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2024 | OriginalPaper | Chapter

2-D Minimal Surface Flow with the Oblique Condition and Translators

Authors : Li Ma, Yuxin Pan

Published in: New Trends in the Applications of Differential Equations in Sciences

Publisher: Springer Nature Switzerland

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Abstract

In this paper, we study evolved surfaces over convex planar domains following the minimal surface flow
$$\begin{aligned}u_{t}= div\left( \frac{Du}{\sqrt{1+|Du|^2}}\right) -H(x,Du).\end{aligned}$$
Here, we specify the angle of contact of the evolved surface to the boundary cylinder. The interesting question is to find translating solitons of the form \(u(x,t)=\omega t+w(x)\) where \(\omega \in \mathbb R\). Under an angle condition on the boundary, we can prove the a priori estimate holds true for the translating solitons (i.e., translator), which makes the solitons exist. Then, we can prove for suitable condition on the function H(x, p) that there is the global solution of the minimal surface flow. Finally, we show that once the translating soliton exists, the global solutions converge to such translator.

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next chapter Graphical Portraits of the Solutions of Binary First Order Nonlinear Ordinary Differential Equation Near Their Singular Point
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Metadata
Title
2-D Minimal Surface Flow with the Oblique Condition and Translators
Authors
Li Ma
Yuxin Pan
Copyright Year
2024
Book
New Trends in the Applications of Differential Equations in Sciences

Print ISBN: 978-3-031-53211-5

Electronic ISBN: 978-3-031-53212-2

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-53212-2_1

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