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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 $Re a_{0} (u, u) = ∥ D u ∥_{L^{2}}^{2}$ .

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Sources

John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter 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 (archived 2025 author manuscript) (standard reference, not scraped)