Dirichlet's principle

Remarks

Proof idea: for $w \in \mathcal{A}$ put $v = w - u$, which vanishes on $\partial\Omega$. Then $E[w] = E[u] + \int_\Omega \nabla u \cdot \nabla v \, dx + E[v]$, and integration by parts gives $\int_\Omega \nabla u \cdot \nabla v \, dx = -\int_\Omega v \, \Delta u \, dx$. If $u$ is harmonic, the middle term is $0$, so $E[w] \ge E[u]$. If $u$ minimizes, the derivative of $t \mapsto E[u + tv]$ at $t = 0$ is $0$ for all such $v$, which forces $\Delta u = 0$. The theorem does not say that a minimizer exists in $\mathcal{A}$; Weierstrass pointed out that this needs proof, and it was later justified by Hilbert. Existence can fail even for continuous $g$: on the unit disc, $g(\theta) = \sum_{k \ge 1} k^{-2} \cos(2^k \theta)$ is continuous, but its harmonic extension $\sum_k k^{-2} r^{2^k} \cos(2^k\theta)$ has energy $\frac{\pi}{2}\sum_k 2^k k^{-4} = \infty$.

Going deeper

For a student

Dirichlet's principle turns the Dirichlet problem into a minimization problem. In modern form, one minimizes the energy over the Sobolev space $H^1$ with given trace. The minimizer exists by the Hilbert space projection theorem, and it is a weak solution of Laplace's equation. Elliptic regularity then shows that the weak solution is a classical harmonic function. The same variational idea gives existence for many other elliptic equations.