TY - JOUR
TI - An extension of the Bianchi-Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions and applications
DO - https://doi.org/doi:10.7282/T3377BXB
PY - 2016
AB - The main result of this dissertation is an extension of a stability estimate of the Sobolev Inequality established by Bianchi and Egnell in cite{BiEg}. Bianchi and Egnell's Stability Estimate answers the question raised by H. Brezis and E. H. Lieb in cite{BrLi}: ``Is there a natural way to bound $| abla varphi |_2^2 - C_N^2 | varphi |_frac{2N}{N-2}^2$ from below in terms of the `distance' of $varphi$ from the manifold of optimizers in the Sobolev Inequality?'' Establishing stability estimates - also known as quantitative versions of sharp inequalities - of other forms of the Sobolev Inequality, as well as other inequalities, is an active topic. See cite{CiFu}, cite{DoTo}, and cite{FiMa}, for stability estimates involving Sobolev inequalities and cite{CaFi}, cite{DoTo}, and cite{FuMa} for stability estimates on other inequalities. In this dissertation, we extend Bianchi and Egnell's Stability Estimate to a Sobolev Inequality for ``continuous dimensions.'' Bakry, Gentil, and Ledoux have recently proved a sharp extension of the Sobolev Inequality for functions on $mathbb{R}_+ imes mathbb{R}^n$, which can be considered as an extension to ``continuous dimensions.'' V. H. Nguyen determined all cases of equality. The dissertation extends the Bianchi-Egnell stability analysis for the Sobolev Inequality to this ``continuous dimensional'' generalization. The secondary result of this dissertation is a sketch of the proof of an extension of a stability estimate of a single case of a sharp Gagliardo-Nirenberg inequality to a whole family of Gagliardo-Nirenberg inequalities, whose sharp constants and extremals were calculated by Del Pino and Dolbeault in cite{DeDo}. The original stability estimate for the Gagliardo-Nirenberg inequality was stated and proved by E. Carlen and A. Figalli in cite{CaFi}. The proof for its extension to the entire class of sharp Gagliardo-Nirenberg inequalities of Del Pino and Dolbeault is a direct application of the extension of the Bianchi-Egnell Stability Estimate to Bakry, Gentil, and Ledoux's extension of the Sobolev Inequality to continuous dimensions.
KW - Mathematics
KW - Functional analysis
LA - eng
ER -