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A note on the Sobolev inequality. (English) Zbl 0755.46014
Let $$M$$ be the set of all functions $$\varphi\in W_ 0^{1,2}$$ for which equality holds in the Sobolev “inequality”, i.e., $$\|\nabla\varphi\|_{L_ 2}=S\|\varphi\|_{L_{2N/(N- 2)}}$$ ($$N\geq 3$$, $$S=$$ best Sobolev constant). The authors show then that the ratio $R(\varphi)={\|\nabla\varphi\|_{L_ 2}-S\|\varphi \|_{L_{2N/(N-2)}} \over \text{dist}(\varphi,M)} \qquad (\varphi\in W_ 0^{1,2})$ is bounded away from zero.

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
##### Keywords:
Sobolev inequality; best Sobolev constant
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##### References:
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