Trace Theorems on Non-Smooth Boundaries for Functional Spaces Related to Maxwell Equations: an Overview
We study tangential vector fields on the boundary of a bounded Lipschitz domain in ℝ3. Our attention is focused on the definition of suitable Hilbert spaces over a range of Sobolev regularity which we try to make as large as possible, and also on the cons
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Summary. We study tangential vector fields on the boundary of a bounded Lipschitz domain in R 3 • Our attention is focused on the definition of suitable Hilbert spaces over a range of Sobolev regularity which we try to make as large as possible, and also on the construction of tangential differential operators. Hodge decompositions are proved to hold for some special choices of spaces which are of interest in the theory of Maxwell equations.
Introduction In the present paper we collect results, observations and open problems as regards to a comprehensive functional theory for Maxwell equations in Lipschitz domains. Many results are known in this field and we refer e.g., to [1],
[10], [11], [12], [13], [20], [21].
The main concern of our research is the construction of a suitable functional setting for non-homogeneous Dirichlet and Neumann problems for time-harmonic Maxwell equations, i.e., curl curl u - k 2 u
=0
D
u x n = g or curl u x n = g'
aD
where D is a Lipschitz-continuous bounded domain, n denotes the outer normal to D, k the wave number, u either the magnetic or the electric field, and g, g' need to be properly chosen. More precisely, we characterize the space of tangential trace (u H u x n) for H( curl, D) as well as more and less regular fields under the assumption that D is a bounded domain with Lipschitz continuous boundary. This will be made precise in the next sections. This work is mainly inspired by [7] and [2] and we aim to extend (in a suitable way) the results contained in these papers. More precisely, we do not succeed in writing a completely general theory, but we present some extensions of the known results and we discuss some open problem. We consider then polyhedral domains. The theory is deduced from the one developed for Lipschitz domains and the results presented in [4], [5] are C. Carstensen et al. (eds.), Computational Electromagnetics © Springer-Verlag Berlin Heidelberg 2003
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Annalisa Buffa
reinterpreted under this point of view. In the case of piecewise regular domains, on one hand the theory should be easier, but on the other hand one expects to have more "explicit" informations. Here we characterize some of the spaces introduced for Lipschitz domains in terms of face by face regularity plus compatibility condition at the edges (i.e., "Ii la Grisvard" [15], [16]). The trace theorems we present here have a direct impact on the application and, more precisely, they are important to properly formulate integral equations for Maxwell equations and to study their approximation by boundary elements. Some pioneering works in this direction are [6], [17], [3], [8].
1 Preliminaries Before stating trace theorems for spaces related to Maxwell equations, we need to define some Sobolev spaces and some differential operator acting on them. We refer to [4, 5, 2] and to [7] for more detail. 1.1 Functional spaces
We denote by D( t.?)3 the space of the 3D vector fields defined as C;?';mp (]R3 ) 1[2. Let D C ]R3 be a bounded Lipschitz-continuous domain in ]R3. We denote by r its boundary
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