the space H^1_0
/ aitch-one-not /
Most boundary-value problems pin a solution down by fixing its value on the edge of the region — for a Dirichlet problem, you say u = 0 on the boundary. But Sobolev functions can be rough and are only defined up to a set of measure zero, and the boundary itself has measure zero, so 'u equals zero on the boundary' is not obviously meaningful. The space H^1_0 is the clean way to encode that boundary condition into the function space itself, so you never have to mention the boundary again.
The construction is a limiting one. Start with the smooth functions that are exactly zero in a neighbourhood of the boundary (they have compact support inside U). Then take H^1_0(U) to be everything you can reach as an H^1-limit of such functions — formally, the closure in the H^1 norm of the smooth compactly supported functions. Intuitively these are the H^1 functions that fade to zero as you approach the boundary; on a nice domain the trace theorem makes that literal, giving H^1_0 = the H^1 functions whose boundary trace is zero. Because the boundary condition is now baked into membership, the weak formulation of a Dirichlet problem just searches for its solution inside H^1_0, and test functions are drawn from H^1_0 too — so all the boundary terms in the integration by parts vanish automatically.
This space is where the magic theorems live. On H^1_0 (over a bounded domain), the Poincare inequality holds, which means the Dirichlet energy integral of |grad u|^2 alone controls the whole H^1 norm — you cannot have a big function with a small gradient if you are forced to vanish on the boundary. That single fact delivers the coercivity that Lax-Milgram needs, which is why the existence theory for the Dirichlet problem is so clean. Homogeneous (zero) boundary data lives directly in H^1_0; nonzero boundary data is handled by subtracting off a fixed function with the right boundary values and solving for the H^1_0 remainder.
On the interval (0, 1), the function u(x) = x(1 - x) is smooth, vanishes at both endpoints, and has integrable derivative, so it lies in H^1_0(0, 1). By contrast the constant function 1 lies in H^1(0, 1) (it is square-integrable with zero derivative) but NOT in H^1_0, because it does not vanish at the ends. H^1_0 is the strictly smaller subspace selected by the boundary condition.
H^1_0 is the part of H^1 that obeys the zero boundary condition — strictly smaller than H^1.
On all of R^n (no boundary) H^1_0 equals H^1 — there is nothing to vanish on. The space encodes ZERO (homogeneous) Dirichlet data only; nonzero boundary values are not represented by H^1_0 itself but by an affine shift of it. And H^1_0 captures the Dirichlet condition, not the Neumann condition, which is enforced differently (it falls out of the weak formulation as a 'natural' boundary condition).