论文标题
Weingarten类型方程的径向解决方案
Radial solutions for equations of Weingarten type
论文作者
论文摘要
在本文中，我们研究了由完全非线性PDE定义的线性Weingarten方程 $$ a \，\ mbox {div} \ frac {du} {\ sqrt {1+ | du |^2}}}+b \， \ frac {\ mbox {det} d^2u} {（1+ | du |^2）^2} = ϕ \ left（\ frac {1} {\ sqrt {\ sqrt {1+ | du |^2}}} \ right） c^1（[ - 1,1]）$和$ a，b \ in \ mathbb {r} $。当$ω$是一个小半径的磁盘时，我们接近径向溶液的存在，当PDE为椭圆类型时，给出了肯定的答案。在双曲线的情况下，我们表明不存在径向溶液，而在抛物线剂中，我们发现所有解决方案。最后，在椭圆情况下，我们证明了有关这种方程问题的唯一性和对称结果。
In this paper we study the linear Weingarten equation defined by the fully non-linear PDE $$a\, \mbox{div}\frac{Du}{\sqrt{1+|Du|^2}}+b\, \frac{\mbox{det}D^2u}{(1+|Du|^2)^2}=ϕ\left(\frac{1}{\sqrt{1+|Du|^2}}\right)$$ in a domain $Ω\subset\mathbb{R}^2$, where $ϕ\in C^1([-1,1])$ and $a,b\in\mathbb{R}$. We approach the existence of radial solutions when $Ω$ is a disk of small radius, giving an affirmative answer when the PDE is of elliptic type. In the hyperbolic case we show that no radial solution exists, while in the parabolic case we find explicitly all the solutions. Finally, in the elliptic case we prove uniqueness and symmetry results concerning the Dirichlet problem of such equation.
