The effect of curvature on the best constant in the Hardy-Sobolev inequalities

@inproceedings{Ghoussoub2004TheEO,
  title={The effect of curvature on the best constant in the Hardy-Sobolev inequalities},
  author={Nassif Ghoussoub and F. Robert},
  year={2004}
}
when 0 < s < 2, 2 := 2∗(s) = 2(n−s) n−2 , and when 0 is on the boundary ∂Ω. This question is closely related to the geometry of ∂Ω, as we extend here the main result obtained in [15] by proving that at least in dimension n ≥ 4, the negativity of the mean curvature of ∂Ω at 0 is sufficient to ensure the attainability of μs(Ω). Key ingredients in our proof are the identification of symmetries enjoyed by the extremal functions correrresponding to the best constant in half-space, as well as a fine… CONTINUE READING

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