arXiv 1108.06212011-08-02math.CAmath-phmath.MP

有向树图上Sturm-Liouville问题的Green函数

Green's functions for Sturm-Liouville problems on directed tree graphs

Jorge M Ramirez

摘要：

设$Γ$是具有$m$边的几何树图，考虑二阶Sturm-Liouville算子$Ł[u]=(-pu')'+qu$作用于在所有$Γ$上连续且在每条边的内部连续可微的函数.假设函数$p$和$q$在每条边上是一致连续的，并且$p$在$Γ$上是严格正的。问题是在$Γ$的节点处具有$2m$附加条件的问题$Ł[f] = h$的解$f:Γ\to \R$。这些节点条件包括在内部节点的连续性，和跳跃条件的衍生物的$f$相对于一个积极的措施$ρ$。节点条件以作用于容许函数空间的线性泛函$ł_1,...,ł_{2m}$的形式给出。给出了与此问题相关的格林函数$G:Γ\times Γ\to \R$的新公式。也就是说，半齐次问题$Ł[f] = h$，$ł_i[f] =0$ for $i=1,...,2m$的解由$f(x) = \int_ΓG(x,y) h(y) \ud ρ$给出。

英文摘要：

Let $Γ$ be geometric tree graph with $m$ edges and consider the second order Sturm-Liouville operator $Ł[u]=(-pu')'+qu$ acting on functions that are continuous on all of $Γ$, and twice continuously differentiable in the interior of each edge. The functions $p$ and $q$ are assumed uniformly continuous on each edge, and $p$ strictly positive on $Γ$. The problem is to find a solution $f:Γ\to \R$ to the problem $Ł[f] = h$ with $2m$ additional conditions at the nodes of $Γ$. These node conditions include continuity at internal nodes, and jump conditions on the derivatives of $f$ with respect to a positive measure $ρ$. Node conditions are given in the form of linear functionals $ł_1,...,ł_{2m}$ acting on the space of admissible functions. A novel formula is given for the Green's function $G:Γ\times Γ\to \R$ associated to this problem. Namely, the solution to the semi-homogenous problem $Ł[f] = h$, $ł_i[f] =0$ for $i=1,...,2m$ is given by $f(x) = \int_ΓG(x,y) h(y) \ud ρ$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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