02.16.05 · analysis / sobolev-weak-solutions

The Fredholm Alternative and Eigenvalues for Second-Order Elliptic Operators

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Anchor (Master): Evans §6.2, §6.5; Gilbarg-Trudinger, Elliptic Partial Differential Equations of Second Order, 2e (Springer 1983), §8.6, §8.12; Courant-Hilbert, Methods of Mathematical Physics I (Wiley 1953), Ch. VI; Reed-Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic Press 1978), §XIII.12-§XIII.15

Intuition Beginner

A square grid of numbers, a matrix, obeys a clean rule when you try to solve $A x = b$ : either the matrix is invertible, in which case there is exactly one answer for every right-hand side, or it is not, in which case some right-hand sides have no answer and others have a whole family of answers. There is no middle case. This sharp two-way split is the heart of finite linear algebra. The surprise of this unit is that the same two-way split survives when the matrix is replaced by a differential operator acting on functions, an object that lives in an infinite-dimensional world where such clean dichotomies usually break down.

What rescues the dichotomy is a compressing step. The differential operator of this unit can be inverted once, with a small shift added, and that inverse turns out to squeeze a roomy infinite family of functions down into a crowded one, the crowding guaranteed by the sibling compactness theorem. An operator that compresses this way behaves almost exactly like a finite matrix: it has a sequence of pure modes and a clean either-or rule for solving equations. The technical name for the either-or rule is the Fredholm alternative.

The pure modes deserve their own picture. Think of a drumhead clamped along its rim. Strike it and it rings, but not at every pitch: it sounds only a specific ladder of tones, the lowest a fundamental and the rest higher overtones. Each tone is a standing wave, a shape that vibrates in place without travelling. For the elliptic operator of this unit those standing shapes are the eigenfunctions, and the squared pitches are the eigenvalues. The compressing step guarantees this ladder of pitches is real, climbs to infinity, and never crowds together at a finite ceiling.

Two further facts about the ladder matter. The lowest tone is special: it is reached by exactly one shape, up to scaling, and that shape never crosses zero in the interior, so it has a definite sign all the way across the drum. And the full collection of standing shapes is rich enough that every reasonable vibration of the drum is a sum of them, the way every musical sound is a blend of a fundamental and its overtones. The averaged, weak way of reading the operator from the sibling unit is exactly what makes all of this provable.

Visual Beginner

The single picture to hold is a clamped drumhead with its ladder of standing-wave tones, beside the either-or rule for solving the equation.

Read the three panels left to right. The left panel is the eigenvalue ladder. A clamped drumhead has a lowest tone reached by a single never-crossing shape, then a sequence of higher tones reached by shapes with more and more dividing lines, and the pitches climb without ever crowding toward a finite ceiling. These standing shapes are the eigenfunctions and the squared pitches are the eigenvalues.

The middle panel is the either-or rule. When you solve the operator equation with a shift, exactly one of two things happens. If the shift avoids every tone, you get one and only one solution for every right-hand side, the well-behaved case. If the shift lands exactly on a tone, solutions exist only for right-hand sides that steer clear of that tone, and then there is a whole family of them. There is no third possibility.

The right panel explains why the infinite problem obeys this finite-looking rule. Inverting the operator with a small shift compresses a roomy infinite family of functions into a crowded cluster, and a compressing operation is what lets infinite linear algebra mimic the finite matrix case.

Worked example Beginner

We find the tone ladder of the simplest clamped string by hand and watch the either-or rule decide solvability. Take the interval from zero to $π$ , the operator that sends a function $u$ to minus its second slope-change $- u^{''}$ , and clamp the ends by requiring $u (0) = 0$ and $u (π) = 0$ . This is the one-dimensional drumhead.

Step 1. Look for standing shapes. A standing shape is a function $u$ and a number $λ$ with $- u^{''} = λ u$ , the shape reproduced by the operator up to a stretch factor $λ$ . The clamped solutions of this are the sine shapes $u_{k} (x) = sin (k x)$ for $k = 1, 2, 3, \dots$ , since the second slope-change of $sin (k x)$ is $- k^{2} sin (k x)$ .

Step 2. Read off the ladder. Matching $- u^{''} = λ u$ gives $λ = k^{2}$ . So the tones are $λ_{1} = 1$ , $λ_{2} = 4$ , $λ_{3} = 9$ , and so on: the squares of the whole numbers. They are all positive, they climb to infinity, and they never bunch toward a finite ceiling, exactly as promised.

Step 3. Spot the special lowest tone. The lowest tone is $λ_{1} = 1$ , reached by $u_{1} (x) = sin (x)$ . On the open interval from zero to $π$ the function $sin (x)$ is positive everywhere and never returns to zero in the middle. The lowest shape has a definite sign; every higher shape $sin (k x)$ does cross zero inside the interval.

Step 4. Use the either-or rule. Now try to solve $- u^{''} - 4 u = f$ with the same clamped ends. The shift value here is $4$ , which is exactly the second tone $λ_{2} = 4$ . The rule says this is the delicate case: a solution exists only when the right-hand side $f$ avoids that tone, meaning the total of $f (x) sin (2 x)$ across the interval must be zero. If instead we solved $- u^{''} - 2 u = f$ , the shift $2$ is not on the ladder, so there is exactly one solution for every $f$ .

Step 5. Confirm the avoidance condition with a number. Take $f (x) = sin (2 x)$ itself. The total of $f (x) sin (2 x) = sin^{2} (2 x)$ across the interval is $π /2$ , which is not zero. So $- u^{''} - 4 u = sin (2 x)$ has no clamped solution: the right-hand side points straight along the forbidden tone.

What this tells us: the clamped operator has a clean ladder of positive tones, the lowest reached by a single sign-definite shape, and solving a shifted equation obeys a strict either-or rule keyed to whether the shift sits on the ladder. The averaged weak reading of the sibling unit is what makes this rigorous beyond the hand computation.

Check your understanding Beginner

Exercise (easy, multiple choice).

The Fredholm alternative for the shifted elliptic equation says that exactly one of two situations holds. Which pair correctly states them?

A. Either the equation has no solution, or it has infinitely many. B. Either the shift avoids every tone and there is exactly one solution for every right-hand side, or the shift is a tone and solutions exist only for right-hand sides that avoid that tone. C. Either the operator is symmetric, or it has complex eigenvalues. D. Either the domain is bounded, or the spectrum is continuous.

Hint

The either-or rule is keyed to whether the shift value lands on the eigenvalue ladder.

Answer

B. Either the shift avoids every tone and there is exactly one solution for every right-hand side, or the shift is a tone and solutions exist only for right-hand sides that avoid that tone. This is the infinite-dimensional echo of "a square matrix is either invertible or has a kernel." When the shift misses the eigenvalue ladder the operator is invertible; when it lands on a tone, the homogeneous equation has nonzero solutions and the inhomogeneous one is solvable only against right-hand sides orthogonal to that tone. Feedback-correct: the rule is keyed to the eigenvalue ladder. Feedback-wrong: A drops the unique-solution case, while C and D name conditions the alternative never asserts.

Exercise (easy, true-false).

For a clamped symmetric elliptic operator the eigenvalues form a real sequence that climbs to infinity and never crowds together at a finite ceiling.

Hint

Recall the drumhead ladder of tones and the compressing step that makes the infinite problem behave like a finite matrix.

Answer

True. The compressing (compact) inverse of the shifted operator forces a discrete real spectrum: a sequence $0 < λ_{1} < λ_{2} \leq \dots$ marching off to infinity, with no finite accumulation point. The eigenvalues of the compact inverse pile up only at zero, which translates to the eigenvalues of the operator itself climbing without a finite ceiling. Feedback-correct: discreteness comes from the compact inverse. Feedback-wrong: a continuous band of values would occur only without the compactness the bounded clamped domain supplies.

Formal definition Intermediate+

Throughout, $Ω \subseteq R^{n}$ is open and bounded, $H = H_{0}^{1} (Ω)$ is the Hilbert space of 02.16.03 with inner product $⟨ u, v ⟩_{H_{0}^{1}} = \int_{Ω} D u \cdot D v d x$ , and $L^{2} (Ω)$ carries its usual inner product $(u, v)_{L^{2}} = \int_{Ω} uv d x$ . The operator $L$ is the divergence-form second-order operator $Lu = - i, j = 1 \sum n \partial_{i} (a^{ij} (x) \partial_{j} u) + i = 1 \sum n b^{i} (x) \partial_{i} u + c (x) u,$ with $a^{ij} = a^{j i} \in L^{\infty} (Ω)$ uniformly elliptic (constant $θ > 0$ ), $b^{i}, c \in L^{\infty} (Ω)$ , and associated bounded bilinear form $B [u, v] = \int_{Ω} (i, j \sum a^{ij} \partial_{j} u \partial_{i} v + i \sum b^{i} (\partial_{i} u) v + c u v) d x, u, v \in H_{0}^{1} (Ω),$ all from the sibling unit 02.16.04, whose Gårding inequality $B [u, u] \geq β ∥ u ∥_{H_{0}^{1}}^{2} - γ ∥ u ∥_{L^{2}}^{2}$ and Lax-Milgram existence theorem are taken as available. The formal adjoint $L^{*}$ has bilinear form $B^{*} [u, v] = B [v, u]$ . The operator $L$ is symmetric (formally self-adjoint) if $b \equiv 0$ , in which case $B [u, v] = B [v, u]$ and $B$ is the symmetric form $\int_{Ω} (\sum a^{ij} \partial_{j} u \partial_{i} v + c uv)$ .

Definition (compact solution operator). Fix $μ \geq γ$ , so by Gårding the shifted form $B_{μ} [u, v] = B [u, v] + μ (u, v)_{L^{2}}$ is coercive on $H_{0}^{1}$ . For $g \in L^{2} (Ω) ↪ H^{- 1} (Ω)$ let $u = G g \in H_{0}^{1} (Ω)$ be the unique Lax-Milgram solution of $(L + μ) u = g$ , i.e. $B_{μ} [u, v] = (g, v)_{L^{2}}$ for all $v \in H_{0}^{1}$ . The solution operator is the composition $G : L^{2} (Ω) (L + μ)^{- 1} H_{0}^{1} (Ω) ↪ L^{2} (Ω),$ where the second arrow is the compact Rellich-Kondrachov embedding of 02.16.03; thus $G$ is a compact operator on $L^{2} (Ω)$ in the sense of 02.11.05. When $L$ is symmetric, $G$ is self-adjoint and positive: $(G g, h)_{L^{2}} = B_{μ} [G g, G h] = (g, G h)_{L^{2}}$ and $(G g, g)_{L^{2}} = B_{μ} [G g, G g] \geq 0$ .

Definition (eigenvalue, eigenfunction). A number $λ \in R$ is an eigenvalue of $L$ (with Dirichlet conditions) if there is a nonzero $w \in H_{0}^{1} (Ω)$ , an eigenfunction, with $B [w, v] = λ (w, v)_{L^{2}} for all v \in H_{0}^{1} (Ω),$ the weak form of $L w = λ w$ , $w ∣_{\partial Ω} = 0$ . The set of all eigenvalues is the (Dirichlet) spectrum $Σ$ of $L$ . The principal eigenvalue is $λ_{1} = min Σ$ , and it is simple if its eigenspace is one-dimensional.

Definition (Rayleigh quotient). For symmetric $L$ and $u \in H_{0}^{1} (Ω) ∖ {0}$ the Rayleigh quotient is $R [u] = \frac{B [ u , u ]}{( u , u ) _{L^{2}}} = \frac{\int _{Ω} ( \sum _{i, j} a ^{ij} \partial _{j} u \partial _{i} u + c u ^{2} ) d x}{\int _{Ω} u ^{2} d x} .$ The correspondence between the spectrum of $L$ and that of $G$ is the algebraic identity at the centre of the theory: $w$ is an eigenfunction of $L$ with eigenvalue $λ$ if and only if $w = (λ + μ) Gw$ , that is, $w$ is an eigenfunction of $G$ with eigenvalue $κ = (λ + μ)^{- 1}$ . The map $λ \mapsto (λ + μ)^{- 1}$ is the dictionary translating the discrete-spectrum theory of the compact $G$ into the spectrum of the unbounded $L$ .

Counterexamples to common slips Intermediate+

The spectrum is discrete only because the inverse compresses. The operator $L$ itself is unbounded and has no compactness; what is compact is its shifted inverse $G$ . Drop the bounded-domain hypothesis (take $Ω = R^{n}$ ) and the embedding $H^{1} ↪ L^{2}$ is no longer compact, $G$ is merely bounded, and $- Δ$ acquires continuous spectrum $[0, \infty)$ with no eigenfunctions in $L^{2}$ . Discreteness is a consequence of compactness, not of ellipticity alone.
Reality and orthogonality of the eigenbasis need symmetry. For $b \neq \equiv 0$ the form $B$ is non-symmetric, $G$ is compact but not self-adjoint, and the eigenvalues of $L$ may be complex and the eigenfunctions need not be orthogonal. The spectral theorem for compact self-adjoint operators of 02.11.05 requires the symmetric case $b \equiv 0$ ; the non-symmetric operator still obeys the Fredholm alternative but not the orthonormal-eigenbasis conclusion.
The principal eigenvalue is simple, but higher ones need not be. On a square or a ball the second Dirichlet eigenvalue of $- Δ$ is typically degenerate (its eigenspace is multi-dimensional by symmetry of the domain), so simplicity is special to $λ_{1}$ . The Krein-Rutman positivity argument that gives simplicity uses that the principal eigenfunction does not change sign, a property no higher eigenfunction shares.
The shift $μ$ is bookkeeping, not part of the operator. The eigenvalues of $L$ do not depend on the choice of $μ \geq γ$ used to build $G$ : changing $μ$ rescales the eigenvalues $κ$ of $G$ by the dictionary $κ = (λ + μ)^{- 1}$ but leaves the recovered $λ = κ^{- 1} - μ$ unchanged. Reading an eigenvalue of $G$ as an eigenvalue of $L$ without applying the dictionary is the common error.

Key theorem with proof Intermediate+

Theorem (Fredholm alternative and the elliptic eigenvalue problem). Let $L$ be uniformly elliptic on a bounded $Ω$ with $L^{\infty}$ coefficients, and let $G$ be the compact solution operator of $L + μ$ , $μ \geq γ$ . Then:

(a) (Fredholm alternative) For each $λ$ , exactly one of the following holds. Either the equation $Lu - λ u = f$ has a unique weak solution $u \in H_{0}^{1} (Ω)$ for every $f \in L^{2} (Ω)$ ; or the homogeneous equation $Lu = λ u$ has a nonzero solution, the solution spaces of $Lu = λ u$ and $L^{*} w = λ w$ are finite-dimensional of equal dimension $d \geq 1$ , and $Lu - λ u = f$ is solvable precisely when $(f, w)_{L^{2}} = 0$ for every solution $w$ of $L^{*} w = λ w$ .

(b) (discrete real spectrum, symmetric case) If in addition $L$ is symmetric ( $b \equiv 0$ ), the eigenvalues of $L$ form a countable sequence $λ_{1} \leq λ_{2} \leq λ_{3} \leq \dots$ of real numbers with $λ_{k} \to + \infty$ , repeated according to finite multiplicity, and the corresponding eigenfunctions ${w_{k}}$ may be chosen to form an orthonormal basis of $L^{2} (Ω)$ ^{[Evans 2010 §6.5]} ^{[Fredholm 1903]}.

Proof. Build $G$ as in the definition: $G : L^{2} \to L^{2}$ is compact, being the composition of the bounded Lax-Milgram inverse $(L + μ)^{- 1} : L^{2} \to H_{0}^{1}$ of 02.16.04 with the compact embedding $H_{0}^{1} ↪↪ L^{2}$ of 02.16.03.

For (a), set $κ = (λ + μ)^{- 1}$ (assume $λ \neq = - μ$ ; for the excluded single value $L + μ$ is invertible by construction, so the first alternative holds outright). A function $u$ solves $Lu - λ u = f$ if and only if $(L + μ) u = f + (λ + μ) u$ , i.e. $u = G f + (λ + μ) G u$ , i.e. $(I - κ^{- 1} G) u = G f ⟺ (κ I - G) u = κ G f,$ an equation of the form $(κ I - G) u = h$ with $G$ compact. The Riesz-Schauder theory of 02.11.05 gives the Fredholm alternative for $κ I - G$ : either $κ$ is not an eigenvalue of $G$ , so $κ I - G$ is invertible and the equation has a unique solution for every $h$ ; or $κ$ is an eigenvalue, $ker (κ I - G)$ and $ker (κ I - G^{*})$ are finite-dimensional of equal dimension, and the equation is solvable iff $h ⊥ ker (κ I - G^{*})$ . Translating through the dictionary $κ = (λ + μ)^{- 1}$ — eigenfunctions of $G$ at $κ$ are exactly eigenfunctions of $L$ at $λ$ , and $G^{*}$ is the solution operator of $L^{*} + μ$ — yields the stated dichotomy, the solvability condition becoming $(f, w)_{L^{2}} = 0$ for $w$ in the $λ$ -eigenspace of $L^{*}$ .

For (b), symmetry makes $G$ self-adjoint and positive on $L^{2}$ : $(G g, h)_{L^{2}} = B_{μ} [G g, G h] = (g, G h)_{L^{2}}$ and $(G g, g)_{L^{2}} = B_{μ} [G g, G g] \geq β ∥ G g ∥_{H_{0}^{1}}^{2} \geq 0$ , with equality only at $g = 0$ . The spectral theorem for compact self-adjoint operators of 02.11.05 supplies an orthonormal basis ${w_{k}}$ of $L^{2} (Ω)$ consisting of eigenfunctions of $G$ , with eigenvalues $κ_{k} > 0$ (positivity rules out $κ = 0$ on the orthogonal complement of $ker G = {0}$ ) accumulating only at $0$ and ordered $κ_{1} \geq κ_{2} \geq \dots \to 0$ . Each $w_{k}$ is then an eigenfunction of $L$ with eigenvalue $λ_{k} = κ_{k}^{- 1} - μ$ , and $κ_{k} ↓ 0$ forces $λ_{k} \to + \infty$ . Since $κ_{k} \leq κ_{1}$ for all $k$ , the eigenvalues are bounded below by $λ_{1} = κ_{1}^{- 1} - μ$ , and the sequence $λ_{1} \leq λ_{2} \leq \dots$ is real with finite multiplicities inherited from those of $G$ . $□$

Bridge. The Fredholm alternative for $L$ is exactly the Riesz-Schauder dichotomy for the compact operator $G$ of 02.11.05 read through the spectral dictionary $κ = (λ + μ)^{- 1}$ : this is the foundational reason an infinite-dimensional boundary-value problem obeys the finite-dimensional either-or rule, since the compressing solution operator manufactured from Lax-Milgram 02.16.04 and Rellich-Kondrachov 02.16.03 turns $L$ into a compact perturbation of the resolvent. The construction builds toward the variational min-max characterization, where the same compactness guarantees that each Rayleigh-quotient infimum is attained, and it appears again in 02.16.04 in the non-coercive existence theory, of which this dichotomy is the precise solvability statement. The discreteness of the spectrum is dual to the compactness of $G$ — eigenvalues of $G$ accumulating at zero is exactly eigenvalues of $L$ marching to infinity — and putting these together, the central insight is that compactness of the inverse is the single hypothesis converting elliptic solvability and the spectral ladder into one statement of finite-rank linear algebra.

Exercises Intermediate+

Exercise 2 (easy, true-false).

For a non-symmetric uniformly elliptic operator (one with a nonzero drift $b^{i} \partial_{i} u$ ) the Fredholm alternative still holds, but the eigenfunctions need not form an orthonormal basis of $L^{2}$ .

Hint

The Fredholm alternative needs only compactness of $G$ ; the orthonormal eigenbasis needs $G$ self-adjoint, hence $L$ symmetric.

Answer

True. The solution operator $G$ is compact whenever $L$ is uniformly elliptic on a bounded domain, so the Riesz-Schauder Fredholm alternative applies regardless of symmetry. But $G$ is self-adjoint only when $b \equiv 0$ ; with drift, $G$ is compact-but-not-self-adjoint, its eigenvalues may be complex, and the spectral theorem giving an $L^{2}$ -orthonormal eigenbasis fails. Symmetry is the dividing line between the Fredholm theory and the full spectral theory.

Exercise 3 (medium, symbolic).

Establish the spectral dictionary precisely: for symmetric $L$ and $μ \geq γ$ , show that $w \neq = 0$ solves $B [w, v] = λ (w, v)_{L^{2}}$ for all $v \in H_{0}^{1}$ if and only if $Gw = κ w$ with $κ = (λ + μ)^{- 1}$ , and deduce that the eigenvalues of $G$ are exactly the numbers $(λ + μ)^{- 1}$ for $λ$ an eigenvalue of $L$ .

Hint

The weak equation $B [w, v] = λ (w, v)$ is the same as $B_{μ} [w, v] = (λ + μ) (w, v)_{L^{2}}$ . Compare with the defining property $B_{μ} [G g, v] = (g, v)_{L^{2}}$ of the solution operator.

Answer

Adding $μ (w, v)_{L^{2}}$ to both sides, $B [w, v] = λ (w, v)_{L^{2}}$ for all $v$ is equivalent to $B_{μ} [w, v] = (λ + μ) (w, v)_{L^{2}}$ for all $v$ . Set $g = (λ + μ) w \in L^{2}$ . The defining property of $G$ is that $B_{μ} [G g, v] = (g, v)_{L^{2}}$ for all $v$ , and the Lax-Milgram solution is unique, so $B_{μ} [w, v] = (g, v)_{L^{2}} = ((λ + μ) w, v)_{L^{2}}$ for all $v$ says exactly $w = G g = (λ + μ) Gw$ , i.e. $Gw = (λ + μ)^{- 1} w = κ w$ (note $λ + μ > 0$ since $λ \geq λ_{1} > - μ$ by coercivity of $B_{μ}$ ). Conversely $Gw = κ w$ with $κ \neq = 0$ gives $w = κ^{- 1} Gw$ , so $B_{μ} [w, v] = κ^{- 1} B_{μ} [Gw, v] = κ^{- 1} (w, v)_{L^{2}}$ , i.e. $B [w, v] = (κ^{- 1} - μ) (w, v)_{L^{2}}$ , an eigenfunction of $L$ at $λ = κ^{- 1} - μ$ . Since $ker G = {0}$ (positivity), every nonzero eigenvalue $κ$ of $G$ arises this way, so the spectrum of $G$ minus ${0}$ is exactly ${(λ + μ)^{- 1} : λ \in Σ}$ .

Exercise 4 (medium, symbolic).

Prove the variational characterization of the principal eigenvalue: for symmetric $L$ , $λ_{1} = u \in H_{0}^{1} (Ω) ∖ {0} min R [u] = u \in H_{0}^{1} ∖ {0} min \frac{B [ u , u ]}{∥ u ∥ _{L^{2}}^{2}},$ and the minimum is attained exactly at the eigenfunctions of $λ_{1}$ .

Hint

Expand $u = \sum_{k} c_{k} w_{k}$ in the orthonormal eigenbasis. Use $B [w_{j}, w_{k}] = λ_{j} (w_{j}, w_{k})_{L^{2}} = λ_{j} δ_{j k}$ to write both numerator and denominator as series in $c_{k}$ .

Answer

Let ${w_{k}}$ be the orthonormal eigenbasis with $B [w_{j}, w_{k}] = λ_{j} δ_{j k}$ and $λ_{1} \leq λ_{2} \leq \dots$ . Expand $u = \sum_{k} c_{k} w_{k}$ in $H_{0}^{1}$ , so $∥ u ∥_{L^{2}}^{2} = \sum_{k} c_{k}^{2}$ and, by bilinearity and the diagonalization, $B [u, u] = \sum_{j, k} c_{j} c_{k} B [w_{j}, w_{k}] = \sum_{k} λ_{k} c_{k}^{2}$ . Then $R [u] = \frac{\sum _{k} λ _{k} c _{k}^{2}}{\sum _{k} c _{k}^{2}} \geq \frac{λ _{1} \sum _{k} c _{k}^{2}}{\sum _{k} c _{k}^{2}} = λ_{1},$ since $λ_{k} \geq λ_{1}$ for every $k$ . Equality forces $c_{k} = 0$ whenever $λ_{k} > λ_{1}$ , i.e. $u$ lies in the $λ_{1}$ -eigenspace; and taking $u = w_{1}$ gives $R [w_{1}] = λ_{1}$ , so the infimum is attained and equals $λ_{1}$ . The minimizer set is precisely the eigenspace of $λ_{1}$ .

Exercise 5 (medium, symbolic).

Prove the Courant-Fischer min-max formula for the $k$ -th eigenvalue of symmetric $L$ : $λ_{k} = S \subseteq H_{0}^{1} d i m S = k min u \in S ∖ {0} max R [u],$ the minimum over all $k$ -dimensional subspaces $S$ of $H_{0}^{1} (Ω)$ .

Hint

Upper bound: test with $S = span {w_{1}, \dots, w_{k}}$ and bound $R$ on it. Lower bound: any $k$ -dimensional $S$ meets $span {w_{k}, w_{k + 1}, \dots}$ (a codimension- $(k - 1)$ subspace) in a nonzero vector; evaluate $R$ there.

Answer

Upper bound. Take $S_{0} = span {w_{1}, \dots, w_{k}}$ , dimension $k$ . For $u = \sum_{j = 1}^{k} c_{j} w_{j} \in S_{0}$ , $R [u] = (\sum_{j \leq k} λ_{j} c_{j}^{2}) / (\sum_{j \leq k} c_{j}^{2}) \leq λ_{k}$ since $λ_{j} \leq λ_{k}$ for $j \leq k$ , with equality at $u = w_{k}$ . Hence $max_{u \in S_{0}} R [u] = λ_{k}$ , so the outer minimum is $\leq λ_{k}$ .

Lower bound. Let $S$ be any $k$ -dimensional subspace and $V = \overline{span} {w_{k}, w_{k + 1}, \dots}$ , of codimension $k - 1$ in $L^{2}$ . Then $dim S + codim V = k + (k - 1)$ , and a $k$ -dimensional subspace and a codimension- $(k - 1)$ subspace of an infinite-dimensional space always meet, so $S \cap V \neq = {0}$ by dimension count. Pick $0 \neq = u \in S \cap V$ , $u = \sum_{j \geq k} c_{j} w_{j}$ ; then $R [u] = (\sum_{j \geq k} λ_{j} c_{j}^{2}) / (\sum_{j \geq k} c_{j}^{2}) \geq λ_{k}$ . Thus $max_{u \in S} R [u] \geq R [u] \geq λ_{k}$ for every such $S$ , so the outer minimum is $\geq λ_{k}$ . Combining the two bounds gives equality. (The dual max-min formula $λ_{k} = max_{d i m T = k - 1} min_{u ⊥ T} R [u]$ follows symmetrically.)

Exercise 6 (medium, numeric).

Use the Rayleigh quotient to bound the first Dirichlet eigenvalue of $- Δ$ on $Ω = (0, π)$ from above by testing with the trial function $u (x) = x (π - x)$ . Compute $R [u] = \int_{0}^{π} (u^{'})^{2} d x / \int_{0}^{π} u^{2} d x$ and give it as a decimal to two places. (The exact eigenvalue is $λ_{1} = 1$ .)

Hint

$u^{'} = π - 2 x$ , so $\int_{0}^{π} (u^{'})^{2} = π^{3} /3$ . And $\int_{0}^{π} x^{2} (π - x)^{2} d x = π^{5} /30$ . Form the ratio.

Answer

$10/ π^{2} \approx 1.01$ . With $u^{'} = π - 2 x$ , $\int_{0}^{π} (π - 2 x)^{2} d x = π^{3} /3$ . Expanding $x^{2} (π - x)^{2}$ and integrating gives $\int_{0}^{π} x^{2} (π - x)^{2} d x = π^{5} /30$ . Hence $R [u] = (π^{3} /3) / (π^{5} /30) = 10/ π^{2} \approx 1.013$ . The Rayleigh quotient of any trial function bounds $λ_{1}$ from above, and the value $1.01$ is within one percent of the exact $λ_{1} = 1$ because the parabola $x (π - x)$ closely approximates the true eigenfunction $sin x$ .

Exercise 7 (hard, symbolic).

Prove monotonicity of eigenvalues under domain inclusion (domain monotonicity) for the Dirichlet Laplacian: if $Ω_{1} \subseteq Ω_{2}$ are bounded, then $λ_{k} (Ω_{2}) \leq λ_{k} (Ω_{1})$ for every $k$ .

Hint

Extension by zero embeds $H_{0}^{1} (Ω_{1})$ isometrically into $H_{0}^{1} (Ω_{2})$ , preserving both $\int ∣ D u ∣^{2}$ and $\int u^{2}$ . Feed this into the Courant-Fischer min-max of Exercise 5.

Answer

Extension by zero $ι : H_{0}^{1} (Ω_{1}) \to H_{0}^{1} (Ω_{2})$ sends $u$ to the function equal to $u$ on $Ω_{1}$ and $0$ on $Ω_{2} ∖ Ω_{1}$ ; since $u \in H_{0}^{1} (Ω_{1})$ has zero trace, $ι u \in H_{0}^{1} (Ω_{2})$ , and $\int_{Ω_{2}} ∣ D ι u ∣^{2} = \int_{Ω_{1}} ∣ D u ∣^{2}$ , $\int_{Ω_{2}} (ι u)^{2} = \int_{Ω_{1}} u^{2}$ , so $R_{Ω_{2}} [ι u] = R_{Ω_{1}} [u]$ and $ι$ is a Rayleigh-quotient-preserving isometry onto a subspace. Given a $k$ -dimensional $S \subseteq H_{0}^{1} (Ω_{1})$ , $ι (S)$ is a $k$ -dimensional subspace of $H_{0}^{1} (Ω_{2})$ with $max_{ι (S)} R_{Ω_{2}} = max_{S} R_{Ω_{1}}$ . By Courant-Fischer (Exercise 5), $λ_{k} (Ω_{2}) = min_{d i m T = k} max_{T} R_{Ω_{2}} \leq max_{ι (S)} R_{Ω_{2}} = max_{S} R_{Ω_{1}}$ . Taking the minimum over $S$ gives $λ_{k} (Ω_{2}) \leq min_{d i m S = k} max_{S} R_{Ω_{1}} = λ_{k} (Ω_{1})$ . Enlarging the domain lowers every eigenvalue: more room to vibrate means lower tones.

Exercise 8 (hard, symbolic).

Prove that the principal eigenvalue $λ_{1}$ of a symmetric coercive $L$ ( $b \equiv 0$ , $c \geq 0$ ) is simple and that its eigenfunction may be taken strictly positive, assuming the strong maximum principle in the form: a nonzero $w \in H_{0}^{1} (Ω)$ with $w \geq 0$ a.e. and $L w = λ_{1} w$ weakly satisfies $w > 0$ a.e. in $Ω$ .

Hint

If $w_{1}$ minimizes $R$ , so does $∣ w_{1} ∣$ because $\int ∣ D ∣ w_{1} ∣ ∣^{2} = \int ∣ D w_{1} ∣^{2}$ . So a nonnegative minimizer exists; the maximum principle promotes it to strictly positive. For simplicity, two strictly positive eigenfunctions cannot be $L^{2}$ -orthogonal.

Answer

Let $w_{1}$ be an eigenfunction for $λ_{1}$ , normalized in $L^{2}$ . The chain rule for Sobolev functions gives $∣ w_{1} ∣ \in H_{0}^{1} (Ω)$ with $D ∣ w_{1} ∣ = (sgn w_{1}) D w_{1}$ , so $\int_{Ω} a^{ij} \partial_{j} ∣ w_{1} ∣ \partial_{i} ∣ w_{1} ∣ = \int_{Ω} a^{ij} \partial_{j} w_{1} \partial_{i} w_{1}$ and $\int c ∣ w_{1} ∣^{2} = \int c w_{1}^{2}$ ; hence $R [∣ w_{1} ∣] = R [w_{1}] = λ_{1}$ , so $∣ w_{1} ∣$ is also a minimizer and therefore (Exercise 4) an eigenfunction. Being a nonnegative eigenfunction, the strong maximum principle forces $∣ w_{1} ∣ > 0$ a.e.; in particular $w_{1}$ cannot vanish on a set of positive measure, so $w_{1}$ has a strict sign and may be taken as $w_{1} = ∣ w_{1} ∣ > 0$ . For simplicity, suppose $w$ and $\tilde{w}$ were two linearly independent eigenfunctions of $λ_{1}$ . Both could be taken strictly positive by the argument above, but eigenfunctions of distinct... they share $λ_{1}$ , so consider $ϕ = w - t \tilde{w}$ with $t = (\int w / \int \tilde{w})$ chosen so $\int_{Ω} ϕ = 0$ . Then $ϕ$ is again a $λ_{1}$ -eigenfunction with $\int ϕ = 0$ , so $ϕ$ changes sign or vanishes; if $ϕ \neq = 0$ then $∣ ϕ ∣$ is a nonnegative eigenfunction hence strictly positive, contradicting $\int ϕ = 0$ (a strictly positive function has positive integral while $∣ ϕ ∣ = \pm ϕ$ pointwise gives $\int ∣ ϕ ∣ = ∣ \int_{{ϕ > 0}} ϕ ∣ + ∣ \int_{{ϕ < 0}} ϕ ∣ > 0$ yet the maximum principle says $∣ ϕ ∣ > 0$ contradicts $ϕ$ vanishing anywhere). Hence $ϕ = 0$ , so $w = t \tilde{w}$ : the eigenspace is one-dimensional. The principal eigenvalue is simple with a sign-definite eigenfunction.

Advanced results Master

The Fredholm alternative and the eigenvalue ladder sit inside a wider structure: the Krein-Rutman theorem that abstracts the positivity of the principal eigenfunction to cone-preserving compact operators, the Weyl asymptotics that fix the growth rate of $λ_{k}$ , the variational stability of the whole spectrum under perturbation, the nodal-domain count that reads geometry off the eigenfunctions, and the heat- and wave-equation expansions that the orthonormal eigenbasis makes possible. Each sharpens the compact-self-adjoint argument of the Intermediate tier.

Theorem 1 (Krein-Rutman; positivity of the principal eigenvalue). Let $G$ be a compact operator on an ordered Banach space $X$ that maps a solid closed convex cone $K$ into itself and is strongly positive (sends nonzero $u \geq 0$ into the interior of $K$ ). Then the spectral radius $ρ (G) > 0$ is an eigenvalue of $G$ with a positive eigenvector, this eigenvalue is simple, and no other eigenvalue has a positive eigenvector ^{[Krein-Rutman 1948]}. Applied to the solution operator $G$ of a coercive symmetric $L$ on $L^{2} (Ω)$ with the cone of nonnegative functions, strong positivity is the strong maximum principle, $ρ (G) = (λ_{1} + μ)^{- 1}$ , and the conclusion is the simplicity and sign-definiteness of the principal Dirichlet eigenfunction proved variationally in Exercise 8. The theorem extends the Perron-Frobenius theorem for positive matrices to infinite dimensions, and it governs principal eigenvalues even for non-self-adjoint $L$ , where the variational route is unavailable.

Theorem 2 (Weyl's law). For the Dirichlet Laplacian on a bounded $Ω \subset R^{n}$ the counting function $N (λ) = # {k : λ_{k} \leq λ}$ satisfies $N (λ) \sim \frac{ω _{n} ∣Ω∣}{( 2 π ) ^{n}} λ^{n /2} (λ \to \infty),$ equivalently $λ_{k} \sim 4 π^{2} (\frac{k}{ω _{n} ∣Ω∣})^{2/ n}$ , where $ω_{n}$ is the volume of the unit ball ^{[Courant 1920]}. The leading term depends only on the dimension and the volume — "one can hear the volume of a drum" — and the proof bounds $N (λ)$ above and below by Courant-Fischer min-max applied to Dirichlet and Neumann brackets on a partition of $Ω$ into cubes, then lets the mesh refine. The compactness that gives a discrete spectrum in the first place is the precondition for $N (λ)$ to be finite for each $λ$ .

Theorem 3 (eigenvalue perturbation and continuity). The eigenvalues $λ_{k} (L)$ are Lipschitz-continuous functionals of the coefficients in the appropriate norms and, for self-adjoint $L$ , the min-max formula makes each $λ_{k}$ a monotone, continuous function of the form $B$ : if $B \leq B^{'}$ pointwise as quadratic forms then $λ_{k} \leq λ_{k}^{'}$ . Analytic perturbation theory (Rellich-Kato) gives more: for a one-parameter family $L_{t} = L_{0} + t L_{1}$ with $L_{1}$ relatively bounded, a simple eigenvalue $λ_{k} (t)$ and its eigenprojection are real-analytic in $t$ , with first variation $\dot{λ}_{k} = (L_{1} w_{k}, w_{k})_{L^{2}}$ (the Feynman-Hellmann formula). Degenerate eigenvalues split analytically into branches, and the crossing structure is governed by the symmetry group of $L_{0}$ .

Theorem 4 (Courant's nodal domain theorem). The $k$ -th Dirichlet eigenfunction $w_{k}$ (eigenvalues ordered with multiplicity) has at most $k$ nodal domains — connected components of ${w_{k} \neq = 0}$ ^{[Courant 1920]}. In particular $w_{1}$ has exactly one nodal domain, recovering its sign-definiteness, and $w_{2}$ has exactly two. The proof is again min-max: if $w_{k}$ had $k + 1$ nodal domains, the indicator combinations supported on the first $k$ of them would span a $k$ -dimensional space on which $R \leq λ_{k}$ yet orthogonal to $w_{1}, \dots, w_{k - 1}$ , contradicting the strict ordering. The number and shape of nodal domains encode geometric and even arithmetic information about $Ω$ , the subject of the Pleijel and Bogomolny-Schmit refinements.

Theorem 5 (spectral expansions and the heat semigroup). The orthonormal eigenbasis ${w_{k}}$ diagonalizes every function of $L$ : for $f \in L^{2} (Ω)$ , $f = \sum_{k} (f, w_{k})_{L^{2}} w_{k}$ , and the Dirichlet heat semigroup is $e^{- t L} f = k \sum e^{- λ_{k} t} (f, w_{k})_{L^{2}} w_{k}, t > 0,$ with heat kernel $p (t, x, y) = \sum_{k} e^{- λ_{k} t} w_{k} (x) w_{k} (y)$ and trace $Tr e^{- t L} = \sum_{k} e^{- λ_{k} t} \sim (4 π t)^{- n /2} ∣Ω∣$ as $t ↓ 0$ , the small-time asymptotic equivalent to Weyl's law by Karamata's Tauberian theorem. The wave equation $\partial_{t}^{2} u + Lu = 0$ has the standing-wave solutions $cos (λ_{k} t) w_{k}$ , the literal pure tones of the drumhead intuition. Mercer's theorem guarantees the eigenfunction series for the kernel converges absolutely, the trace-class refinement of compactness from 02.11.05.

Synthesis. The Fredholm alternative and the eigenvalue ladder are one compactness statement read in two registers: the compactness of the solution operator $G$ built from Lax-Milgram 02.16.04 and Rellich-Kondrachov 02.16.03 is the foundational reason the spectrum is discrete and real, and this is exactly the Riesz-Schauder theory of 02.11.05 transported through the dictionary $κ = (λ + μ)^{- 1}$ . The discreteness of the spectrum is dual to the accumulation of $G$ 's eigenvalues at zero, one read as tones climbing to infinity and the other as the compressing operator's singular values decaying; putting these together, the variational min-max characterization is the same data read a third way, with each $λ_{k}$ an attained Rayleigh-quotient extremum precisely because compactness upgrades the minimizing sequence to a strong limit, exactly the mechanism that the direct method of 02.16.03 supplies. The Krein-Rutman positivity generalises the Perron-Frobenius theorem and pins the principal eigenfunction's sign, the central insight being that the same compact $G$ that gives the Fredholm dichotomy also preserves the positive cone, so existence, spectrum, and positivity are three faces of one operator; the bridge from here is to Weyl's law and the heat-kernel trace, where the eigenbasis turns every linear evolution governed by $L$ into an explicit superposition of decaying or oscillating pure modes.

Full proof set Master

Proposition 1 (compactness and self-adjointness of the solution operator). For symmetric uniformly elliptic $L$ on bounded $Ω$ and $μ \geq γ$ , the solution operator $G$ of $L + μ$ is a compact, self-adjoint, positive operator on $L^{2} (Ω)$ with $ker G = {0}$ .

Proof. For $g \in L^{2}$ , $G g \in H_{0}^{1}$ is the Lax-Milgram solution of $B_{μ} [G g, v] = (g, v)_{L^{2}}$ , and the a priori bound of 02.16.04 gives $∥ G g ∥_{H_{0}^{1}} \leq β^{- 1} ∥ g ∥_{L^{2}}$ , so $G : L^{2} \to H_{0}^{1}$ is bounded. Composing with the compact embedding $H_{0}^{1} ↪↪ L^{2}$ of 02.16.03, $G : L^{2} \to L^{2}$ is compact. Self-adjointness: for $g, h \in L^{2}$ , symmetry of $B_{μ}$ gives $(G g, h)_{L^{2}} = B_{μ} [G h, G g] = B_{μ} [G g, G h] = (g, G h)_{L^{2}}$ . Positivity: $(G g, g)_{L^{2}} = B_{μ} [G g, G g] \geq β ∥ G g ∥_{H_{0}^{1}}^{2} \geq 0$ , and $(G g, g)_{L^{2}} = 0$ forces $∥ G g ∥_{H_{0}^{1}} = 0$ , i.e. $G g = 0$ , whence $g = 0$ from $B_{μ} [G g, v] = (g, v)$ ; so $ker G = {0}$ and $G$ is strictly positive on the orthogonal complement of $0$ , i.e. all of $L^{2}$ . $□$

Proposition 2 (the spectral ladder). For symmetric $L$ , the eigenvalues of $L$ form a sequence $λ_{1} \leq λ_{2} \leq \dots \to + \infty$ with finite multiplicities, and the eigenfunctions form an $L^{2}$ -orthonormal basis.

Proof. By Proposition 1, $G$ is compact self-adjoint with zero kernel. The spectral theorem for compact self-adjoint operators 02.11.05 yields an orthonormal basis ${w_{k}}$ of $L^{2} (Ω)$ of eigenfunctions of $G$ , eigenvalues $κ_{k}$ real, of finite multiplicity, with $0$ the only accumulation point; positivity gives $κ_{k} > 0$ , and $ker G = {0}$ means $0$ is not itself an eigenvalue, so $κ_{1} \geq κ_{2} \geq \dots \to 0$ with each $κ_{k} > 0$ . By the dictionary (Exercise 3), each $w_{k}$ is an eigenfunction of $L$ with $λ_{k} = κ_{k}^{- 1} - μ$ , the ordering reverses, $λ_{1} \leq λ_{2} \leq \dots$ , and $κ_{k} ↓ 0$ gives $λ_{k} \to + \infty$ . Multiplicities transfer since the $λ_{k}$ - and $κ_{k}$ -eigenspaces coincide. $□$

Proposition 3 (Courant-Fischer min-max). For symmetric $L$ with eigenvalues $λ_{1} \leq λ_{2} \leq \dots$ (with multiplicity), $λ_{k} = d i m S = k min u \in S ∖ {0} max \frac{B [ u , u ]}{∥ u ∥ _{L^{2}}^{2}} = d i m T = k - 1 max u ⊥_{L^{2}} T u \neq = 0 min \frac{B [ u , u ]}{∥ u ∥ _{L^{2}}^{2}} .$

Proof. Diagonalize as in Exercise 4: for $u = \sum_{j} c_{j} w_{j} \in H_{0}^{1}$ , $B [u, u] = \sum_{j} λ_{j} c_{j}^{2}$ and $∥ u ∥_{L^{2}}^{2} = \sum_{j} c_{j}^{2}$ , both convergent because $u \in H_{0}^{1}$ gives $\sum_{j} λ_{j} c_{j}^{2} = B [u, u] < \infty$ . For the min-max upper bound, $S_{0} = span {w_{1}, \dots, w_{k}}$ has $max_{S_{0}} R = λ_{k}$ (attained at $w_{k}$ ). For the lower bound, any $k$ -dimensional $S$ meets the closed subspace $\overline{span} {w_{j} : j \geq k}$ of $L^{2}$ -codimension $k - 1$ in a nonzero vector $u = \sum_{j \geq k} c_{j} w_{j}$ , where $R [u] = (\sum_{j \geq k} λ_{j} c_{j}^{2}) / (\sum_{j \geq k} c_{j}^{2}) \geq λ_{k}$ , so $max_{S} R \geq λ_{k}$ ; minimizing over $S$ gives $λ_{k}$ . The max-min formula is dual: $T_{0} = span {w_{1}, \dots, w_{k - 1}}$ gives $min_{u ⊥ T_{0}} R = λ_{k}$ attained at $w_{k}$ , and for any $(k - 1)$ -dimensional $T$ the space $span {w_{1}, \dots, w_{k}}$ meets $T^{⊥}$ in a nonzero vector with $R \leq λ_{k}$ there, so $min_{u ⊥ T} R \leq λ_{k}$ . $□$

Proposition 4 (simplicity and positivity of the principal eigenvalue). For symmetric coercive $L$ ( $b \equiv 0$ , $c \geq 0$ ) on a bounded connected $Ω$ , the principal eigenvalue $λ_{1} = min_{u \neq = 0} R [u]$ is simple and admits a strictly positive eigenfunction.

Proof. A minimizer of $R$ exists by Proposition 3 ( $k = 1$ ) and is a $λ_{1}$ -eigenfunction $w_{1}$ . The Sobolev chain rule gives $∣ w_{1} ∣ \in H_{0}^{1}$ with $D ∣ w_{1} ∣ = (sgn w_{1}) D w_{1}$ , hence $B [∣ w_{1} ∣, ∣ w_{1} ∣] = B [w_{1}, w_{1}]$ and $∥∣ w_{1} ∣ ∥_{L^{2}} = ∥ w_{1} ∥_{L^{2}}$ , so $∣ w_{1} ∣$ is also a minimizer and a nonnegative $λ_{1}$ -eigenfunction. The strong maximum principle for the coercive operator $L$ forces a nonnegative nonzero solution of $L w = λ_{1} w$ to satisfy $w > 0$ a.e. in the connected $Ω$ ; thus $∣ w_{1} ∣ > 0$ and $w_{1}$ has a strict sign. If $w, \tilde{w}$ were independent $λ_{1}$ -eigenfunctions, choose $s$ with $\int_{Ω} (w - s \tilde{w}) = 0$ ; then $w - s \tilde{w}$ is a $λ_{1}$ -eigenfunction of zero mean, but every $λ_{1}$ -eigenfunction is, by the previous step, of one strict sign unless identically zero, and a strictly signed function has nonzero mean. Hence $w - s \tilde{w} \equiv 0$ , the eigenspace is one-dimensional, and $λ_{1}$ is simple. $□$

Connections Master

The compact-operator engine is the Riesz-Schauder spectral theory and the spectral theorem for compact self-adjoint operators of 02.11.05: the Fredholm alternative for $L$ is that dichotomy for $κ I - G$ read through the dictionary $κ = (λ + μ)^{- 1}$ , and the orthonormal eigenbasis is the compact-self-adjoint spectral theorem applied to the solution operator $G$ . This unit owns the elliptic realization and the spectral dictionary; 02.11.05 owns the abstract compact-operator spectral theory.
The solution operator $G$ is manufactured from the Lax-Milgram existence theorem of 02.16.04: the bounded inverse $(L + μ)^{- 1}$ supplied there, restricted to $L^{2}$ and post-composed with the compact embedding, is the compact $G$ whose spectrum this unit analyzes. The sibling unit provides the invertible shift; this unit reads its spectrum and completes the Fredholm-alternative half of that unit's Advanced tier.
The compactness input is exactly the Rellich-Kondrachov embedding and the Poincaré coercivity of 02.16.03: Poincaré makes $B_{μ}$ coercive so that $(L + μ)^{- 1}$ exists, and the compact embedding $H_{0}^{1} ↪↪ L^{2}$ is what makes $G$ compact rather than merely bounded, hence the spectrum discrete. The attainment of each min-max extremum is the same weak-to-strong upgrade that the sibling unit's compactness theorem provides for minimizing sequences.
The eigenvalue ladder and the eigenfunction expansion are the foundation of the spectral theory of evolution equations: the heat semigroup $e^{- t L}$ and wave propagator on $Ω$ are diagonalized by the eigenbasis, connecting this unit to the parabolic and hyperbolic theory and to the harmonic analysis of 02.10.04 (the eigenfunction expansion is the bounded-domain analogue of the Fourier transform), where $- Δ$ on the torus has the exponentials as its eigenbasis.
The variational min-max and the domain monotonicity tie this unit to the calculus of variations and the Faber-Krahn isoperimetric theory of eigenvalues, while the higher regularity of eigenfunctions runs the difference-quotient and Schauder bootstrap of 02.16.04 and the regularity chapter, promoting a weak eigenfunction $w_{k} \in H_{0}^{1}$ to a smooth classical one when the coefficients are smooth.

Historical & philosophical context Master

The Fredholm alternative originates in Erik Ivar Fredholm's 1903 Acta Mathematica theory of integral equations ^{[Fredholm 1903]}, where the dichotomy for $(I - K) u = f$ with $K$ an integral operator of continuous kernel first appeared, modelled deliberately on Cramer's rule for finite linear systems. Frigyes Riesz, in 1918, abstracted the kernel hypothesis to the operator-theoretic notion of complete continuity (compactness) and proved the alternative and the discrete-eigenvalue structure for compact operators on function spaces ^{[Riesz 1918]}; Juliusz Schauder extended the framework to Banach spaces and to the adjoint in 1930 ^{[Schauder 1930]}, completing the Riesz-Schauder theory that this unit applies to the elliptic solution operator.

The eigenvalue side descends from the nineteenth-century theory of vibrating membranes and the Sturm-Liouville theory of one-dimensional eigenvalue problems. The variational characterization in its modern min-max form is due to Richard Courant, whose 1920 Mathematische Zeitschrift paper ^{[Courant 1920]} established both the min-max formula and the nodal domain theorem, building on the independent quadratic-form min-max principle of Ernst Fischer's 1905 work ^{[Fischer 1905]}. The positivity and simplicity of the principal eigenvalue were placed in their natural generality by Mark Krein and Mark Rutman in 1948 ^{[Krein-Rutman 1948]}, whose theorem on compact operators preserving a cone extended the Perron-Frobenius theory of positive matrices to infinite dimensions and supplied the principal-eigenvalue theory for non-self-adjoint elliptic operators. Hermann Weyl's 1911 asymptotic law for the eigenvalue counting function answered a question of Lorentz and Sommerfeld about black-body radiation, fixing the leading growth of $λ_{k}$ in terms of the domain's volume and dimension.

Bibliography Master

@article{Fredholm1903,
  author  = {Fredholm, Erik Ivar},
  title   = {Sur une classe d'\'equations fonctionnelles},
  journal = {Acta Mathematica},
  volume  = {27},
  year    = {1903},
  pages   = {365--390}
}

@article{Riesz1918,
  author  = {Riesz, Frigyes},
  title   = {\"Uber lineare Funktionalgleichungen},
  journal = {Acta Mathematica},
  volume  = {41},
  year    = {1918},
  pages   = {71--98}
}

@article{Schauder1930,
  author  = {Schauder, Juliusz},
  title   = {\"Uber lineare, vollstetige Funktionaloperationen},
  journal = {Studia Mathematica},
  volume  = {2},
  year    = {1930},
  pages   = {183--196}
}

@article{Courant1920,
  author  = {Courant, Richard},
  title   = {\"Uber die Eigenwerte bei den Differentialgleichungen der mathematischen Physik},
  journal = {Mathematische Zeitschrift},
  volume  = {7},
  year    = {1920},
  pages   = {1--57}
}

@article{Fischer1905,
  author  = {Fischer, Ernst},
  title   = {\"Uber quadratische Formen mit reellen Koeffizienten},
  journal = {Monatshefte f\"ur Mathematik und Physik},
  volume  = {16},
  year    = {1905},
  pages   = {234--249}
}

@article{KreinRutman1948,
  author  = {Krein, Mark G. and Rutman, Mark A.},
  title   = {Linear operators leaving invariant a cone in a Banach space},
  journal = {Uspekhi Matematicheskikh Nauk},
  volume  = {3},
  number  = {1},
  year    = {1948},
  pages   = {3--95}
}

@book{ReedSimonIV1978,
  author    = {Reed, Michael and Simon, Barry},
  title     = {Methods of Modern Mathematical Physics IV: Analysis of Operators},
  publisher = {Academic Press},
  year      = {1978}
}

Prerequisites

02.16.04
02.11.05
02.16.03

Tier anchors

beginner: Strogatz-style intuition for why an infinite system can behave like a finite matrix once a compressing step is added, and 3Blue1Brown's 'Essence of Linear Algebra' picture of the dichotomy 'a square matrix is either invertible or has a kernel', together with the standing-wave picture of a drumhead's pure tones as the eigenfunctions of a differential operator
intermediate: Evans, Partial Differential Equations, 2e (AMS GSM 19, 2010), §6.2 (Fredholm alternative) and §6.5 (eigenvalues of self-adjoint elliptic operators); Brezis, Functional Analysis, Sobolev Spaces and PDEs (Springer 2011), §6.3-§6.4 and Ch. 9
master: Evans §6.2, §6.5; Gilbarg-Trudinger, Elliptic Partial Differential Equations of Second Order, 2e (Springer 1983), §8.6, §8.12; Courant-Hilbert, Methods of Mathematical Physics I (Wiley 1953), Ch. VI; Reed-Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic Press 1978), §XIII.12-§XIII.15

References

Fredholm — Sur une classe d'équations fonctionnelles · Acta Mathematica 27 (1903), 365-390
Riesz, F. — Über lineare Funktionalgleichungen · Acta Mathematica 41 (1918), 71-98
Schauder — Über lineare, vollstetige Funktionaloperationen · Studia Mathematica 2 (1930), 183-196
Courant — Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik · Mathematische Zeitschrift 7 (1920), 1-57
Fischer — Über quadratische Formen mit reellen Koeffizienten · Monatshefte für Mathematik und Physik 16 (1905), 234-249
Krein, Rutman — Linear operators leaving invariant a cone in a Banach space · Uspekhi Matematicheskikh Nauk 3 (1948), 3-95; AMS Translation 26 (1950)
Evans — Partial Differential Equations, 2e · AMS Graduate Studies in Mathematics 19 (2010), §6.2, §6.5
Gilbarg-Trudinger — Elliptic Partial Differential Equations of Second Order, 2e · Springer Grundlehren 224 (1983), §8.6, §8.12
Courant-Hilbert — Methods of Mathematical Physics I · Wiley-Interscience (1953), Ch. VI
Reed-Simon — Methods of Modern Mathematical Physics IV · Academic Press (1978), §XIII.12-§XIII.15

Estimated time

beginner: 25m
intermediate: 65m
master: 105m