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Constant Dirichlet forcing is square summable but outside the heat-generator domain

Statement

For the constant forcing $f (x) : = π / 4$ on $(0, π)$ , define its Fourier sine coefficient by ${\hat{f}}_{k} : = \frac{2}{π} \int_{0}^{π} f (x) \sin (k x) d x .$ Then ${\hat{f}}_{k} = g_{k}$ from Diagonal Dirichlet sine-mode heat model with constant forcing. Thus $g \in H$ but $g$ is outside the generator domain of $A$ . The constant forcing has nonzero endpoint value, while every Dirichlet sine mode vanishes at both endpoints.

Facts & Assumptions

Given: The modal space, generator domain, forcing coefficients, and Dirichlet sine modes in the statement.

[F1]

Diagonal Dirichlet sine-mode heat model with constant forcing and The diagonal heat semigroup has the stated generator domain: $H = ℓ^{2} ({k \in N : k \geq 1}, R)$ , and the generator of $E_{t}$ has domain $D (A) = {u : \sum_{k \geq 1} k^{4} ∣ u_{k} ∣^{2} < \infty}$ ; $g_{k} = 1 / k$ for odd $k$ and $0$ for even $k$ .

[F2]

The p-series for a real exponent p converges exactly when p is greater than one: $\sum_{k \geq 1} k^{- p}$ converges exactly when $p > 1$ .

[F3]

The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ : if $G^{'} = q$ and $q$ is integrable, then $\int_{a}^{b} q = G (b) - G (a)$ .

[F4]

The derivatives of sine and cosine are cosine and minus sine: $\frac{d}{d x} \cos (k x) = - k \sin (k x)$ .

[F5]

Quarter-turn values and shifts by pi/2 and pi: $\sin (k π) = 0$ and $\cos (k π) = (- 1)^{k}$ for each integer $k$ .

[F6]

Pi as twice the smallest positive zero of cosine: $π$ is strictly positive.

Proof

1.1givenF1F3F4F5algebra

By [F4], an antiderivative of $\sin (k x)$ is $- \cos (k x) / k$ ; applying [F3] and [F5] gives ${\hat{f}}_{k} = \frac{2}{π} \cdot \frac{π}{4} \cdot \frac{1 - (- 1)^{k}}{k} = \frac{1 - (- 1)^{k}}{2 k} .$ This is $1 / k$ for odd $k$ and $0$ for even $k$ , exactly the sequence in [F1].

2.1F1F2step 1.1algebra

The squared coefficients sum to $\sum_{j \geq 0} (2 j + 1)^{- 2} \leq \sum_{k \geq 1} k^{- 2} < \infty$ by [F2], so $g \in H$ .

2.2F1step 1.1algebra

The generator-domain sum is $\sum_{k \geq 1} k^{4} ∣ g_{k} ∣^{2} = \sum_{j \geq 0} (2 j + 1)^{2}$ , whose partial sums are at least the number of terms and hence unbounded. Therefore $g \notin D (A)$ by [F1].

3.1givenF5F6algebra∎

By [F6], the constant forcing $f = π / 4$ is nonzero. By [F5], every sine mode $\sin (k x)$ vanishes at both $x = 0$ and $x = π$ . Hence the forcing is incompatible with the homogeneous Dirichlet endpoint values. This modal observation supplies no general PDE error theorem.

Depends on

Used by

Boundary-incompatible forcing reduces Strang order to five quarters in a Dirichlet modal model Example

Dependency tree · two levels

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Sources

Richard S. Laugesen, Spectral Theory of Partial Differential Equations, complete 120-page graduate lecture notes (standard reference, not scraped)