A note on strong-form stability for the Sobolev inequality

By Robin Neumayer. Calculus of Variations and Partial Differential Equations, 2019

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In this note, we establish a strong form of the quantitive Sobolev inequality in Euclidean space for p∈(1,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in (1,n)$$\end{document}. Given any function u∈W˙1,p(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u \in \dot{W}^{1,p}({\mathbb {R}}^n)$$\end{document}, the gap in the Sobolev inequality controls ‖∇u-∇v‖p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \nabla u -\nabla v\Vert _{p}$$\end{document}, where v is an extremal function for the Sobolev inequality.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fa6eb13a62c550615422671956274767a2a2eaf8

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