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A Y-junction satisfies Kirchhoff flux balance

Example

Take three unit edges oriented from a central vertex v to leaves a,b,c, with unit diffusivity and prescribed leaf values u(a)=2, u(b)=0, and u(c)=1. Define on the three edge coordinates u1(x)=1+x, u2(x)=1−x, and u3(x)=1 for 0≤x≤1. The common central trace is 1, and the total outward diffusive flux at v is zero.

The arrows give the coordinate orientations from the central vertex; the drawing is schematic and the proof uses the exact endpoint values below.

\begin{center} \begin{tikzpicture}[>=stealth, every node/.style={font=\small}] \node[circle,draw] (v) at (0,0) {v:1}; \node[circle,draw] (a) at (-2,1.4) {a:2}; \node[circle,draw] (b) at (2,1.4) {b:0}; \node[circle,draw] (c) at (0,-2) {c:1}; \draw[->] (v) -- node[above left] {u1(x)=1+x} (a); \draw[->] (v) -- node[above right] {u2(x)=1−x} (b); \draw[->] (v) -- node[right] {u3(x)=1} (c); \end{tikzpicture} \end{center}

Facts & Assumptions

Given: The three unit intervals with a common central endpoint, their orientation from v to the leaves, unit diffusivity, the displayed edge functions, and zero interior load.

[F1]

A metric graph has edgewise interval coordinates and common endpoint traces at each vertex (Anchored finite metric graph and its H1 energy space).

[F2]

For edgewise H2 functions, the signed endpoint terms in Green's identity group by common vertex trace and outward flux; zero test traces remove Dirichlet endpoints (Green identity and Kirchhoff flux cancellation on a metric graph).

[A1]

The general endpoint-representative interface assumes the Axiom of Choice (The Axiom of Choice, One-dimensional W1,p functions have unique absolutely continuous representatives).

Verification

Given: The oriented unit Y-graph and the three edge functions in the Example.

1.1A1F1givenalgebra

At the central endpoint x=0, all three values equal 1. At the leaves x=1, the values are 2,0,1, respectively. Hence the functions define a continuous metric-graph function with the specified Dirichlet data.

1.2F2givenalgebra

The outward derivative at the tail x=0 is −ui′(0). The three values are −1,+1,0, whose sum is 0. Each edge has ui′′=0. Therefore the edge equation has zero load and the free central vertex satisfies Kirchhoff balance.

2.1F2step 1.2given∎

For any test function with zero trace at the three leaves, [F2] shows that the edge bilinear form has only the central boundary contribution, which is its test trace times the sum −1+1+0=0. Thus the displayed function is also a weak harmonic field for these Dirichlet data.

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