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\(H(\text{div,rot},\Omega)\) spaces in a plane polygonal domain. (Espaces \(H(\text{div,rot},\Omega)\) dans un polygône plan.) (French. Abridged English version) Zbl 0852.46034
Summary: For a plane polygonal domain \(\Omega\) with boundary \(\Gamma\) we compare the two spaces: \[ V= \{\vec u\in H^1(\Omega)^2;\;\vec u\cdot \vec n= 0\text{ on } \Gamma\} \] and \[ W= \{\vec u\in L^2(\Omega)^2; \text{ div } \vec u\in L^2(\Omega),\text{ rot } \vec u\in L^2(\Omega),\;\vec u\cdot \vec n= 0\text{ on } \Gamma\}. \] We prove that \(V= W\) iff \(\Omega\) is convex. Otherwise, \(V\) is a finite codimensional closed subspace of \(W\) and we give the dimension and a basis of its orthogonal.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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