as the thickness goes to zero, of equations in thin layers or shells. Cf. [7, §3] for the case of elasticity, and [29] and [30] for the case of Navier-Stokes equations.A hypersurface S in R n has the natural structure of a (n − 1)-dimensional Riemannian manifold and the aforementioned PDE's are not the immediate analogues of the ones corresponding to the flat, Euclidean case, since they have to take into consideration geometric characteristics of S such as curvature. Inherently, these PDE's are originally written in local coordinates, intrinsic to the manifold structure of S.The main aim of this paper is to explore the extent to which the most basic partial differential operators (PDO's), as well as their associated boundary value problems, on a hypersurface S in R n , can be expressed globally, in terms of the standard spatial coordinates in R n . It turns out that a convenient way to carry out this program is by employing the so-called Günter derivatives (cf. [11], [14] and [17]): D := (D 1 , D 2 , . . . , D n ) . (0.1)Here, for each 1 ≤ j ≤ n, the first-order differential operator D j is the directional derivative along πe j , where π : R n → T S is the orthogonal projection onto the tangent plane to S and, as usual, e j = (δ jk ) 1≤k≤n ∈ R n , with δ jk denoting the Kronecker symbol. The operator D is globally defined on (as well as tangential to) S, and *

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On the transmission problems for the Oseen and Brinkman systems on Lipschitz domains in compact Riemannian manifolds

Gutt

Kohr

Pintea

et al. 2015

Mathematische Nachrichten

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Key words Brinkman and Oseen systems, Lipschitz domain, Riemannian manifold, Robin-transmission problem, Layer potentials, Sobolev spaces MSC (2010) 42B20, 76D, 46E35, 5J25, 76MThe purpose of this work is to show the well-posedness in L 2 -Sobolev spaces of the Poisson-transmission problem for the Oseen and Brinkman systems on complementary Lipschitz domains in a compact Riemannian manifold. The Oseen system appears as a perturbation of order one of the Stokes system, given in terms of the Levi-Civita connection, while the Brinkman system is a zero order perturbation of the Stokes system. The technical details of this paper rely on the layer potential theory for the Stokes system and the invertibility of some perturbed zero index Fredholm operators by a first order differential operator given in terms of the LeviCivita connection. The compactness of this differential operator requires to restrict ourselves to low dimensional compact Riemannian manifolds.

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Potential Theory on Lipschitz Domains in Riemannian Manifolds: Sobolev–Besov Space Results and the Poisson Problem

Cited by 121 publications

References 31 publications

Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

Differential operators and boundary value problems on hypersurfaces

On the transmission problems for the Oseen and Brinkman systems on Lipschitz domains in compact Riemannian manifolds

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