Liouville theorems for self-similar solutions of heat flows

• Volume: 011, Issue: 1, page 207-221
• ISSN: 1435-9855

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Abstract

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Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere on $N$ is a harmonic map from $\left({ℝ}^{m},{e}^{{|x|}^{2}/2\left(m-2\right)}/d{s}_{0}^{2}\right)$ to $N$ ($m\ge 3$) with finite energy ([LnW]). Here $ds{2}_{0}$ is the Euclidean metric in ${ℝ}^{m}$. Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$. We also derive gradient estimates and Liouville theorems for positive quasi-harmonic functions.

How to cite

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Li, Jiayu, and Wang, Meng. "Liouville theorems for self-similar solutions of heat flows." Journal of the European Mathematical Society 011.1 (2009): 207-221. <http://eudml.org/doc/277485>.

@article{Li2009,
abstract = {Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere on $N$ is a harmonic map from $(\mathbb \{R\}^m,e^\{|x|^2/2(m-2)\}/ds^2_0)$ to $N$ ($m\ge 3$) with finite energy ([LnW]). Here $ds2^_0$ is the Euclidean metric in $\mathbb \{R\}^m$. Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$. We also derive gradient estimates and Liouville theorems for positive quasi-harmonic functions.},
author = {Li, Jiayu, Wang, Meng},
journal = {Journal of the European Mathematical Society},
keywords = {harmonic sphere; self-similar solution; quasi-harmonic sphere; heat flow; harmonic sphere; self-similar solution; quasi-harmonic sphere; heat flow},
language = {eng},
number = {1},
pages = {207-221},
publisher = {European Mathematical Society Publishing House},
title = {Liouville theorems for self-similar solutions of heat flows},
url = {http://eudml.org/doc/277485},
volume = {011},
year = {2009},
}

TY - JOUR
AU - Li, Jiayu
AU - Wang, Meng
TI - Liouville theorems for self-similar solutions of heat flows
JO - Journal of the European Mathematical Society
PY - 2009
PB - European Mathematical Society Publishing House
VL - 011
IS - 1
SP - 207
EP - 221
AB - Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere on $N$ is a harmonic map from $(\mathbb {R}^m,e^{|x|^2/2(m-2)}/ds^2_0)$ to $N$ ($m\ge 3$) with finite energy ([LnW]). Here $ds2^_0$ is the Euclidean metric in $\mathbb {R}^m$. Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$. We also derive gradient estimates and Liouville theorems for positive quasi-harmonic functions.
LA - eng
KW - harmonic sphere; self-similar solution; quasi-harmonic sphere; heat flow; harmonic sphere; self-similar solution; quasi-harmonic sphere; heat flow
UR - http://eudml.org/doc/277485
ER -

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