Stability for the Sobolev inequality: Existence of a minimizer

  • Tobias König

    Goethe-Universität Frankfurt am Main, Frankfurt am Main, Germany
Stability for the Sobolev inequality: Existence of a minimizer cover
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Abstract

We prove that the quantitative stability inequality associated to Sobolev’s inequality, due to Bianchi and Egnell, admits a minimizer attaining the best constant for every . Our proof consists in an appropriate refinement of a classical strategy going back to Brezis and Lieb. As a crucial ingredient, we establish the strict inequality , which means that a sequence of two asymptotically non-interacting bubbles cannot be minimizing. Our arguments cover in fact the analogous stability inequality for the fractional Sobolev inequality for arbitrary fractional exponent and dimension .

Cite this article

Tobias König, Stability for the Sobolev inequality: Existence of a minimizer. J. Eur. Math. Soc. (2025), published online first

DOI 10.4171/JEMS/1582