Bubbling Analysis at Free Boundary for Approximate H-surfaces with Tension Fields Bounded in

L^p

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}

Abstract

Given a sequence of approximate H-surfaces

{\{u_n\}}_{n\in \mathbb{N}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{u_n\}_{n\in \mathbb {N}}$$\end{document}

from a Riemann surface M to a compact Riemannian manifold N with free boundaries

u_n(\partial M)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$u_n(\partial M)$$\end{document}

lying on a given supporting submanifold

\mathcal{K}\subset N

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {K}\subset N$$\end{document}

, admitting uniform energy bound and tension fields uniformly bounded in

L^p(M,TN)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p(M, TN)$$\end{document}

for

p\ge 6/5

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p \ge {6}/{5}$$\end{document}

, we employ the doubling annulus Pohozaev-type identity to establish energy identity and no-neck property during bubbling process near free boundaries for

{\{u_n\}}_{n\in \mathbb{N}}.

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{u_n\}_{n\in \mathbb {N}}.$$\end{document}

Keywords

Approximate H-surfaces / Energy Identity / No Neck / Free boundary / 58E20 / 35J57

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