the direct method of the calculus of variations
If a PDE is secretly the Euler-Lagrange equation of an energy, there are two ways to attack it. The old way: solve the differential equation directly. The direct method flips this around — instead of solving the PDE, prove that the energy actually attains its minimum, and then the minimiser automatically solves the equation. It is 'direct' because it goes straight for a minimiser without ever touching the differential equation along the way.
The logic copies the most basic fact of calculus — a continuous function on a closed bounded interval attains its minimum — and lifts it to infinitely many dimensions, where it is no longer free. You take a minimising sequence: functions u_k whose energies E[u_k] decrease toward the infimum. You want to extract a limit u that is itself a minimiser. Two ingredients make this work. First, coercivity: the energy must grow without bound as the function grows, E[u] tends to infinity as the norm of u tends to infinity, so the minimising sequence stays in a bounded set and (by weak compactness of the reflexive Sobolev space) has a weakly convergent subsequence u_k tending weakly to some u. Second, weak lower semicontinuity: the energy of the limit must not exceed the limiting energy, E[u] is not greater than the liminf of E[u_k]. Put together: E[u] is not greater than liminf E[u_k] = the infimum, so u attains the minimum. Coercivity catches a candidate; lower semicontinuity certifies it.
This is the existence engine of modern PDE. It proves solutions exist for enormous classes of elliptic problems — including nonlinear ones — without any explicit formula and without first establishing regularity, which is then recovered afterward by elliptic regularity theory. The crucial subtlety is that the limit is only WEAK, and weak limits are treacherous: a weakly convergent sequence need not converge pointwise, energies can leak away in the limit, and that is precisely why lower semicontinuity (rather than full continuity) is the right and necessary hypothesis. Convexity of the Lagrangian in the gradient is the standard guarantee that lower semicontinuity holds.
To prove - Laplacian u = f has a solution, do not solve it; minimise E[u] = the integral of (one-half |grad u|^2 minus f u) over the space of functions vanishing on the boundary. The Poincare inequality makes E coercive (it grows with the size of u), and convexity of |grad u|^2 makes E weakly lower semicontinuous. So a minimiser exists, and its Euler-Lagrange equation is exactly - Laplacian u = f. Existence is won without ever writing down a solution.
Coercivity corrals a minimising sequence; lower semicontinuity lets its weak limit inherit the minimum.
The method proves a minimiser exists but does NOT, by itself, prove it is smooth — the minimiser arrives only as a weak Sobolev function, and you need separate elliptic regularity to upgrade it to a classical solution. Also, coercivity and weak lower semicontinuity are both essential: drop either and the infimum may simply not be attained (a minimising sequence can escape to infinity or oscillate without converging).