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The Fredholm alternative for weak elliptic Dirichlet problems

Statement

Assume the Axiom of Choice and Countable Choice. Let $Ω \subseteq R^{n}$ be bounded open, $K \in {R, C}$ , and let $L, a$ be as in Uniformly elliptic divergence-form operators and their sesquilinear forms with ellipticity constant $θ$ and coefficient bounds $M_{a}, M_{b}, M_{c}$ . Let $a^{*}$ be the adjoint form of The formal adjoint and the adjoint weak Dirichlet problem. Consider the weak Dirichlet problem $a (u, v) = (f, v)_{L^{2}}$ for all $v \in H_{0}^{1} (Ω)$ , with datum $f \in L^{2} (Ω)$ (Weak Dirichlet solutions for a divergence-form operator). Then exactly one of the following alternatives holds. (1) The homogeneous problem $a (u, v) = 0$ for all $v \in H_{0}^{1} (Ω)$ has only the solution $u = 0$ . Then for every $f \in L^{2} (Ω)$ the problem has exactly one weak solution $u \in H_{0}^{1} (Ω)$ . (2) The homogeneous problem has a nonzero solution. Then both homogeneous solution spaces $N : = {u \in H_{0}^{1} (Ω) : a (u, v) = 0 \forall v}, N^{*} : = {v \in H_{0}^{1} (Ω) : a^{*} (v, w) = 0 \forall w}$ are finite-dimensional and nontrivial with $\dim N = \dim N^{*}$ ; for $f \in L^{2} (Ω)$ the problem has a solution if and only if $(f, v)_{L^{2}} = 0$ for every $v \in N^{*}$ ; and whenever a solution exists the solution set is an affine translate of $N$ , so uniqueness fails. The data class is $L^{2} (Ω)$ ; the weaker class $H^{- 1} (Ω)$ is deliberately not treated here.

Facts & Assumptions

Given: the Axiom of Choice and Countable Choice; a bounded open set $Ω \subseteq R^{n}$ ; the divergence-form operator $L$ and form $a$ with constants $θ, M_{a}, M_{b}, M_{c}$ ; the adjoint form $a^{*}$ ; a fixed $μ \geq β$ ; the shifted solution operator $K_{μ}$ and its adjoint $K_{μ}^{*}$ ; and $f \in L^{2} (Ω)$ .

[F1]

Operator reduction: for $u \in H_{0}^{1} (Ω)$ the weak equation $a (u, v) = (f, v)_{L^{2}}$ for all $v \in H_{0}^{1} (Ω)$ is equivalent to $(I - μ K_{μ}) u = K_{μ} f$ in $L^{2} (Ω)$ , and both sides of that equation lie in $H_{0}^{1} (Ω)$ (On bounded domains, the unshifted equation is an identity-minus-compact equation).

[F2]

Compactness and the abstract alternative: $A : = I - μ K_{μ}$ is an identity-minus-compact operator on the Banach space $L^{2} (Ω)$ , the homogeneous spaces satisfy $N = \ker A$ and, under the Riesz identification, $N^{*} = \ker A^{*}$ , and $K_{μ} f \in ran A$ if and only if $(f, v)_{L^{2}} = 0$ for every $v \in N^{*}$ (The adjoint solution operator solves the adjoint form problem, The elliptic Fredholm range condition is orthogonality to the adjoint kernel, Fredholm alternative for identity minus compact, Kernel of identity minus compact is finite dimensional, Compact linear operator, The Axiom of Choice).

[F3]

Abstract Fredholm alternatives: for a compact $K$ on a Banach space and $A = I - K$ , either $A$ is injective, in which case it is bijective with bounded inverse, or $\ker A$ and the cokernel are finite dimensional and nontrivial with equal dimensions (Fredholm alternative for identity minus compact). The range of $A = I - μ K_{μ}$ is closed by Range of identity minus compact is closed; AC supplies its DC premise by AC supplies the countable and dependent choices used in Banach integration. By Orthogonal decomposition by a closed subspace, $L^{2} = ran A \oplus (ran A)^{⊥}$ . The adjoint identity gives $(ran A)^{⊥} = \ker (I - μ K_{μ}^{*})$ , and $v \mapsto v + ran A$ restricts to a linear bijection from this orthogonal kernel onto the cokernel.

[F4]

Data conventions: weak solutions are $H_{0}^{1}$ classes, the datum $f \in L^{2} (Ω)$ acts through the conjugate-linear pairing $v \mapsto (f, v)_{L^{2}}$ , and the adjoint form is $a^{*} (v, w) = \overline{a (w, v)}$ (Weak Dirichlet solutions for a divergence-form operator, The formal adjoint and the adjoint weak Dirichlet problem, The space $L^{p} (μ)$ as the quotient by null functions, Zero-boundary Sobolev space as a norm closure, The shifted elliptic solution operator, Uniformly elliptic divergence-form operators and their sesquilinear forms, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1F1F2F4given

Identification of the spaces. By [F1] with $f = 0$ , a class $u \in H_{0}^{1} (Ω)$ solves the homogeneous problem $a (u, v) = 0$ for all $v$ if and only if $(I - μ K_{μ}) u = 0$ ; since $ran K_{μ} \subseteq H_{0}^{1} (Ω)$ , this identifies $N$ with $\ker A$ , where $A = I - μ K_{μ}$ is bounded on $L^{2} (Ω)$ . By [F2] the adjoint homogeneous space $N^{*}$ is, under the Riesz identification of $L^{2} (Ω)$ with its dual, exactly the kernel of the transpose $A^{*}$ , and for $f \in L^{2} (Ω)$ the solvability of the weak problem is equivalent to $K_{μ} f \in ran A$ , hence to $(f, v)_{L^{2}} = 0$ for every $v \in N^{*}$ .

2.1F2F3step 1.1given

The two alternatives. Since $μ K_{μ}$ is compact, [F3] gives exactly the following dichotomy for $A = I - μ K_{μ}$ : either $A$ is injective, hence bijective with bounded inverse, or $\ker A \neq {0}$ and both $\ker A$ and the cokernel are finite dimensional with equal dimensions. In the first case $\ker A = N = {0}$ by step 1.1.

3.1F1step 2.1given

Case (1). Assume $N = {0}$ . Then $A$ is injective, so by step 2.1 it is bijective and boundedly invertible; for every $f \in L^{2} (Ω)$ the equation $A u = K_{μ} f$ has the unique solution $u = A^{- 1} K_{μ} f \in L^{2} (Ω)$ ; the equation gives $u = K_{μ} f + μ K_{μ} u \in H_{0}^{1} (Ω)$ , and then [F1] shows that it solves the weak problem, and by the equivalence [F1] any weak solution gives a solution of $A u = K_{μ} f$ , so the weak solution is unique. This proves alternative (1).

3.2F1F2step 1.1step 2.1givenalgebra

Case (2). Assume $N \neq {0}$ . Then $\ker A = N$ is nontrivial and finite dimensional, and by step 2.1 its dimension equals that of the cokernel, which under the Riesz identification is $\dim \ker A^{*} = \dim N^{*}$ ; so $N$ and $N^{*}$ are finite-dimensional and nontrivial with $\dim N = \dim N^{*}$ . By step 1.1 the weak problem is solvable exactly when $(f, v)_{L^{2}} = 0$ for every $v \in N^{*}$ . If $u_{0}$ is one solution, then for any $u$ the class $u - u_{0}$ satisfies the homogeneous problem, i.e. lies in $N$ , and conversely $u_{0} + n$ with $n \in N$ is a solution; hence the solution set is the affine translate $u_{0} + N$ , which is not a singleton because $N \neq {0}$ , so uniqueness fails. This proves alternative (2).

4.1F2F3step 3.1step 3.2given∎

Exhaustiveness. Steps 3.1 and 3.2 cover the two mutually exclusive possibilities of step 2.1, so exactly one of the alternatives holds; the datum class is $L^{2} (Ω)$ throughout, no $H^{- 1} (Ω)$ data are used, and the Axiom of Choice is inherited only through the compactness of $K_{μ}$ and the abstract Fredholm alternative.

Depends on

Used by

The elliptic kernel and cokernel are finite dimensional Corollary
The global H² estimate without the L² term under uniqueness Corollary
Uniqueness implies existence for the elliptic Dirichlet problem Corollary
Elliptic Fredholm solvability can fail at an eigenvalue Counterexample

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Sources

John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page notes) (standard reference, not scraped)
Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, 2020, complete 158-page graduate notes) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, complete 392 pages) (standard reference, not scraped)