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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