LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Coercivity of the principal Dirichlet form

Statement

Assume the Axiom of Choice, inherited through the Poincaré supplier named below, together with Countable Choice. Let $Ω \subseteq R^{n}$ be open and bounded in one direction, and let $a_{0} (u, v) = \int_{Ω} a^{i j} D_{j} u \overline{D_{i} v} d x$ be the principal part of a uniformly elliptic form with constants $θ$ and $M_{a}$ (Uniformly elliptic divergence-form operators and their sesquilinear forms), restricted to $u, v \in H_{0}^{1} (Ω)$ . Then $a_{0}$ is a bounded sesquilinear form on $H_{0}^{1} (Ω)$ and $Re a_{0} (u, u) = \int_{Ω} Re (a^{i j} D_{j} u \overline{D_{i} u}) d x \geq θ ∥ D u ∥_{L^{2}}^{2} \geq \frac{θ}{1 + C_{P}^{2}} ∥ u ∥_{H_{0}^{1}}^{2} (u \in H_{0}^{1} (Ω)),$ where $C_{P}$ is the Poincar'e constant of The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction for $p = 2$ . Hence $a_{0}$ is coercive on $H_{0}^{1} (Ω)$ with constant $α = θ / (1 + C_{P}^{2})$ , and $Re a_{0} (u, u) = θ ∥ D u ∥^{2}$ in the model case $a^{i j} = δ^{i j}$ (so $θ = 1$ ).

Facts & Assumptions

Given: The Axiom of Choice and Countable Choice; an open $Ω \subseteq R^{n}$ bounded in one direction; uniformly elliptic coefficients $a^{i j}$ with constants $θ > 0$ and $M_{a}$ (Uniformly elliptic divergence-form operators and their sesquilinear forms); and the principal form $a_{0} (u, v) = \int_{Ω} a^{i j} D_{j} u \overline{D_{i} v} d x$ on $H_{0}^{1} (Ω)$ .

[F1]

Uniform ellipticity: for almost every $x \in Ω$ and every $ξ \in C^{n}$ , $Re (\sum_{i, j} a^{i j} (x) ξ_{j} \overline{ξ_{i}}) \geq θ ∣ ξ ∣^{2}$ , and $∣ a^{i j} ∣ \leq M_{a}$ a.e. (Uniformly elliptic divergence-form operators and their sesquilinear forms).

[F2]

Absolute convergence and boundedness: the principal term is absolutely convergent for $u, v \in H^{1} (Ω)$ and $∣ a_{0} (u, v) ∣ \leq n M_{a} ∥ u ∥_{H_{0}^{1}} ∥ v ∥_{H_{0}^{1}}$ on $H_{0}^{1} (Ω)$ ; in particular $a_{0}$ is a bounded sesquilinear form (The elliptic form is well defined and bounded on $H^{1}$ ).

[F3]

Poincar'e at $p = 2$ : $∥ u ∥_{L^{2} (Ω)} \leq C_{P} ∥ D u ∥_{L^{2} (Ω)}$ for every $u \in H_{0}^{1} (Ω)$ , where $D u = (D_{1} u, \dots, D_{n} u)$ and $C_{P}$ is the constant of the cited theorem at $p = 2$ ; hence $∥ u ∥_{H_{0}^{1}}^{2} = ∥ u ∥_{L^{2}}^{2} + ∥ D u ∥_{L^{2}}^{2} \leq (1 + C_{P}^{2}) ∥ D u ∥_{L^{2}}^{2}$ (The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction, Integer-order Sobolev spaces and their norms, Zero-boundary Sobolev space as a norm closure).

[F4]

Nonnegative measurable functions have nonnegative integrals, and the integral of a nonnegative function is monotone under pointwise comparison; the real part of an integral of a complex-valued integrable function is the integral of its real part (Monotone convergence for the integral, Integral over a measurable subset, Real and imaginary parts, complex conjugation, and modulus).

[F5]

Coercivity of a sesquilinear form means $Re a (u, u) \geq α ∥ u ∥^{2}$ for all $u$ and some $α > 0$ (Bounded, coercive and symmetric sesquilinear forms).

Proof

1.1F1given

Pointwise bound: substituting $ξ = D u (x)$ in the ellipticity condition of [F1] and taking real parts gives, for almost every $x \in Ω$ , $Re (a^{i j} (x) D_{j} u (x) \overline{D_{i} u (x)}) \geq θ ∣ D u (x) ∣^{2} \geq 0.$

1.2F3

Poincar'e bound: by [F3], $∥ u ∥_{H_{0}^{1}}^{2} \leq (1 + C_{P}^{2}) ∥ D u ∥_{L^{2}}^{2}$ for every $u \in H_{0}^{1} (Ω)$ , that is $∥ D u ∥_{L^{2}}^{2} \geq ∥ u ∥_{H_{0}^{1}}^{2} / (1 + C_{P}^{2})$ .

2.1F1F2F4step 1.1

Integrating the pointwise bound: the function $x \mapsto Re (a^{i j} D_{j} u \overline{D_{i} u})$ is measurable and its negative part is bounded by $(n M_{a} + n M_{a}) ∣ D u ∣^{2}$ a.e., so $Re a_{0} (u, u) = \int_{Ω} Re (a^{i j} D_{j} u \overline{D_{i} u}) d x$ ; the difference from $θ ∣ D u ∣^{2}$ is nonnegative and measurable, so its integral is nonnegative and $Re a_{0} (u, u) \geq θ \int_{Ω} ∣ D u ∣^{2} d x = θ ∥ D u ∥_{L^{2}}^{2}$ .

3.1F2F5step 2.1step 1.2algebra∎

Coercivity and the model case: combining steps 2.1 and 1.2 gives $Re a_{0} (u, u) \geq θ ∥ D u ∥_{L^{2}}^{2} \geq \frac{θ}{1 + C_{P}^{2}} ∥ u ∥_{H_{0}^{1}}^{2}$ for every $u \in H_{0}^{1} (Ω)$ , so $a_{0}$ is coercive with constant $α = θ / (1 + C_{P}^{2})$ ; it is bounded by [F2]. In the model case $a^{i j} = δ^{i j}$ , $θ = 1$ , the pointwise identity $Re (a^{i j} D_{j} u \overline{D_{i} u}) = ∣ D u ∣^{2}$ gives $Re a_{0} (u, u) = ∥ D u ∥_{L^{2}}^{2}$ .

Coercivity of the principal Dirichlet form

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources