a weak solution
What does it mean to 'solve' a PDE when the thing you are solving for is too rough to have the derivatives the equation literally asks for? A weak solution is the answer: it is a function that satisfies the equation in an averaged, tested sense rather than pointwise. Instead of demanding minus Laplacian u = f at every point, you demand that the equation hold after you multiply by any test function and integrate — the integrated identity is the relaxed standard a weak solution must meet.
Concretely, a weak solution of minus Laplacian u = f with zero boundary data is a function 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 — exactly the weak formulation, now read as a definition of 'solution'. The crucial relaxation is that u carries only one derivative, so u need not be twice differentiable, or even continuous in high dimensions — it only has to be in the Sobolev space H^1_0 and make the tested identity true. A classical solution (one with honest second derivatives solving the equation everywhere) is automatically a weak solution, by integration by parts; the converse is the deep direction, and only holds after regularity theory does its work.
Why settle for less? Because existence of a weak solution is provable when existence of a classical one is not — that is the whole strategic point of the modern theory. Lax-Milgram delivers a weak solution under mild, checkable conditions on the bilinear form; you then run elliptic regularity to bootstrap that weak solution up to a classical one when the data is smooth. Two honest caveats. First, for elliptic equations the weak solution is UNIQUE, but for nonlinear conservation laws (shocks) it is NOT — there a weak solution alone is ambiguous and you must add an entropy condition to single out the physical one. Second, a weak solution can be genuinely non-classical and stay that way: on a domain with a sharp inward corner, the weak solution of the Dirichlet problem really does have unbounded gradient at the corner, and no amount of regularity theory makes it C^1 there. The weak solution is the honest object; the classical solution is the lucky special case.
On the interval (-1, 1), the tent function u(x) = 1 - |x| is the weak solution of minus u'' = 2 delta (a point load at 0) with u(-1) = u(1) = 0. It is not twice differentiable at 0 — its second classical derivative does not exist there — yet for every test function v one checks integral of u' v' = 2 v(0), which is exactly the weak equation. So u solves the problem weakly while being only H^1, not C^2.
The tent function solves a point-load problem weakly, despite having no classical second derivative.
Existence and uniqueness of a weak solution is a clean win for ELLIPTIC problems — but the word 'weak solution' alone is not enough for nonlinear conservation laws, where infinitely many weak solutions can exist and the entropy condition is required to restore uniqueness. Never assume a weak solution is unique without checking the structure of the equation.