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The differentiated weak equation with coefficient commutators

Statement

Assume Countable Choice. Let $Ω \subseteq R^{n}$ be open, $K \in {R, C}$ , let $a^{i j} \in W_{l o c}^{2, \infty} (Ω)$ and $b^{i}, c \in W_{l o c}^{1, \infty} (Ω)$ with $∣ a^{i j} ∣ \leq M_{a}$ , $∣ b^{i} ∣ \leq M_{b}$ , $∣ c ∣ \leq M_{c}$ and $∣ D_{k} a^{i j} ∣, ∣ D_{k} b^{i} ∣, ∣ D_{k} c ∣ \leq M_{1}$ almost everywhere, let $f \in H_{l o c}^{1} (Ω)$ , and let $u \in H^{1} (Ω) \cap H_{l o c}^{2} (Ω)$ be a local weak solution of $L u = f$ on $Ω$ (Local weak solutions of a divergence-form operator). Then for every coordinate direction $k$ and every $φ \in C_{c}^{\infty} (Ω)$ , writing $v : = D_{k} u \in H_{l o c}^{1} (Ω)$ (Weak derivative of a locally integrable function), with $f_{k} : = D_{k} f$ , $α_{k}^{i j} : = D_{k} a^{i j}$ , $β_{k}^{i} : = D_{k} b^{i}$ and $γ_{k} : = D_{k} c$ , $\int_{Ω} a^{i j} D_{j} v \overline{D_{i} φ} d x + \int_{Ω} b^{i} D_{i} v \overline{φ} d x + \int_{Ω} c v \overline{φ} d x = \int_{Ω} f_{k} \overline{φ} d x - \int_{Ω} α_{k}^{i j} D_{j} u \overline{D_{i} φ} d x - \int_{Ω} β_{k}^{i} D_{i} u \overline{φ} d x - \int_{Ω} γ_{k} u \overline{φ} d x .$ Thus on every bounded open $U ⋐ Ω$ , the restriction $v ∣_{U}$ is a local weak solution of the equation with the same principal part $a^{i j}$ and the same bounded first- and zero-order coefficients; its datum $g_{k} : = D_{k} f + D_{i} ((D_{k} a^{i j}) D_{j} u) - (D_{k} b^{i}) D_{i} u - (D_{k} c) u$ lies in $L_{l o c}^{2} (Ω)$ . More generally, if $∣ α ∣ = j \geq 1$ , $u \in H_{l o c}^{j + 1}$ , $f \in H_{l o c}^{j}$ , $a^{i j} \in W_{l o c}^{j + 1, \infty}$ and $b^{i}, c \in W_{l o c}^{j, \infty}$ , then $D^{α} u$ satisfies the same-principal-part compact-test equation on $Ω$ , and is a local weak solution on each such $U$ , with datum $g_{α} : = D^{α} f + \sum_{0 < β \leq α} (\binom{α}{β}) D_{i} ⁣ ((D^{β} a^{i j}) D_{j} D^{α - β} u) - \sum_{0 < β \leq α} (\binom{α}{β}) ((D^{β} b^{i}) D_{i} D^{α - β} u + (D^{β} c) D^{α - β} u) \in L_{l o c}^{2} (Ω) .$ The principal coefficient derivatives through order $j + 1$ ensure that the divergence commutators are genuine $L_{l o c}^{2}$ functions, not merely $H_{l o c}^{- 1}$ functionals. The scaffold assumed only $u \in H^{1} (Ω)$ ; the integral defining $\int a^{i j} D_{j} v \overline{D_{i} φ}$ requires $v \in H_{l o c}^{1}$ , equivalently $u \in H_{l o c}^{2} (Ω)$ , which is the regularity available in every induction step that consumes this lemma.

Facts & Assumptions

Given: Countable Choice; the open set $Ω$ ; the coefficients with their bounds; the data $f$ and $u$ ; and the weak equation $a (u, φ) = \int_{Ω} f \overline{φ} d x$ for every $φ \in C_{c}^{\infty} (Ω)$ .

[F1]

Local weak solution: $a (u, φ) = \int_{Ω} f \overline{φ} d x$ for every $φ \in C_{c}^{\infty} (Ω)$ , and $D_{k} φ \in C_{c}^{\infty} (Ω)$ for every such $φ$ . (Local weak solutions of a divergence-form operator)

[F2]

Regularity of the data: $f \in H_{l o c}^{1}$ gives $f_{k} = D_{k} f \in L_{l o c}^{2} (Ω)$ ; $u \in H_{l o c}^{2}$ gives $v = D_{k} u \in H_{l o c}^{1}$ and $D_{j} v, D_{j} D_{i} u \in L_{l o c}^{2}$ . The lower-order derivatives $β_{k}^{i}, γ_{k}$ are locally bounded, and $a^{i j} \in W_{l o c}^{2, \infty}$ makes both $α_{k}^{i j}$ and $D_{i} α_{k}^{i j}$ locally bounded. Thus $D_{i} (α_{k}^{i j} D_{j} u) \in L_{l o c}^{2}$ as required for $g_{k}$ . (The notation $H^{k}$ and the reserved zero-boundary symbol, Integer-order Sobolev spaces and their norms, Uniformly elliptic divergence-form operators and their sesquilinear forms, The cutoff difference-quotient commutator estimate).

[F3]

Second weak derivatives commute: if $w \in H_{l o c}^{2} (Ω)$ , then $D_{k} D_{j} w = D_{j} D_{k} w$ almost everywhere, both being represented by the same $L_{l o c}^{2}$ class. This is the distributional identity $\partial_{k} \partial_{j} T_{w} = \partial_{j} \partial_{k} T_{w}$ together with the injectivity of the regular-distribution map. (Linearity, locality, and commutation of weak derivatives, Locally integrable functions as regular distributions)

[F4]

Hölder and Cauchy--Schwarz bounds: for $g \in L_{l o c}^{2}$ , a bounded coefficient $q$ and a compactly supported test function, all the pairings below are absolutely convergent with the bounds read off from $∥ g ∥_{L^{2} (supp φ)}$ and $∥ q ∥_{\infty}$ . (Holder's inequality for integrals, including the endpoint cases)

Proof

technique · direct

1.1F2F4

All objects in the display are defined and the pairings are finite: $v = D_{k} u \in H_{l o c}^{1}$ with $D_{j} v \in L_{l o c}^{2}$ , $f_{k} \in L_{l o c}^{2}$ , and $α_{k}^{i j}, β_{k}^{i}, γ_{k}$ are bounded; every term pairs an $L_{l o c}^{2}$ class with a bounded coefficient and a compactly supported test function, so [F4] bounds it.

2.1F1F2F3step 1.1algebra

Replacement and integration by parts. Since $D_{k} φ \in C_{c}^{\infty} (Ω)$ , [F1] gives $a (u, D_{k} φ) = \int_{Ω} f \overline{D_{k} φ} d x$ , and moving the derivative off the test function term by term (the boundary terms vanish because $φ$ is compactly supported) gives $\int_{Ω} a^{i j} D_{j} u \overline{D_{i} D_{k} φ} d x = - \int_{Ω} a^{i j} D_{j} v \overline{D_{i} φ} d x - \int_{Ω} α_{k}^{i j} D_{j} u \overline{D_{i} φ} d x,$ where $D_{k} D_{j} u = D_{j} D_{k} u = D_{j} v$ by [F3], and likewise $\int_{Ω} b^{i} D_{i} u \overline{D_{k} φ} d x = - \int_{Ω} b^{i} D_{i} v \overline{φ} d x - \int_{Ω} β_{k}^{i} D_{i} u \overline{φ} d x, \int_{Ω} c u \overline{D_{k} φ} d x = - \int_{Ω} c v \overline{φ} d x - \int_{Ω} γ_{k} u \overline{φ} d x,$ while $\int_{Ω} f \overline{D_{k} φ} d x = - \int_{Ω} f_{k} \overline{φ} d x$ . Substitution into the original identity and multiplication by $- 1$ gives the corrected signs in the Statement.

3.1step 2.1F2algebra

In the weak equation of step 2.1, the left-hand side is $a_{k} (v, φ) : = \int_{Ω} (a^{i j} D_{j} v \overline{D_{i} φ} + b^{i} D_{i} v \overline{φ} + c v \overline{φ}) d x$ and the distributional right-hand side is $D_{k} f + D_{i} (α_{k}^{i j} D_{j} u) - β_{k}^{i} D_{i} u - γ_{k} u$ . By [F2] this distribution is represented by the claimed $L_{l o c}^{2}$ function. On each bounded $U ⋐ Ω$ , one has $v \in H^{1} (U)$ , so the identity makes $v ∣_{U}$ a local weak solution in the cited definition. No global $H^{1} (Ω)$ membership of $v$ is asserted.

4.1step 3.1F2algebra

Higher-order commutators. For any multi-index $α$ of length $j$ , differentiate the distributional equation by $D^{α}$ and apply the proved Sobolev multiplier rule of The cutoff difference-quotient commutator estimate repeatedly. The principal commutators are divergences $D_{i} ((D^{β} a^{i j}) D_{j} D^{α - β} u)$ ; expanding each divergence shows that its terms involve coefficient derivatives through order $∣ β ∣ + 1 \leq j + 1$ and derivatives of $u$ through order $j - ∣ β ∣ + 2 \leq j + 1$ . The lower-order commutators use derivatives of $b, c$ through order $j$ and derivatives of $u$ through order at most $j$ . Under $a \in W_{l o c}^{j + 1, \infty}$ , $b, c \in W_{l o c}^{j, \infty}$ , $u \in H_{l o c}^{j + 1}$ and $f \in H_{l o c}^{j}$ , every term in $g_{α}$ is therefore in $L_{l o c}^{2}$ , as asserted in the Statement.

5.1step 3.1step 4.1∎

Conclusion. For every coordinate direction $k$ and every $φ \in C_{c}^{\infty} (Ω)$ the identity displayed in the Statement holds, so $v = D_{k} u$ satisfies the differentiated compact-test equation on $Ω$ and is a local weak solution on each bounded $U ⋐ Ω$ , with the same principal part and bounded first- and zero-order coefficients; in particular no consumer may claim that a derivative of a weak solution solves the identical equation, since the commutator terms $α_{k}^{i j} D_{j} u$ , $β_{k}^{i} D_{i} u$ and $γ_{k} u$ are exactly the correction.

Source notes

Teschl's proof of Corollary 10.17 (printed p. 241) differentiates the equation and exhibits the coefficient commutators; Hunter's remark before Theorem 4.28 (printed p. 114) performs the same formal differentiation. Both use the regularity $u \in H_{l o c}^{2}$ at the first differentiation, and the induction of the sources proceeds exactly as in step 4.1. The scaffold's hypothesis $u \in H^{1} (Ω)$ alone leaves $D_{j} v$ undefined as a function; the item assumes $u \in H_{l o c}^{2} (Ω)$ , which every consuming induction step supplies.

Depends on

Used by

Dependency tree · two levels

55 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, complete 392 pages) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter graduate notes) (standard reference, not scraped)