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The Eigenvalues of an Elliptic Operator

A vibrating drum has its own set of pure tones, and so does any elliptic operator. This guide turns the energy method of this rung into a machine that finds those tones one by one, names the smallest with the Rayleigh quotient, and reveals the min-max formula that pins down every eigenvalue without ever solving the equation.

From solving to listening

The first three guides of this rung taught you to solve: given a forcing f, the Lax-Milgram theorem hands you a unique weak solution of an elliptic problem, and regularity then smooths that solution back into a classical one. This guide asks a different, more musical question. Forget the forcing for a moment. Does the elliptic operator L itself have preferred shapes — functions it merely stretches rather than mixes? We are no longer pushing the drum and watching it respond; we are tapping it once and listening to the tones it rings at on its own.

The precise object is the Dirichlet eigenvalue problem: find a number lambda and a nonzero function u, vanishing on the boundary, with L u = lambda u. For the cleanest case L = -Laplacian on a clamped membrane this reads -Laplacian u = lambda u with u = 0 on the edge. A solution pair is an eigenvalue lambda and its eigenfunction u — a standing wave the operator scales by lambda without deforming. The eigenfunctions are the membrane's pure modes of vibration; the eigenvalues are the squares of their frequencies, the actual pitches you would hear. The whole game of this guide is to prove these tones exist, that there are infinitely many, and to find a formula for them.

The lowest note: the Rayleigh quotient

Where do we even begin to find lambda without solving the equation? The energy method of guide 1 already handed us the tool. Recall the symmetric bilinear form a(u, u) — for L = -Laplacian it is just the integral of |grad u|^2, the stretching energy stored in the deformed membrane. Compare that against the integral of u^2, the mass of the shape. Their ratio is the Rayleigh quotient: R(u) = a(u, u) / (integral of u^2) = (energy) / (mass). It measures how much it costs, per unit of material, to bend the membrane into the shape u. A taut, wiggly shape has large R; a relaxed, gently bulging one has small R.

         a(u, u)        integral of |grad u|^2        bending energy
R(u) = ----------- = ------------------------- =  ----------------
       (u, u)_L2        integral of u^2                  mass

  lambda_1  =  min over all u in H^1_0 , u not 0  of  R(u)

  The minimiser u_1 is the principal eigenfunction;  L u_1 = lambda_1 u_1 .
The Rayleigh quotient as energy-per-mass, and the variational fact that drives the whole guide: the lowest eigenvalue is simply the smallest value the ratio can take.

Here is the beautiful claim. The smallest eigenvalue, the principal eigenvalue lambda_1, is exactly the minimum of R(u) over all admissible shapes u in H^1_0. The drum's lowest, deepest tone is the cheapest possible shape: the one that stores the least bending energy per unit of mass. You do not solve the eigenvalue equation to get lambda_1; you minimise an energy ratio, which is exactly the kind of problem the next guide's calculus of variations is built to attack. The minimiser u_1 is the fundamental mode — for a circular drum, the broad symmetric bulge with no internal nodal lines.

Why is this minimum even attained, rather than approached but never reached? Because of exactly the machinery this rung assembled. The Poincare inequality guarantees R(u) is bounded below by a positive number, so the bending energy cannot be made arbitrarily cheap — there is a genuine floor. And the compactness packed into the Sobolev embedding lets a minimising sequence converge to an actual minimiser inside H^1_0, rather than escaping to a rougher limit. The same direct-method ingredients that the next guide formalises are what make lambda_1 a real attained value and u_1 a real function.

Climbing the spectrum, one note at a time

One tone is not a chord. How do we reach the second eigenvalue lambda_2, the first overtone? The idea is disarmingly simple: minimise the same Rayleigh quotient, but only over shapes that are orthogonal to the fundamental mode u_1 — shapes whose L^2 inner product with u_1 is zero. We have forbidden the cheapest shape, so the cheapest remaining one costs more, and that cost is lambda_2 with minimiser u_2. Repeat: for lambda_3, minimise over shapes orthogonal to both u_1 and u_2. Each new constraint locks out the modes already found and forces the search up to the next available note.

  1. Minimise R(u) over all of H^1_0; the minimum is lambda_1 and the minimiser is u_1, the deepest tone with no internal nodes.
  2. Minimise R(u) again, but only over shapes orthogonal to u_1; this cheapest remaining shape gives lambda_2 and u_2.
  3. At each stage k, minimise R(u) over shapes orthogonal to all of u_1, ..., u_(k-1) already found; this yields lambda_k and u_k.
  4. The eigenvalues stack up as 0 < lambda_1 <= lambda_2 <= lambda_3 <= ... marching to infinity, and the eigenfunctions form a complete orthogonal basis of L^2.

Run this forever and you generate an increasing sequence 0 < lambda_1 <= lambda_2 <= lambda_3 <= ... that marches off to infinity, never bunching up at a finite ceiling. The gap between the first two, lambda_2 - lambda_1, is the spectral gap, and it is not idle bookkeeping: it sets how quickly a heat distribution on the domain relaxes toward its fundamental mode, since higher modes decay faster by exactly their eigenvalue. A large gap means the drum forgets its initial detail fast and rings almost purely at its lowest tone; a small gap means the overtones linger.

Min-max: the formula that needs no eigenfunctions

The orthogonality recipe has a flaw of dependence: to reach lambda_k you must already know u_1, ..., u_(k-1). That is awkward if you only want, say, lambda_5 and a clean bound on it. The cure is one of the most elegant results in spectral theory, the min-max characterization, also called the Courant-Fischer principle. It computes lambda_k while mentioning no eigenfunctions at all. Instead of picking the previous modes, you range over every k-dimensional subspace S of trial functions, and inside each you find the worst (largest) Rayleigh quotient. Then you take the best such worst-case over all choices of S.

  lambda_k  =   min          max        R(u)
             dim S = k     u in S, u not 0

  S ranges over every k-dimensional subspace of trial functions.
  Inside each S take the WORST (largest) Rayleigh quotient;
  then take the BEST (smallest) such worst-case over all S.
The Courant-Fischer min-max formula for the k-th eigenvalue. It never names an eigenfunction, which is exactly what makes it a practical tool for bounding eigenvalues.

This double extremum looks abstract, so feel the payoff. Because lambda_k is the minimum over subspaces, any particular k-dimensional space S you happen to write down gives an honest upper bound: lambda_k is no larger than the worst Rayleigh quotient inside your S. So a single clever guess of a few trial functions instantly bounds an eigenvalue from above, with no equation solved — this is precisely the engine behind the Rayleigh-Ritz method and finite-element eigenvalue codes. Equally, the formula makes monotonicity transparent: shrink the domain or stiffen the operator and every lambda_k can only rise, because every subspace's worst case rises. This is why a smaller drum sounds higher — a one-line consequence of min-max.

Why the tones exist at all, and where this rung points next

Step back and notice that nothing here was lucky — it all rests on properties of L proved earlier in the rung. The eigenvalues are real and the eigenfunctions orthogonal because L is symmetric, sitting in self-adjoint form; a non-symmetric operator can have complex eigenvalues and the clean min-max breaks. The spectrum is discrete — separated points marching to infinity, not a continuous smear — because the inverse of L is a compact operator, courtesy of the same Sobolev compactness that gave existence. And the Fredholm alternative from guide 2 is exactly the statement that L - lambda is invertible except at these eigenvalues, where the homogeneous problem suddenly admits a nonzero solution. The spectrum is precisely the set of forcings the alternative warns you about.

Notice, finally, what this guide quietly did: it solved a PDE eigenvalue problem entirely by minimising an energy, never differentiating the operator directly. That is the tell. The principal eigenvalue is the minimum of a Rayleigh quotient; the eigenfunction is the minimiser; the eigenvalue equation L u = lambda u is just the condition that this minimiser must satisfy. That last fact has a name and a destination — it is the Euler-Lagrange equation of the energy, and the direct method is what proved the minimum is attained. Guide 5 takes this exact correspondence and makes it the whole subject: every elliptic PDE you have met is the Euler-Lagrange equation of some energy you can minimise instead of solve.