Dirichlet's principle
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Nature, when settling into equilibrium, tends to minimize energy — a stretched membrane sags into the shape that stores the least elastic energy, and that shape turns out to be harmonic. Dirichlet's principle turns this physical intuition into a method: among all functions with the given boundary values, the harmonic one is exactly the one that minimizes a certain energy. So you can find the solution of the Dirichlet problem not by solving the equation directly but by minimizing energy.
Precisely, define the Dirichlet energy of a function u as the integral over the region of the squared length of its gradient, the integral of |grad u|^2. Dirichlet's principle states that among all sufficiently smooth functions taking the prescribed values g on the boundary, the one that minimizes this energy is the solution of the Dirichlet problem — it is harmonic. The link runs through the calculus of variations: setting the first variation of the energy to zero (the Euler-Lagrange equation for this functional) gives exactly Laplacian u = 0. So harmonic functions are the energy minimizers, and minimizing energy is an alternative, variational characterization of solving Laplace's equation. For Poisson's equation the energy gains a source term and the minimizer solves Laplacian u = f.
This variational viewpoint is enormously powerful. It is the conceptual seed of the finite element method (minimize the energy over a finite-dimensional space of trial functions) and of the modern existence theory for elliptic equations (the direct method: take a minimizing sequence and extract a limit). But it carries a famous cautionary tale. Riemann used Dirichlet's principle as if a minimizer obviously exists; Weierstrass pointed out that an energy bounded below need not attain its minimum, so the principle was not rigorous as stated. It was only rescued decades later (by Hilbert and the direct method in the right function spaces). The lesson endures: existence of a minimizer is a real theorem requiring the right setting, not a free lunch.
To find the steady temperature on a plate with prescribed edge temperatures, replace 'solve Laplacian u = 0' with 'among all interior temperature profiles matching the edges, find the one of least Dirichlet energy'. Discretize that minimization over a mesh of simple basis functions and you have, essentially, the finite element method.
Solving Laplace's equation equals minimizing the gradient energy over admissible functions.
Weierstrass's critique is the famous caveat: an energy bounded below need not attain its minimum, so 'the minimizer exists' is a genuine theorem, not a triviality — it requires the right function space (today, Sobolev spaces). Riemann's original use was a gap later filled by Hilbert.