12.5 Rayleigh quotients and energy decay

The same variational ideas give eigenvalues. For \(-\Delta \phi = \lambda \phi \) in \(\Omega \) with \(\phi = 0\) on \(\partial \Omega \), the lowest eigenvalue is \[ \lambda _1 = \min _{w \in H^1_0,\ w \neq 0}R[w], \qquad R[w] = \frac {\int _\Omega |\nabla w|^2}{\int _\Omega w^2} , \] the Rayleigh quotient: expanding \(w\) in eigenfunctions, \(R[w] = \sum \lambda _kc_k^2/\sum c_k^2 \geq \lambda _1\), with equality for the first eigenfunction. Any trial function gives an upper bound: \(w = x(1 - x)\) on \((0, 1)\) gives \(R = 10\) against \(\lambda _1 = \pi ^2 = 9.8696\), an error of \(1.3\%\) from a crude guess, because errors in \(w\) enter \(R\) only to second order. The Ritz method minimises \(R\) over a subspace, which leads to the generalised eigenvalue problem \(Kc = \lambda Mc\) with \(M_{ij} = \int \phi _i\phi _j\). The Poincaré inequality is the statement \(\lambda _1 \geq (\pi /d)^2\).

For evolution equations, energy gives decay and uniqueness without formulas. If \(u_t = \kappa u_{xx}\) on \((0, L)\) with \(u = 0\) at the ends, then \(\frac {d}{dt}\int u^2 = -2\kappa \int u_x^2 \leq -\frac {2\kappa \pi ^2}{L^2}\int u^2\) by Poincaré, so \[ \int _0^Lu(x, t)^2\,dx \leq e^{-2\kappa \pi ^2t/L^2}\int _0^Lu(x, 0)^2\,dx , \] the exact rate of the slowest mode; and the same argument works for variable coefficients and any shape of region, where no series is available.

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