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Weak shifted Dirichlet form and energy bound on an open set

Statement

Assume AC. Let $U \subseteq R^{n}$ be nonempty and open, $H = L^{2} (U; C)$ , and $Q = H_{0}^{1} (U)$ . The form $q (u, v) = \int_{U} (u \overline{v} + \nabla u \cdot \overline{\nabla v})$ is densely defined, closed and satisfies $q [u] \geq ∥ u ∥_{2}^{2}$ . Its self-adjoint representing operator $A \geq I$ has $D (A) = {u \in H_{0}^{1} (U) : Δ u \in L^{2} (U)}, A u = u - Δ u .$ Here the Laplacian is distributional, and $∥ u ∥_{H^{1}} \leq ∥ A u ∥_{2}$ for $u \in D (A)$ . Moreover $D (A^{1 / 2}) = Q$ and $q [u] = ∥ A^{1 / 2} u ∥_{2}^{2}$ on $Q$ .

Facts & Assumptions

Given: The hypotheses and conventions of the Statement.

[F1]

Assume the Axiom of Choice The Axiom of Choice; its countable and dependent instances follow from AC supplies the countable and dependent choices used in Banach integration. These supply the choice hypotheses of the Sobolev, Hilbert-space, distributional and form-representation inputs.

[F2]

The weak derivatives and $H^{1}$ norm are those of Integer-order Sobolev spaces and their norms; $H_{0}^{1} (U)$ is the $H^{1}$ closure of $C_{c}^{\infty} (U)$ by Zero-boundary Sobolev space as a norm closure. Completeness is supplied by Integer-order Sobolev spaces are Banach, and density in $L^{2}$ by Smooth compactly supported functions of an open set are dense in L².

[F3]

The Dirichlet Laplacian generates an analytic heat semigroup gives the self-adjoint nonpositive form operator $Δ_{D}$ on any nonempty open set; its weak domain is defined by the Dirichlet energy identity. Only clause (1) is used.

[F4]

Semibounded closed-form representation and perturbation is a mathematics-domain supplier with inner products linear in the first variable. At $c = 0$ its statement gives the self-adjoint representing operator, $D (A^{1 / 2}) = Q$ , and $q (u, v) = ⟨ A u, v ⟩$ for $u \in D (A)$ and $v \in Q$ . The square-root form identity on all of $Q$ is deduced locally below from these stated interfaces and the spectral form-core input [F10].

[F5]

By L² with the integral pairing is a Hilbert space, $H = L^{2} (U; C)$ is a complex Hilbert space for $⟨ u, v ⟩ = \int_{U} u \overline{v}$ , linear in the first variable, under Countable Choice supplied by [F1].

[F6]

Distributional derivatives use the bilinear test pairing and signed-transpose convention of Distributional derivative. By Weak derivatives are represented distributional derivatives, weak derivatives agree with represented distributional derivatives under Countable Choice. Thus for $u \in H^{1} (U)$ , $Δ u = \sum_{j = 1}^{n} \partial_{j} (\partial_{j} u)$ is a distribution, with $⟨ Δ u, ψ ⟩ = - \sum_{j} \int_{U} \partial_{j} u \partial_{j} ψ$ for every test $ψ$ . Here $Δ u \in L^{2}$ means that this distribution is represented by an $L^{2}$ class.

[F7]

Cauchy–Schwarz in a complex inner-product space is supplied by Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs. Applying it to the $L^{2}$ pairings from [F5] and to the finite-dimensional coordinate pairing gives $∣ q (u, v) ∣ \leq ∥ u ∥_{H^{1}} ∥ v ∥_{H^{1}}$ , and $∣ ⟨ h, v ⟩ ∣ \leq ∥ h ∥_{2} ∥ v ∥_{2} \leq ∥ h ∥_{2} ∥ v ∥_{H^{1}}$ for $h \in H$ and $u, v \in Q$ .

[F8]

For a nonnegative symmetric form, Closed sectorial form and its associated operator defines closedness by completeness in the shifted form norm $(q [u] + ∥ u ∥_{2}^{2})^{1 / 2}$ , with $M = 0$ and $θ = 0$ in its sector condition. Its form-domain boundedness and continuous inclusion requirements are checked below. Only the closedness definition is used; the positive representing-operator convention is that of [F4].

[F9]

Under AC, Spectral theorem for unbounded self-adjoint operators (PVM form) supplies the spectral projection-valued measure $E$ of a self-adjoint operator on a nonzero complex Hilbert space. Unbounded Borel functional calculus: domains, products, spectral mapping supplies the Borel calculus, addition of constants and spectral support. By The unbounded PVM integral is densely defined, closed and normal, $∥ f (E) u ∥^{2} = \int ∣ f (t) ∣^{2} d E_{u} (t)$ on its squared-integrability domain, where $E_{u} (S) = ⟨ E (S) u, u ⟩$ and $E_{u} (R) = ∥ u ∥^{2}$ . Square roots use the nonnegative square-root function on the spectral support. The zero Hilbert space has the unique zero operator and all these norm identities are zero.

[F10]

Under AC, Spectral form domain and core of a semibounded operator applies to a self-adjoint operator $A \geq 0$ on a complex Hilbert space. With its lower-bound parameter $c = 0$ , it states that $D (A)$ is dense in $D (A^{1 / 2})$ for the norm $(∥ z ∥_{2}^{2} + ∥ A^{1 / 2} z ∥_{2}^{2})^{1 / 2}$ and that $⟨ A z, z ⟩ = ∥ A^{1 / 2} z ∥_{2}^{2}$ for $z \in D (A)$ . This includes its zero-space convention. The density is in the spectral graph norm; its conversion into density in the given $H^{1}$ form norm is proved below.

Proof

Given: The hypotheses and data in the Statement.

Goal: Prove the conclusions in the statement under its stated hypotheses.

1.1F1F2F3F4F5F7F8givenalgebra

By [F5], $H$ is the required complex Hilbert space. The form $q$ is symmetric sesquilinear and bounded on $Q$ by [F7], and $q [u] = ∥ u ∥_{H^{1}}^{2} \geq ∥ u ∥_{2}^{2}$ . The space $Q$ is a closed subspace of the complete space $H^{1} (U)$ , hence complete in $q [u]^{1 / 2}$ . The shifted norm required by [F4] at $c = 0$ satisfies $q [u] \leq q [u] + ∥ u ∥_{2}^{2} \leq 2 q [u]$ , so it is complete as well. The inclusion $Q ↪ H$ is continuous because $∥ u ∥_{2} \leq ∥ u ∥_{H^{1}}$ , and the nonnegative real diagonal satisfies the sector condition with $M = 0$ and $θ = 0$ . Thus [F8], together with boundedness from [F7], identifies this completeness as closedness of $q$ . The inclusion $Q \subseteq H$ is dense because $C_{c}^{\infty} (U)$ is dense in $L^{2} (U)$ . With the choice hypotheses supplied by [F1], the closed-form theorem gives the self-adjoint representative. The weak identity identifies its domain and action with $I - Δ_{D}$ from the any-open realization: $q (u, v) = ⟨ f, v ⟩$ is equivalent to $\int_{U} \nabla u \cdot \overline{\nabla v} = ⟨ f - u, v ⟩$ . Since $Δ_{D} \leq 0$ , this gives $A \geq I$ .

2.1F2F4F5F6F7step 1.1

If $u \in D (A)$ and $A u = f$ , test the form identity with $ϕ = \overline{ψ}$ for $ψ \in C_{c}^{\infty} (U)$ . By [F6], $⟨ Δ u, ψ ⟩ = - \sum_{j} \int_{U} \partial_{j} u \partial_{j} ψ = \int_{U} (u - f) ψ$ , so $Δ u = u - f$ in distributions. Conversely if $u \in Q$ and $Δ u = g \in L^{2}$ , use $ψ = \overline{ϕ}$ in this distributional identity to obtain $q (u, ϕ) = ⟨ u - g, ϕ ⟩$ for smooth compactly supported $ϕ$ . Both sides are continuous in the $H^{1}$ norm of $ϕ$ by [F7], so the defining density of $Q$ extends the equality to every $ϕ \in Q$ . The form-operator domain characterization gives $u \in D (A)$ and $A u = u - g$ .

2.2F1F4F5F9F10step 1.1

If $H = {0}$ the square-root claims are immediate. Otherwise $A \geq I \geq 0$ by step 1.1, so [F10] applies under [F1] with $c = 0$ . Its spectral square root is the one in [F9], and [F4] identifies $D (A^{1 / 2}) = Q$ . For every $z \in D (A)$ , the representing identity and the operator-domain identity in [F10] give $q [z] = ⟨ A z, z ⟩ = ∥ A^{1 / 2} z ∥_{2}^{2}$ . Thus the two forms agree on $D (A)$ before any extension to $Q$ is made.

3.1F4F5F7step 1.1step 2.1algebra

For $u \in D (A)$ the form identity and Cauchy–Schwarz yield $∥ u ∥_{H^{1}}^{2} = ⟨ A u, u ⟩ \leq ∥ A u ∥_{2} ∥ u ∥_{2} \leq ∥ A u ∥_{2} ∥ u ∥_{H^{1}}$ . Division when $u \neq 0$ , with the zero case immediate, proves the bound.

4.1F1F2F4F7F10step 1.1step 2.2choosealgebra∎

Fix $u \in Q = D (A^{1 / 2})$ . Choose, using [F1] and the density assertion in [F10], a sequence $u_{k} \in D (A)$ such that $∥ u_{k} - u ∥_{2} + ∥ A^{1 / 2} (u_{k} - u) ∥_{2} \to 0$ . For differences in $D (A)$ , step 2.2 gives $∥ u_{k} - u_{j} ∥_{H^{1}}^{2} = q [u_{k} - u_{j}] = ∥ A^{1 / 2} (u_{k} - u_{j}) ∥_{2}^{2} \to 0$ as $k, j \to \infty$ . Completeness of $Q$ in the $H^{1}$ norm from step 1.1 yields $v \in Q$ with $∥ u_{k} - v ∥_{H^{1}} \to 0$ . Since $∥ u - v ∥_{2} \leq ∥ u - u_{k} ∥_{2} + ∥ u_{k} - v ∥_{H^{1}} \to 0$ , this limit is $v = u$ . Continuity of the two norms and step 2.2 now give $q [u] = \lim_{k} q [u_{k}] = \lim_{k} ∥ A^{1 / 2} u_{k} ∥_{2}^{2} = ∥ A^{1 / 2} u ∥_{2}^{2}$ . This proves the full form-domain identity and also that $D (A)$ is dense in $Q$ for the original form norm. Expanding the identity at $u + v$ and $u + i v$ and comparing real and imaginary parts gives $q (u, v) = ⟨ A^{1 / 2} u, A^{1 / 2} v ⟩$ for all $u, v \in Q$ .

Source notes

The any-open realization, the closed-form definition and representation theorem, the spectral-calculus norm identity, and the spectral form-core lemma are used here. The full square-root form identity is proved by completion from the operator domain. No bounded-domain spatial regularity, spectral gap, compact resolvent or eigenfunction expansion is assumed. The density of tests in L² and Sobolev completeness supply the form-domain density and completeness requirements.

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