the weak formulation
A PDE in its usual 'strong' form, like minus Laplacian u = f, demands that u have two honest derivatives at every point. The weak formulation is a clever rephrasing that asks for much less — it never differentiates u twice. The trick is to test the equation against arbitrary smooth probe functions and integrate by parts, shoving derivatives off u and onto the probe. What comes out is an equivalent statement that a far rougher u can satisfy.
Walk it through for minus Laplacian u = f on a domain U with u = 0 on the boundary. Multiply both sides by a test function v (smooth, vanishing on the boundary) and integrate over U: integral of (minus Laplacian u) v = integral of f v. Now integrate by parts on the left (Green's identity): integral of (minus Laplacian u) v = integral of grad u . grad v minus a boundary term, and the boundary term dies because v = 0 on the boundary. So the equation becomes: find u in H^1_0 such that integral of grad u . grad v = integral of f v, for every test function v in H^1_0. Notice what happened — u now appears with only ONE derivative, perfectly matched against v's one derivative, so u need only be in H^1, not C^2. This is the weak (or variational) formulation; the left side is a bilinear form a(u, v) and the right side a linear functional, so it reads a(u, v) = L(v) for all v.
Why is this not cheating? Because the two formulations are equivalent for smooth solutions (integration by parts is reversible), so a classical solution is always a weak solution — and the weak form simply admits more candidates. The payoff is enormous: the weak form lives entirely in the Hilbert space H^1_0, where the Lax-Milgram theorem guarantees a unique solution under mild conditions, and where the Galerkin method approximates it by finite-dimensional subspaces (the doorway to finite elements). Boundary conditions split into two kinds here: Dirichlet conditions are 'essential', built into the space H^1_0, while Neumann conditions are 'natural', emerging on their own from the boundary term you keep. The weak formulation is, in short, the move that converts a PDE into a problem in functional analysis.
For minus u'' = f on (0, 1) with u(0) = u(1) = 0, multiply by v and integrate by parts once: integral from 0 to 1 of u' v' dx = integral from 0 to 1 of f v dx, the boundary term [u' v] vanishing since v(0) = v(1) = 0. The weak problem is: find u in H^1_0(0, 1) with integral of u' v' = integral of f v for all v in H^1_0. A solution u need only have one square-integrable derivative — for instance the response to a point load f = delta is the continuous tent function, which is in H^1 but not in C^2.
Integrating by parts once turns a second-order PDE into a balance of single derivatives.
Equivalence runs only one way for free: a classical solution is always a weak solution, but a weak solution is a classical one only after regularity theory upgrades it — and on a domain with a re-entrant corner, the weak solution may genuinely be non-smooth there and have NO classical counterpart. Also, the choice of test space encodes the boundary conditions, so the SAME PDE has different weak formulations for Dirichlet vs. Neumann data.