the Poincare inequality
/ pwankah-RAY /
Pin a function to zero on the boundary of a region, then ask: how big can the function get in the interior? Intuitively, if its slopes (gradient) are small, it cannot climb very far from the zero it starts at on the edge before it has to come back down. The Poincare inequality makes this precise: on a bounded region, a function that vanishes on the boundary cannot be large unless its gradient is large. Controlling the slopes controls the function itself.
The clean statement is for the space H^1_0(U) on a bounded domain U: there is a constant C, depending only on the domain (its size and shape), such that integral of u^2 over U is at most C times integral of |grad u|^2 over U, for every u in H^1_0. In words, ||u||_{L^2} is at most a constant times ||grad u||_{L^2}. The closely related Friedrichs inequality is essentially the same statement, often quoted for functions with zero boundary trace; the two names tend to be used interchangeably in PDE. The constant cannot be removed — it genuinely grows with the diameter of the domain (a longer string can bow further for the same maximal slope). Crucially, the inequality FAILS without a boundary condition: a nonzero constant function has zero gradient but is not zero, so you cannot bound its L^2 norm by its (vanishing) gradient. That is exactly why the boundary anchoring in H^1_0 is essential.
Why this small inequality is a giant lever: it says that on H^1_0 the Dirichlet energy integral of |grad u|^2 (the H^1 seminorm) is, all by itself, equivalent to the full H^1 norm. That single fact is what delivers coercivity for the Dirichlet problem — the bilinear form a(u, u) = integral of |grad u|^2 bounds ||u||_{H^1}^2 from below — and coercivity is the hypothesis Lax-Milgram needs to hand you existence and uniqueness of a weak solution. So the Poincare inequality is the quiet workhorse standing behind nearly every existence theorem for elliptic boundary-value problems, and it also pins down the smallest Dirichlet eigenvalue: the best (smallest) constant C is one over the first eigenvalue of minus the Laplacian on U.
On (0, L), for u with u(0) = u(L) = 0, the inequality reads integral of u^2 at most (L/pi)^2 times integral of (u')^2, and the constant (L/pi)^2 is sharp — equality is hit by u(x) = sin(pi x/L), the lowest eigenfunction. Double the length L and the constant quadruples: a longer string, anchored at both ends, can bow four times as much in mean-square for the same gradient energy.
The sharp Poincare constant on an interval is (L/pi)^2, attained by the first sine mode.
The inequality needs SOME way to kill constants — either vanishing on (part of) the boundary (the H^1_0 version) or having mean value zero (the version on all of H^1). With neither, it is false: constants are the counterexample. And it requires a bounded domain (at least in some direction); on all of R^n it fails because there is room to spread out at no gradient cost.