A note on the moving hyperplane method

A Bernstein Type Theorem For Self-similar Shrinkers

In this note, we prove that smooth self-shrinkers in $\Real^{n+1}$, that are entire graphs, are hyperplanes. Previously Ecker and Huisken showed that smooth self-shrinkers, that are entire graphs and have at most polynomial growth, are hyperplanes. The point of this note is that no growth assumption at infinity is needed.

关于 Sobolev 映射的移动标架与弱调和映射正则性的注记

A Note on Moving Frames along Sobolev Maps and the Regularity of Weakly Harmonic Maps

本文证明，对弱可微映射，梯度 Morrey 范数的小量条件保证拉回切丛被有限能量标架平凡化；对弱调和映射，BMO 半范数的小量条件保证完全正则性。

Global existence of bounded weak solutions to degenerate cross-diffusion equations in moving domain

The aim of this note is to present preliminary existence results for a system of cross-diffusion equations defined on a domain with moving boundaries, which model the evolution of the concentrations of different chemical species in a solid during a Chemical Vapor Deposition process. The system of equations, when the domain remains fixed over time, can be seen formally as a gradient flow system which can be analyzed using the boundedness by entropy method introduced by Burger and Jüngel. Preliminary existence results are presented in the case of a one-dimensional moving boundary domain.

A Remark on Gauge Transformations and the Moving Frame Method

In this note we give a shorter proof of recent regularity results by Riviere and Riviere-Struwe. We differ from the mentioned articles only in using the direct method of Helein's moving frame to construct a suitable gauge transformation. Though this is neither new nor surprising, it enables us to describe a proof of regularity using besides the duality of Hardy- and BMO-space only elementary arguments of calculus of variations and algebraic identities. Moreover, we remark that in order to prove Hildebrandt's conjecture one can avoid the Nash-Moser imbedding theorem. There are no new results pr…

分数阶 $p(x,\cdot)$-Laplacian 的极值原理与移动平面法

Maximum principles and moving planes method for the fractional $p(x,\cdot)$-Laplacian

本文针对含变指数分数阶 Laplacian 的非线性方程，首次建立极值原理并运用直接移动平面法证明了解的单调性，填补了变指数局部与非局部方程单调性研究的空白。