$2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II

Meiyue  Jiang; Juncheng Wei

doi:10.3934/dcds.2016.36.785

Article Contents

2016, Volume 36, Issue 2: 785-803. Doi: 10.3934/dcds.2016.36.785

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$2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II

Meiyue Jiang^1, and
Juncheng Wei^2,

1.
LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871
2.
Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

Received: March 31, 2014

Published: May 2017

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Abstract

The existence of $2\pi$-periodic positive solutions of the equation $$ u'' + u = \displaystyle{\frac{a(x)}{u^3}} $$ is studied, where $a$ is a positive smooth $2\pi$-periodic function. Under some non-degenerate conditions on $a$, the existence of $2\pi$-periodic solutions to the equation is established.

Keywords:

Self-similar solutions,
anisotropic affine curve shortening problem.

Mathematics Subject Classification: Primary: 34B15, 34B16; Secondary: 44J99.

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References

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