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The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 868–901 | Cite as

On a New Harmonic Heat Flow with the Reverse Hölder Inequalities

  • Kazuhiro HorihataEmail author
Article
  • 74 Downloads

Abstract

This paper first proposes a new approximate scheme to construct a harmonic heat flow u between Q and \({\mathbb {S}}^D\)\(\subset \)\({\mathbb {R}}^{D+1}\) with Q\(=\)\((0,\infty )\)\(\times \)\({\mathbb {B}}^d\) and positive integers d and D: We agree that harmonic heat flow u means a solution of
$$\begin{aligned} \frac{\partial u}{\partial t} \, - \, \triangle u \, - \, | \nabla u|^2 u \; = \; 0. \end{aligned}$$
Its scheme is crucially given by
$$\begin{aligned} \frac{\partial u_\lambda }{\partial t} \, - \, \triangle u_\lambda \, + \, \lambda ^{1-\kappa } \bigl ( | u_\lambda |^2 \, - \, 1 \bigr ) u_\lambda \; = \; 0, \end{aligned}$$
where the unknown mapping \(u_\lambda \) is from Q to \({\mathbb {R}}^{D+1}\) with positive number \(\lambda \) and \(\kappa (t)\)\(=\)\(\arctan (t)/\pi \)\((0 \le t)\). The benefit to introduce a time-dependent parameter \(\lambda ^{1-\kappa }\) is readily to see
$$\begin{aligned} \int _{Q} \lambda ^{1-\kappa } ( | u_\lambda |^2 \, - \, 1 )^2 \, \mathrm{d}t \mathrm{d}x \; \le \; \frac{C}{\log \lambda } \end{aligned}$$
for some positive constant C independent of \(\lambda \). Next, making the best of it, we prove that a passing to the limits \(\lambda \nearrow \infty \) ( modulo sub-sequence of \(\lambda \) ) brings the existence of a harmonic heat flow into spheres with (i) a global energy inequality, (ii) a monotonicity for the scaled energy, (iii) a reverse Poincaré inequality. These inequalities (i), (ii) and (iii) improve the estimates on its singular set of a harmonic heat flow by Chen and Struwe (Math Z 201(1):83–103, 1989), i.e. I show that a singular set of the new harmonic heat flows into spheres has at most finite \((d-\epsilon _0)\)-dimensional Hausdorff measure with respect to the parabolic metric whereupon \(\epsilon _0\) is a small positive number. We finally prove that if the harmonic heat flows is a constant at the boundary, then \(u_\lambda (t)\) strongly converges to the constant \(t \nearrow \infty \) in \(H^{1,2} ({\mathbb {B}}^d;{\mathbb {S}}^{D+1})\). We call this the parabolic constancy theorem. We restrict ourselves a harmonic heat flow from the unit ball into a sphere to avoid confusion of notation. But it is readily seen that our results can be extended to it between compact Riemannian manifolds using a distance function combined with Nash’s imbedding theorem.

Keywords

Harmonic heat flow A monotonicity inequality A reverse Poincaré inequality 

Mathematics Subject Classification

35K10 35K55 58J35 

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.6-3, Aramaki, Aoba-kuSendaiJapan

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