Riesz decomposition for super-polyharmonic functions in the punctured unit ball

2014 ◽  
Vol 51 (1) ◽  
pp. 1-16
Author(s):  
Toshihide Futamura ◽  
Yoshihiro Mizuta ◽  
Takao Ohno

We consider a Riesz decomposition theorem for super-polyharmonic functions satisfying certain growth condition on surface integrals in the punctured unit ball. We give a condition that super-polyharmonic functions u have the bound \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$u\left( x \right) = O\left( {\mathcal{R}_2 \left( x \right)} \right),$$ \end{document} where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{R}_2$$ \end{document} denotes the fundamental solution for −Δu in ℝn.

2014 ◽  
Vol 66 (5) ◽  
pp. 1110-1142
Author(s):  
Dong Li ◽  
Guixiang Xu ◽  
Xiaoyi Zhang

AbstractWe consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator and give a robust algorithm to prove sharp L1 → L∞ dispersive estimates. We showcase the analysis in dimensions n = 5, 7. As an application, we obtain global well–posedness and scattering for defocusing energy-critical NLS on with Dirichlet boundary condition and radial data in these dimensions.


1980 ◽  
Vol 173 (2) ◽  
pp. 105-109
Author(s):  
Myron Goldstein ◽  
Wellington H. Ow

2011 ◽  
Vol 83 (3) ◽  
pp. 810-810
Author(s):  
Joel M. Cohen ◽  
Flavia Colonna ◽  
David Singman

2008 ◽  
Vol 38 (1) ◽  
pp. 37-50
Author(s):  
I. Bajunaid ◽  
J. M. Cohen ◽  
F. Colonna ◽  
D. Singman

2014 ◽  
Vol 44 (2) ◽  
pp. 235-245
Author(s):  
Toshihide Futamura ◽  
Yoshihiro Mizuta ◽  
Takao Ohno

2007 ◽  
Vol 75 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Joel M. Cohen ◽  
Flavia Colonna ◽  
David Singman

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