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An energy functional assigns an integral energy to a function. Minimization of a suitable energy is the basis of the Dirichlet principle. For the modified Helmholtz equation, a natural example is

Put $h=u-\varphi -C$. Then $h=0$ on $S$, and the constant does not change a gradient, so $\nabla u=\nabla \varphi +\nabla h$. Expand the unnormalized Dirichlet energy:The cross term is zero by Green's first identity:Thus the useful stronger identity isEquality holds exactly when $h$ has zero gradient. It is then constant on every connected component and zero on that component’s boundary, hence $h=0$. Therefore equality holds precisely when $u=\varphi +C$ throughout $V$. This is the Dirichlet principle; an additive boundary constant does not alter the minimizing energy. The energy here omits the optional factor $1/2$, which does not affect minimizers or the inequality.

Green's first identity and the boundary condition giveThus $\nabla u=0$, so $u$ is constant on each connected component of $V$; its zero boundary value forces $\boxed{u=0}$.

Since $w-v=0$ on $S$ and ${\nabla }^{2}v=0$, another application of Green's first identity givesConsequentlyExpanding $\nabla w=\nabla v+\nabla (w-v)$ and using the vanishing cross term yields the Dirichlet principle

For a spherically symmetric function $\Phi (r)$, the Laplacian in spherical coordinates isIntegrating from zero to $r$ and using regularity at the origin gives

For the stated density, integration and the boundary condition $\Phi (a)=0$ giveThe two pieces and their first derivatives agree at $r=b$. If another solution existed, its difference from $\Phi$ would be harmonic with zero boundary data, so part (a) proves Uniqueness of the Dirichlet problem.

On the shell $U(b,a)$, normalize the outer radial part to obtain the harmonic functionBy the Dirichlet principle, this function minimizes the energy among all $w$ with the same boundary values. Sinceits energy isTherefore

Let $v$ be the voltage with $v(a)=1$, $v(b)=0$, and harmonic values at every other vertex. For any other admissible $f$, write $f=v+g$, where $g(a)=g(b)=0$. Its discrete Dirichlet energy expands asDiscrete summation by parts turns the cross term intobecause $v$ is harmonic in the interior and $g$ vanishes at the boundary. Thus $v$ minimizes the energy. Under a unit voltage drop, its energy equals the total current from $a$ to $b$, namely the effective conductance $1/R_{\mathrm{eff}}(a,b)$. Therefore the Dirichlet principle gives

The identity follows immediately from the product rule:If ${\varphi }_1,{\varphi }_2$ have the same boundary data, put $\psi ={\varphi }_1-{\varphi }_2$. Integrating the identity with $\varphi =\psi$ givesbecause the volume equation and boundary term vanish. Positivity of $\kappa$ makes $\psi$ constant, and its boundary value makes it zero, proving uniqueness.

For any $w=\varphi +\psi$ with $\psi =0$ on the boundary, expansion givesbecause the cross term vanishes by the same integration by parts. This proves the inequality and the Dirichlet principle.

Since $q=-\kappa \nabla T$ and $\nabla \cdot q=H$, the divergence theorem givesSpherical symmetry gives $(r^2\kappa T^{\prime }{)}^{\prime }=-r^{2}H$. Regularity at zero and decay at infinity yieldFor $\alpha \le -1$, the exterior solution cannot tend to zero: it is logarithmic at $-1$ and grows below it.

On the exterior domain, the energy-minimizing solution with boundary value one is $\varphi =r^{-\alpha -1}$. Applying the Dirichlet principle to any admissible $w$ gives