Rayleigh quotients and min–max principles

For a nonzero form-domain function, divide its integrated squared gradient by its squared square-integrable norm. Multiplying the trial function by any nonzero constant leaves this quotient unchanged. The infimum is the first eigenvalue, which is zero for a closed connected manifold because constants are admissible. To estimate its first positive eigenvalue, impose orthogonality to constants, equivalently zero mean. Forgetting that constraint gives a formally correct bound on zero instead of information about the spectral gap.

Dirichlet trials belong to $H_0^1$, while Neumann trials belong to $H^1$ and need not have zero normal derivative. The latter condition belongs to the operator domain and arises from the variational equation at a critical point. Requiring it of every trial can unnecessarily restrict the space. Conversely, allowing a trial with nonzero trace in a Dirichlet problem invalidates the intended bound because it is not admissible in that problem's energy domain.

A single trial function gives an upper bound for the first eigenvalue in the relevant constrained space. For the kth eigenvalue, a k-dimensional trial subspace gives an upper bound only after maximizing the quotient over all its nonzero combinations. Checking several basis vectors separately is insufficient because mixed energy terms can increase that maximum. The subspace formulation is the conceptual source of the Rayleigh–Ritz method, where the maximum becomes a generalized matrix eigenvalue problem.

An orthonormal eigenbasis makes the principle transparent. The quotient of a finite combination is a weighted average of its participating eigenvalues, with weights proportional to squared coefficients. The span of the first k eigenfunctions has maximal quotient exactly the kth eigenvalue. Every k-dimensional competitor contains a nonzero vector orthogonal to the preceding k−1 eigenfunctions, whose quotient is at least that value. This proof counts multiplicity automatically because it counts dimensions, not distinct numerical levels.

When a metric is multiplied by the square of a length factor, the energy numerator scales by length to the dimension minus two, while the denominator scales by length to the dimension. Their ratio therefore scales inversely with squared length. This gives a quick consistency test for proposed geometric bounds. A bound proportional to inverse volume without a dimension-dependent power cannot have the correct general scaling for Laplace eigenvalues.

Variational bounds are often useful when exact eigenfunctions are unknown. A function almost constant with opposite signs on two large regions, changing only across a thin connector, can have small gradient energy and substantial mass. After arranging zero mean, it bounds the first positive eigenvalue from above. This is a geometric construction of slow relaxation. It does not by itself give a matching lower bound or identify the true first eigenfunction; those require additional analysis.