elliptic regularity
When you solve an elliptic equation by the energy method, you first get only a weak solution — an object that has just one derivative in a mean-square sense and might, for all you know, be jagged. Elliptic regularity is the remarkable discovery that this fear is unfounded: an elliptic equation forces its solutions to be far smoother than you put in. The solution is automatically as smooth as the data and the boundary allow, and if everything is infinitely smooth, so is the solution. Equilibrium irons out wrinkles.
The engine is a gain-of-derivatives estimate. The interior version says: if L u = f and L is elliptic with smooth coefficients, then u has TWO more derivatives than f, controlled by f. In the Sobolev scale, if f is in H^k then u is in H^(k+2) locally, with a bound like the H^(k+2) norm of u not greater than a constant times (the H^k norm of f plus the L^2 norm of u). The heuristic is clean: the equation reads (top-order derivatives of u) = f minus (lower-order junk), so two derivatives of u are as good as f is. Now bootstrap: start with u merely in H^1; the equation gives u in H^2; feed that back and you earn H^3; iterate. Each pass buys two more derivatives until you run out of smoothness in f. If f is in C-infinity, u is in C-infinity; if the data is real-analytic, u is even analytic.
This is why it is legitimate to hunt for weak solutions first and worry about classical ones later — regularity theory converts a weak solution into a classical one for free, provided the data cooperate. The story has three parallel scales, each with its own great theorem: the Sobolev H^k estimates, the Holder Schauder estimates (for Holder-continuous coefficients), and the L^p Calderon-Zygmund estimates. Up to the boundary the same gain holds provided the boundary is smooth enough. The one place the clean story breaks is rough coefficients — there the gain of two full derivatives fails, and you need the deeper De Giorgi-Nash-Moser theory.
Suppose - Laplacian u = f on a smooth domain with f continuous but nowhere differentiable. You might fear u is barely twice-differentiable, yet the Schauder estimate (if f is Holder continuous) makes u genuinely C^2 up to the boundary, and if you smooth f a little — make it C^infinity — then u is C^infinity. The wrinkles in f are never amplified; they are softened by two whole orders of differentiation.
Two derivatives in, two derivatives out — the elliptic gain, iterated, is the bootstrap to full smoothness.
The gain of two derivatives needs the coefficients themselves to be smooth enough; with only bounded measurable coefficients you do NOT get C^2 solutions, only Holder continuity (De Giorgi-Nash-Moser). Also, regularity is local and depends on the boundary: a re-entrant corner in the domain can pin a singularity in the solution no matter how smooth f is.