Resumo

RAMIREZ, JORGE M.. Green's Functions for Sturm-Liouville Problems on Directed Tree Graphs. Rev.colomb.mat. [online]. 2012, vol.46, n.1, pp.15-25. ISSN 0034-7426.

Let Γ be geometric tree graph with m edges and consider the second order Sturm-Liouville operator L[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 continuous on each edge, and p strictly positive on Γ. The problem is to find a solution f:Γ → R to the problem L[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 \l₁,…,\l_2m acting on the space of admissible functions. A novel formula is given for the Green's function G:Γ\times Γ → R associated to this problem. Namely, the solution to the semi-homogenous problem L[f] = h, \l_i[f] =0 for i=1,…,2m is given by f(x) = \int_Γ G(x,y) h(y)\,dρ.

Palavras-chave : Problema Sturm-Liouville en grafo; función de Green.

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