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