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Sobolev spaces; Poincaré’s inequality; existence of intermediate derivates
Sobolev's original definition of his spaces $L^{m,p}(\Omega)$ is revisited. It only assumed that $\Omega\subseteq\Bbb R^n$ is a domain. With elementary methods, essentially based on Poincare's inequality for balls (or cubes), the existence of intermediate derivates of functions $u\in L^{m,p}(\Omega)$ with respect to appropriate norms, and equivalence of these norms is proved.
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