论文标题
紧凑的黎曼表面上平均场方程的边界值问题
Boundary value problem for the mean field equation on a compact Riemann surface
论文作者
论文摘要
令$（σ，g）$为平滑边界$ \partialς$，$Δ_g$是Laplace-Beltrami操作员，而$ h $是一个积极的平滑功能。使用Djadli-Malchiodi（2006）和Djadli（2008）引入的Min-Max方案，我们证明，如果$σ$不可扣除，那么对于任何$ρ\ in（8kπ，8（k+1）π） $ \ left \ {\ oken {array} {lll}Δ_gu =ρ\ frac {he^u} {\int_σhe^udv_g}＆{\ rm in} 解决方案。这概括了欧几里得领域中丁 - 乔斯特·瓦（Ding-Jost-Li-Wang（1999）和Chen-lin（2003）的早期存在结果。 另外，我们考虑相应的Neumann边界价值问题。如果$ h $是一个积极的光滑功能，则对于任何$ρ\ in（4kπ，4（k+1）π）$，带有$ k \ in \ mathbb {n}^\ ast $，均值field方程$ $ \ left \ left \ left {\ begin u =ρ\ left（\ frac {he^u} {\int_σhe^udv_g} - \ frac {1} {|σ|} \ right）＆{\ rm in}＆之一on}＆\partialς\ end {array} \ right。$$有一个解决方案，其中$ \ mathbf {v} $表示$ \partialς$上的单位正常向外向量。请注意，在这种情况下，我们不需要表面是不可摘除的。
Let $(Σ,g)$ be a compact Riemann surface with smooth boundary $\partialΣ$, $Δ_g$ be the Laplace-Beltrami operator, and $h$ be a positive smooth function. Using a min-max scheme introduced by Djadli-Malchiodi (2006) and Djadli (2008), we prove that if $Σ$ is non-contractible, then for any $ρ\in(8kπ,8(k+1)π)$ with $k\in\mathbb{N}^\ast$, the mean field equation $$\left\{\begin{array}{lll} Δ_g u=ρ\frac{he^u}{\int_Σhe^udv_g}&{\rm in}&Σ\\[1.5ex] u=0&{\rm on}&\partialΣ\end{array}\right.$$ has a solution. This generalizes earlier existence results of Ding-Jost-Li-Wang (1999) and Chen-Lin (2003) in the Euclidean domain. Also we consider the corresponding Neumann boundary value problem. If $h$ is a positive smooth function, then for any $ρ\in(4kπ,4(k+1)π)$ with $k\in\mathbb{N}^\ast$, the mean field equation $$\left\{\begin{array}{lll} Δ_g u=ρ\left(\frac{he^u}{\int_Σhe^udv_g}-\frac{1}{|Σ|}\right)&{\rm in}&Σ\\[1.5ex] \partial u/\partial{\mathbf{v}}=0&{\rm on}&\partialΣ\end{array}\right.$$ has a solution, where $\mathbf{v}$ denotes the unit normal outward vector on $\partialΣ$. Note that in this case we do not require the surface to be non-contractible.
