角点处毛细表面的径向极限
偏微分方程分析
2017-04-19 v1
摘要
考虑预设平均曲率方程 d i v ( ∇ f 1 + ∣ ∇ f ∣ 2 ) = 2 H ( x , f ) i n Ω ⊂ R 2 , {\rm div}\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^{2}}}\right)=2H(x,f) \ \ \ \ {\rm in} \ \ \Omega\subset R^{2}, div ( 1 + ∣∇ f ∣ 2 ∇ f ) = 2 H ( x , f ) in Ω ⊂ R 2 , f ∈ C 2 ( Ω ) f\in C^{2}(\Omega) f ∈ C 2 ( Ω ) Ω \Omega Ω O = ( 0 , 0 ) ∈ ∂ Ω {\cal O}=(0,0)\in\partial\Omega O = ( 0 , 0 ) ∈ ∂ Ω 2 α 2\alpha 2 α α ∈ ( 0 , π ) \alpha\in (0,\pi) α ∈ ( 0 , π ) sup x ∈ Ω ∣ f ( x ) ∣ \sup_{x\in\Omega} |f(x)| sup x ∈ Ω ∣ f ( x ) ∣ sup x ∈ Ω ∣ H ( x , f ( x ) ) ∣ \sup_{x\in\Omega} |H(x,f(x))| sup x ∈ Ω ∣ H ( x , f ( x )) ∣ α > π 2 \alpha>\frac{\pi}{2} α > 2 π f f f O {\cal O} O R f ( θ ) = lim r ↓ 0 f ( r cos ( θ ) , r sin ( θ ) ) Rf(\theta) = \lim_{r\downarrow 0} f(r\cos(\theta),r\sin(\theta)) R f ( θ ) = r ↓ 0 lim f ( r cos ( θ ) , r sin ( θ )) f f f ∂ Ω \partial\Omega ∂ Ω α ∈ ( π 4 , π 2 ] \alpha\in \left(\frac{\pi}{4},\frac{\pi}{2}\right] α ∈ ( 4 π , 2 π ] f f f ∂ Ω \partial\Omega ∂ Ω γ ( ⋅ ) \gamma(\cdot) γ ( ⋅ ) O {\cal O} O γ 2 \gamma_{2} γ 2 π − 2 α < γ 2 < 2 α . \pi-2\alpha < \gamma_{2} <2\alpha. π − 2 α < γ 2 < 2 α . f f f O {\cal O} O f f f ∂ Ω \partial\Omega ∂ Ω α ∈ ( 0 , π 2 ] \alpha\in \left(0,\frac{\pi}{2}\right] α ∈ ( 0 , 2 π ]
引用
@article{arxiv.1604.01836,
title = {Radial Limits of Capillary Surfaces at Corners},
author = {Mozhgan and Entekhabi and Kirk Lancaster},
journal= {arXiv preprint arXiv:1604.01836},
year = {2017}
}
备注
To be submitted to the Pacific Journal of Mathematics. arXiv admin note: text overlap with arXiv:1510.05288