论文标题
某些部分偏微分方程的组不变解决方案
Group invariant solutions of certain partial differential equations
论文作者
论文摘要
让$ m $成为一个完整的riemannian歧管，$ g $一个$ m $的谎言子组在$ M上自由地作用。 } {\ left \ vert \ nabla u \ right \ vert} \ nabla u \ right） c^{0} \ left（\ partial \overlineΩ\ right）$ a $ g- $不变函数。这个家庭中包含两个经典的PDE：$ p- $ laplacian $（a（s）= s^{p-1}，$ $ $ $ p> 1）$和最小的表面方程$（a（s）= s/\ sqrt {1+s^{2}}）。在有界域上无界域上的dirichlet问题。
Let $M$ be a complete Riemannian manifold and $G$ a Lie subgroup of the isometry group of $M$ acting freely and properly on $M.$ We study the Dirichlet Problem \begin{align*} \operatorname{div}\left( \frac{a\left( \left\Vert \nabla u\right\Vert \right) }{\left\Vert \nabla u\right\Vert }\nabla u\right) & =0\text{ in }Ω\\ u|\partialΩ& =φ\end{align*} where $Ω$ is a $G-$invariant domain of $C^{2,α}$ class in $M$ and $φ\in C^{0}\left( \partial\overlineΩ\right) $ a $G-$invariant function. Two classical PDE's are included in this family: the $p-$Laplacian $(a(s)=s^{p-1},$ $p>1)$ and the minimal surface equation $(a(s)=s/\sqrt {1+s^{2}}).$ Our motivation is to present a method in studying $G$-invariant solutions for noncompact Lie groups which allows the reduction of the Dirichlet problem on unbounded domains to one on bounded domains.
