We prove a quantitative Sobolev inequality in cones of Bianchi-Egnell type, which implies a stability property. Our result holds for any cone as long as the minimizers of the Sobolev quotient are nondegenerate. When the minimizers are the classical bubbles we have more precise results. Finally, we show that local estimates are not enough to get the optimal constant for the quantitative Sobolev inequality.
Stability for the Sobolev inequality in cones / G. Ciraolo, F. Pacella, C.C. Polvara. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 433:(2025 Jul 15), pp. 113325.1-113325.30. [10.1016/j.jde.2025.113325]
Stability for the Sobolev inequality in cones
G. CiraoloPrimo
;C.C. PolvaraUltimo
2025
Abstract
We prove a quantitative Sobolev inequality in cones of Bianchi-Egnell type, which implies a stability property. Our result holds for any cone as long as the minimizers of the Sobolev quotient are nondegenerate. When the minimizers are the classical bubbles we have more precise results. Finally, we show that local estimates are not enough to get the optimal constant for the quantitative Sobolev inequality.
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