The Hodge-Laplacian. Boundary value problems on Riemannian manifolds
type: Item
instance of: scholarly article
title: The Hodge-Laplacian. Boundary value problems on Riemannian manifolds
zbMATH Open document ID: 1357.58001
DOI: 10.1515/9783110484380
author: Dorina Mitrea
author: Irina Mitrea
author: Marius Mitrea
author: Michael Taylor
published in: De Gruyter Studies in Mathematics
publication date: 2016-08-02T00:00:00Z
review text: From the authors' introduction: In their influential paper [Acta Math. 141, 165--186 (1978; Zbl 0402.31009)] on layer potential techniques for boundary value problems, \textit{E. Fabes}, \textit{M. Jodeit} and \textit{N. Rivière} have showed that for a bounded domain \(\Omega\subset \mathbb{R}^n\) of class \(\mathcal{C}^1,\) with outward unit normal \(\nu\) and boundary surface measure \(\sigma,\) the harmonic double layer \[ Kf(x):=\lim_{\varepsilon\to0^+} \frac{1}{\omega_{n-1}}\int\limits_{\underset{|x-y|>\varepsilon}{y\in\partial\Omega}} \frac{\langle \nu(y),y-x\rangle}{|x-y|^n}f(y)\text{d}\sigma(y),\quad x\in \partial\Omega,\eqno(1) \] is a compact operator on \(L^p(\partial\Omega)\) for each \(p\in(1,\infty),\) where \(\langle\cdot,\cdot\rangle\) is the standard inner product, and \(\omega_{n-1}\) denotes the area of the unit sphere in \(\mathbb{R}^n\). This continued a long line of work, originating with Erik Ivar Fredholm in the late 1800's and early 1900's whose motivation for the development of Fredholm theory was the use of such compactness results in order to treat boundary value problems via integral equations methods. In particular, this development made it possible to solve in [loc. cit.] boundary value problems for the Laplacian in a bounded \(\mathcal{C}^1\) domain \(\Omega\subset\mathbb{R}^n\) equipped with Dirichlet or Neumann boundary conditions. In the formulation of these problems, the boundary data is selected from Lebesgue spaces, the boundary traces are taken in a pointwise nontangential sense, and the size of the solution was measured through the membership of the nontangential maximal function to \(L^p(\partial\Omega)\). Concerning the compactness of \(K\) on \(L^p(\partial\Omega),\) the demand that \(\Omega\subset\mathbb{R}^n\) is of class \(\mathcal{C}^1\) may be significantly relaxed and, in [\textit{S. Hofmann} et al., Int. Math. Res. Not. 2010, No. 14, 2567--2865 (2010; Zbl 1221.31010)], the authors have identified the natural class of…
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 58-02
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 45B05
Mathematics Subject Classification ID: 35G30
zbMATH DE Number: 6608462
zbMATH Keywords: Hodge-Laplacian
zbMATH Keywords: boundary value problems on Riemannian manifolds
zbMATH Keywords: harmonic double layer
zbMATH Keywords: regular SKT domain
zbMATH Keywords: \(L^p\)-solvability
zbMATH Keywords: Fredholmness
zbMATH Keywords: de Rham-Hodge formalism
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110484380
OpenAlex ID: W4245378310
description: scientific article; zbMATH DE number 6608462