Well-posed weak elliptic diffusion on an anchored metric graph

The form $a$ is symmetric and bilinear. By Cauchy–Schwarz on the finite edge product, $\mathrm{\mid }a(w,v)\mathrm{\mid }\le {\kappa }_{\max }\mathrm{\parallel }{w}^{\mathrm{\prime }}{\mathrm{\parallel }}_H\mathrm{\parallel }{v}^{\mathrm{\prime }}{\mathrm{\parallel }}_H\le {\kappa }_{\max }\mathrm{\parallel }w{\mathrm{\parallel }}_{V_D}\mathrm{\parallel }v{\mathrm{\parallel }}_{V_D}\mathrm{.}$ Thus it is bounded with constant $M={\kappa }_{\max }$.

By [F2], $\mathrm{\parallel }v{\mathrm{\parallel }}_{V_D}^2=\mathrm{\parallel }v{\mathrm{\parallel }}_H^2+\mathrm{\parallel }{v}^{\mathrm{\prime }}{\mathrm{\parallel }}_H^2\le (1+L_{\mathrm{\Gamma }}^2)\mathrm{\parallel }{v}^{\mathrm{\prime }}{\mathrm{\parallel }}_H^2\mathrm{.}$ Consequently $a(v,v)\ge {\kappa }_{\min }\mathrm{\parallel }{v}^{\mathrm{\prime }}{\mathrm{\parallel }}_H^2\ge \frac{{\kappa }_{\min }}{1+L_{\mathrm{\Gamma }}^2}\mathrm{\parallel }v{\mathrm{\parallel }}_{V_D}^2\mathrm{.}$ The form is coercive with $\alpha ={\kappa }_{\min }\mathrm{/}(1+L_{\mathrm{\Gamma }}^2)>0$.

The load $F(v)={\sum }_e\int f_e{v}_e$ is linear and satisfies $\mathrm{\mid }F(v)\mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_H\mathrm{\parallel }v{\mathrm{\parallel }}_H\le L_{\mathrm{\Gamma }}\mathrm{\parallel }f{\mathrm{\parallel }}_H\mathrm{\parallel }{v}^{\mathrm{\prime }}{\mathrm{\parallel }}_H\le L_{\mathrm{\Gamma }}\mathrm{\parallel }f{\mathrm{\parallel }}_H\mathrm{\parallel }v{\mathrm{\parallel }}_{V_D}$ by [F5] and [F2]. Hence $F$ is bounded on $V_D$.

Apply [F3] to the real Hilbert space [F1], the bounded coercive form from steps 1.1–1.2, and the bounded functional from step 1.3. It gives exactly one $u\in V_D$ satisfying $a(u,v)=F(v)$ for all $v\in V_D$. If this $u$ is edgewise H2, [F6] and the weak equation give, for every $v\in V_D$, ${\sum }_e{\int }_0^{{\mathrm{\ell }}_e}(-{\kappa }_e{u}_e^{\mathrm{\prime }\mathrm{\prime }}-f_e)v_{e}d{x}_e+{\sum }_{z\mathrm{\notin }D}v(z){\sum }_{\iota \in \mathcal{I}(z)}{\kappa }_{e(\iota )}{\mathrm{\partial }}_{{\nu }_{\iota }}{u}_{e(\iota )}(z)=0.$

Fix an edge and take $v$ smooth with compact support in its interior and zero on every other edge. The vertex sum vanishes, so the integral of $-{\kappa }_e{u}_e^{\mathrm{\prime }\mathrm{\prime }}-f_e$ against every such test is zero. This residual is in $L^2(0,{\mathrm{\ell }}_e)$; density from [F7] and continuity of the pairing by [F5] imply $-{\kappa }_e{u}_e^{\mathrm{\prime }\mathrm{\prime }}=f_e$ almost everywhere on that edge. Now fix a free vertex $z$. Prescribe endpoint-incidence value $1$ at every incidence at $z$ and $0$ at incidences at every other vertex, and interpolate linearly on each edge. Both incidences of a loop at $z$ receive value $1$, so this defines a trace-matching test function in $V_D$. Substitution in step 2.1 and the interior equation just proved leaves only the flux sum at $z$, so it is zero. This proves the stated natural Kirchhoff condition.