a Sobolev space
/ SOH-buh-lyef /
Classical PDE theory insists that a solution be smooth enough to differentiate everywhere — but most of the functions you actually need (a plucked string with a corner, a temperature with a kink at a material join) are not that nice, and forcing smoothness shuts the door on them. A Sobolev space is the room you move into so those rougher functions are allowed in. It is the natural home for the modern, functional-analytic way of doing PDE: instead of asking a solution to have honest pointwise derivatives, you ask only that it and a few of its weak derivatives have finite integral size.
Precisely, fix a region (a domain) U in R^n, a number p between 1 and infinity, and an integer k of how many derivatives you want. The Sobolev space W^{k,p}(U) is the set of functions u for which u itself and every weak (distributional) derivative of u up to order k lie in L^p(U) — that is, each such derivative is p-th-power integrable, integral of |derivative|^p over U is finite. The single most important case is p = 2, written H^k(U) = W^{k,2}(U); because the L^2 norm comes from an inner product, H^k is a Hilbert space, which is exactly the structure the Lax-Milgram theorem and the Galerkin method run on. The simplest example, H^1(U), holds functions u that are square-integrable and whose first weak derivatives u_x1, ..., u_xn are also square-integrable — finite energy, roughly speaking.
Why bother enlarging the space? Because existence of solutions is far easier to prove in a big complete space than in the cramped space of smooth functions. The strategy of modern PDE is two-step: first find a weak solution living in a Sobolev space (cheap, thanks to Hilbert-space theorems), then use regularity theory to show that, when the data is nice, this weak solution is secretly smooth after all and solves the equation classically. Sobolev spaces are the workshop where the first step happens. They are complete (a Banach space, or Hilbert space when p = 2), which is what lets limits of approximating sequences stay inside the space — the property smooth functions fatally lack.
The hat function on the interval (-1, 1) — u(x) = 1 - |x|, equal to 0 at the ends and rising to a peak at x = 0 — has a corner, so it has no classical derivative at 0. Yet its weak derivative is the step that is +1 on (-1, 0) and -1 on (0, 1), which is bounded and square-integrable. So u and u' both lie in L^2, meaning u is in H^1(-1, 1) even though it is not differentiable in the old sense.
A function with a corner can fail to be classically differentiable yet still live happily in H^1.
A Sobolev space depends jointly on the region U, the order k, and the exponent p — H^1(U) is not the same space as L^2(U), and changing p or k changes who is allowed in. Saying just 'Sobolev space' without naming these three is genuinely ambiguous.