论文标题
liouville方程式的刚性结果
Rigidity results on Liouville equation
论文作者
论文摘要
我们提供了从liouville方程$$的界限的解决方案的完整分类 - ΔU= e^{2U} \ quad \ mbox {in} \ quad {\ quad {\ sathbf {r}}}^2。$ $ $ $更广泛地，class $ n：= \ limsup ftity fty fty ftinity，更广泛地解决方案。 u（z）/\ log | z |：= k（u）<\ infty \} $$。结果，我们获得了五个刚性结果。首先，$ k（u）$只能采用离散值：$ k = -2 $或$ 2K $是一个非负整数。其次，$ u \ to- \ to infty $ as $ z \ to \ infty $，并且仅当$ u $在某个点上是径向的。第三，如果$ u $相对于$ x $和$ y $ axes和$ u_x <0，\; u_y <0 $在第一个象限中，$ u $是径向对称的。第四，如果$ u $是凹面并且从上方界限，则$ u $是一维的。第五，如果$ u $从上方界限，而$ {\ mathbf {r}}^2 $的直径与公制$ e^{2u}δ$是$π$，其中$Δ$是欧几里得公制，那么$ u $是$ u $，则是点宽度或一级或一级。 此外，我们在更高维度的liouville方程式上扩展了凹度刚度。
We give a complete classification of solutions bounded from above of the Liouville equation $$-Δu=e^{2u}\quad\mbox{in}\quad {\mathbf{R}}^2.$$ More generally, solutions in the class $$N:=\{ u:\limsup_{z\to\infty} u(z)/\log|z|:=k(u)<\infty\}$$ are described. As a consequence, we obtain five rigidity results. First, $k(u)$ can take only a discrete set of values: either $k=-2$, or $2k$ is a non-negative integer. Second, $u\to-\infty$ as $z\to\infty$, if and only if $u$ is radial about some point. Third, if $u$ is symmetric with respect to $x$ and $y$ axes and $u_x<0,\; u_y<0$ in the first quadrant then $u$ is radially symmetric. Fourth, if $u$ is concave and bounded from above, then $u$ is one-dimensional. Fifth, if $u$ is bounded from above, and the diameter of ${\mathbf{R}}^2$ with the metric $e^{2u}δ$ is $π$, where $δ$ is the Euclidean metric, then $u$ is either radial about a point or one-dimensional. In addition, we extend the concavity rigidity result on Liouville equation in higher dimensions.
