12.2 The Poincaré inequality
The cure is to enlarge the class until minimising sequences converge in it. The size of a function is to be measured by its energy, \(\int |\nabla w|^2\), and the first step is to show that this controls the function itself.
Theorem 12.2.1 (Poincaré inequality). If \(u \in C^1[0, L]\) with \(u(0) = u(L) = 0\), then \(\int _0^Lu^2\,dx \leq (L/\pi )^2\int _0^Lu'^2\,dx\), with equality for \(u = \sin (\pi x/L)\). If \(\Omega \) lies in a slab \(0 < x_1 < d\) and \(u \in C^1(\overline \Omega )\) vanishes on \(\partial \Omega \), then \(\int _\Omega u^2 \leq (d/\pi )^2\int _\Omega |\nabla u|^2\).
Proof. Expand \(u = \sum b_k\sin (k\pi x/L)\); then \(u' = \sum b_k(k\pi /L)\cos (k\pi x/L)\), and by Parseval for the sine and cosine systems, \(\int u^2 = \frac L2\sum b_k^2 \leq \frac L2\sum b_k^2(k/1)^2 = (L/\pi )^2\int u'^2\), with equality only if \(b_k = 0\) for \(k \geq 2\). For \(\Omega \), extend \(u\) by \(0\) to the slab and apply the one-dimensional inequality on each line parallel to the \(x_1\)-axis, then integrate over the other variables; \(|\partial _1u| \leq |\nabla u|\). □
So on functions vanishing on the boundary, \(\|w\|_E = (\int |\nabla w|^2)^{1/2}\) is a norm, the energy norm, and it comes from the inner product \((v, w)_E = \int \nabla v\cdot \nabla w\). The space \(C^1_0(\overline \Omega )\) is not complete in this norm (Weierstrass’s sequence is Cauchy, with no limit in it), and its completion is the Sobolev space \(H^1_0(\Omega )\): functions which, with their first derivatives in a generalised sense, are square-integrable, and which vanish on the boundary in a generalised sense. By the Poincaré inequality, every element of \(H^1_0\) is a genuine square-integrable function.
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