Sharp Hodge decompositions, Maxwell's equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds
type: Item
instance of: scholarly article
title: Sharp Hodge decompositions, Maxwell's equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds
zbMATH Open document ID: 1073.31006
DOI: 10.1215/S0012-7094-04-12322-1
author: Marius Mitrea
published in: Duke Mathematical Journal
publication date: 2005-03-07T00:00:00Z
review text: From the introduction of the paper: ``We solve three basic potential theoretic problems: (I) Hodge decompositions for vector fields, (II) Poisson problems for the Hodge Laplacian, and (III) inhomogeneous Maxwell equations in Lipschitz subdomains of a smooth, compact, boundaryless, three-dimensional Riemannian manifold \({\mathcal M}\). They are all considered in the context of Sobolev-Besov spaces, that is, when the global smoothness of both the data and the solutions is measured on these scales. In hindsight, the problems (I)--(III) above turn out to be closely related. A manifestation of this is that they share a common, (asymptotically) sharp ``well-posedness region'', stemming from necessary limitations on the indices \(s\) (smoothness) and \(p\) (integrability), of the Sobolev-Besov spaces for which these partial differential equations (PDEs) have unique solutions, continuously dependent on the given data. In turn, this region, call it \({\mathcal R}_\Omega\), is entirely determined by the geometric characteristics of the underlying domain \(\Omega\subset {\mathcal M}\). More concretely, \[ \left(s,\frac{1}{p}\right)\in {\mathcal R}_\Omega \Longleftrightarrow\begin{cases} 0<\frac{ 1}{p}<1,\;-1+\frac{1}{p}<s<\frac{1}{p},\\ \frac{ 2}{3}(1-\frac{1}{p_\Omega})<\frac{1}{p}-\frac{s}{3} <\frac{1}{3}(\frac{2}{p_\Omega}+1).\end{cases} \] Here, \(p_\Omega\) is further defined in terms of the critical exponents intervening in the (regular) Dirichlet and Neumann problems for the Laplace-Beltrami operator in \(\Omega\) (as well as its complement) when optimal \(L^p\)-estimates for the associated nontangential maximal function are sought. A precise definition is given in \S 4. For the purpose of this introduction, we note that, generally speaking, \(1\leq p_\Omega <2\) (cf.\ \textit{B. E. J. Dahlberg} [Arch. Ration. Mech. Anal. 65, 272--288 (1977; Zbl 0406.28009) and Stud. Math. 66, 13--24 (1979; Zbl 0422.31008)], \textit{D. S. Jerison} and \textit{C. E. Kenig} [Bull. Am. Ma…
reviewed by: Eleutherius Symeonidis
Mathematics Subject Classification ID: 31C12
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 58A14
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 31B10
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 2141301
zbMATH Keywords: Hodge decomposition
zbMATH Keywords: Hodge Laplacian
zbMATH Keywords: Poisson problem
zbMATH Keywords: Maxwell equation
zbMATH Keywords: Lipschitz domain
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Sobolev-Besov space
Wikidata QID: Q115240304
MaRDI profile type: Publication
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