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Injections de type Sobolev. (Injections of Sobolev type.). (French) Zbl 0602.46031

We consider some open subsets \(\Omega \hookrightarrow {\mathbb{R}}^ 3\) with radial symmetry and a degeneracy at the boundary. We consider some Sobolev spaces of radially symmetric functions defined in the sequence of open subsets. We prove the following results: \[ W_ 0^{1,p}(\Omega,\quad axisymmetric)\subset L^ q(\Omega,\quad axisymmetric)\;if\;\frac{1}{q}>\frac{1}{p}-\frac{1}{\alpha} \] and \[ W_ 0^{1,p}(\Omega,\quad axisymmetric)\not\subset L^ q(\Omega,\quad axisymmetric)\;if\;\frac{1}{q}<\frac{1}{p}-\frac{1}{\beta} \] where \(\alpha\) and \(\beta\) depend on the open subset form and satisfy: \(2\leq \alpha \leq \beta \leq 3\) (generally \(\beta <3)\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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