Attainability of the best Sobolev constant in a ball

  title={Attainability of the best Sobolev constant in a ball},
  author={Norisuke Ioku},
  journal={Mathematische Annalen},
  • Norisuke Ioku
  • Published 30 June 2018
  • Mathematics
  • Mathematische Annalen
The best constant in the Sobolev inequality in the whole space is attained by the Aubin–Talenti function; however, this does not happen in bounded domains because of the break down of the dilation invariance. In this paper, we investigate a new scale invariant form of the Sobolev inequality in a ball and show that its best constant is attained by functions of the Aubin–Talenti type. Generalization to the Caffarelli–Kohn–Nirenberg inequality in a ball is also discussed. 
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