A special class of nonlinear hypoelliptic equations on spheres View Full Text


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Article Info

DATE

2017-03-04

AUTHORS

Mayukh Mukherjee

ABSTRACT

We study nonlinear second-order equations of the form -Δu+X2u+iλXu+σu=K|u|p-1u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta u + X^2 u + i\lambda X u + \sigma u = K|u|^{p - 1}u$$\end{document} on Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^n$$\end{document} with the usual round metric, where X is a Killing field. The case of principal interest is when X has length ≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\le }1$$\end{document}, which leads to hypoelliptic operators with loss of one derivative. After establishing existence of solutions via variational methods, we carry out the subelliptic analysis on Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^n$$\end{document} utilizing their homogeneous coset space properties. Writing Lα=Δ-X2-iαX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\alpha = \Delta - X^2 - i\alpha X$$\end{document}, we establish the optimal range of p such that the embedding D((-Lα)1/2)↪Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}((-L_\alpha )^{1/2}) \hookrightarrow L^p$$\end{document} is compact, which gives sharp versions of results in Taylor (Houst J Math 42(1):143–165, 2016) for all dimensions. Such subelliptic phenomena have no parallel in the setting of flat spaces. More... »

PAGES

15

References to SciGraph publications

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    http://scigraph.springernature.com/pub.10.1007/s00030-017-0439-9

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    http://dx.doi.org/10.1007/s00030-017-0439-9

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