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Localisation of a weak solution up to a bounded first-order term

Statement

Assume Countable Choice. Let $Ω \subseteq R^{n}$ be open, $K \in {R, C}$ , let $L, a$ be as in Uniformly elliptic divergence-form operators and their sesquilinear forms, let $f \in L_{l o c}^{2} (Ω)$ , let $u \in H^{1} (Ω)$ be a local weak solution of $L u = f$ (Local weak solutions of a divergence-form operator) and let $ζ \in C_{c}^{\infty} (Ω; R)$ . Then $ζ u \in H^{1} (Ω)$ and for every $v \in H_{0}^{1} (Ω)$ $a (ζ u, v) = \int_{Ω} ζ f \overline{v} d x + \int_{Ω} a^{i j} (D_{j} ζ) u \overline{D_{i} v} d x + \int_{Ω} b^{i} u (D_{i} ζ) \overline{v} d x - \int_{Ω} a^{i j} (D_{i} ζ) D_{j} u \overline{v} d x .$ The three commutator terms are bounded by $(2 n M_{a} + \sqrt{n} M_{b}) ∥ D ζ ∥_{\infty} ∥ u ∥_{H^{1} (Ω)} ∥ v ∥_{H^{1} (Ω)}$ and form a bounded sesquilinear form in $(u, v)$ with coefficients of first and zero order bounded by $3 n M_{a} ∥ D ζ ∥_{\infty}$ and $\sqrt{n} M_{b} ∥ D ζ ∥_{\infty}$ , while $ζ f \in L^{2} (Ω)$ with $∥ ζ f ∥_{L^{2}} \leq ∥ ζ ∥_{\infty} ∥ f ∥_{L^{2} (supp ζ)}$ . The identity supplies an $H^{- 1}$ commutator for bounded coefficients; it does not by itself supply an $L^{2}$ datum for a regularity theorem. If additionally $a^{i j} \in W^{1, \infty} (Ω)$ , the localized datum is $g_{ζ} = ζ f - (D_{i} ζ) a^{i j} D_{j} u - D_{i} (a^{i j} u D_{j} ζ) + b^{i} u D_{i} ζ \in L^{2} (Ω),$ with $∥ g_{ζ} ∥_{2} \leq C (∥ ζ ∥_{W^{2, \infty}}, M_{a}, M_{b}, ∥ D a ∥_{\infty}, n) (∥ f ∥_{L^{2} (supp ζ)} + ∥ u ∥_{H^{1} (Ω)})$ . Expanding the divergence uses the second derivatives of $ζ$ and the first derivatives of $a$ ; these costs cannot be omitted.

Facts & Assumptions

Given: Countable Choice; the open set $Ω$ ; the scalar field $K$ ; the operator $L$ and its form $a$ with ellipticity constant $θ$ and bounds $M_{a}, M_{b}, M_{c}$ ; the datum $f \in L_{l o c}^{2} (Ω)$ ; the local weak solution $u \in H^{1} (Ω)$ of $L u = f$ ; and the real cutoff $ζ \in C_{c}^{\infty} (Ω; R)$ .

[F1]

Local weak solution: $a (u, v) = \int_{Ω} f \overline{v} d x$ for every $v \in C_{c}^{\infty} (Ω)$ , and by the equivalences of the definition also for every $v \in H_{0}^{1} (Ω_{2})$ with $Ω_{2} ⋐ Ω$ bounded open; the identity for a class $v$ supported in such an $Ω_{2}$ reads $a (u, v) = \int_{Ω_{2}} f \overline{v} d x$ . (Local weak solutions of a divergence-form operator)

[F2]

The form and its coefficients: $a (w, z) = \int_{Ω} (a^{i j} D_{j} w \overline{D_{i} z} + b^{i} D_{i} w \overline{z} + c w \overline{z}) d x$ with $∣ a^{i j} ∣ \leq M_{a}$ , $∣ b^{i} ∣ \leq M_{b}$ , $∣ c ∣ \leq M_{c}$ almost everywhere, and $a$ is linear in the first slot and conjugate-linear in the second. (Uniformly elliptic divergence-form operators and their sesquilinear forms)

[F3]

Weak Leibniz rule: if $w \in H^{1} (Ω)$ and $η \in C_{c}^{\infty} (Ω)$ , then $η w \in H^{1} (Ω)$ with $D_{j} (η w) = η D_{j} w + w D_{j} η$ a.e.; moreover $η w$ has compact support in $Ω$ . (Weak Leibniz rule with a smooth factor)

[F4]

The form is bounded on $H^{1} (Ω)$ : $∣ a (w, z) ∣ \leq (n M_{a} + n M_{b} + M_{c}) ∥ w ∥_{H^{1} (Ω)} ∥ z ∥_{H^{1} (Ω)}$ , and the same bound holds on $H_{0}^{1} (Ω)$ . (The elliptic form is well defined and bounded on $H^{1}$ )

[F5]

$H_{0}^{1} (Ω) = \overline{C_{c}^{\infty} (Ω)}$ in the $H^{1} (Ω)$ norm; in particular $C_{c}^{\infty} (Ω) \subseteq H_{0}^{1} (Ω)$ , and a function of $H^{1} (Ω)$ with compact support in $Ω$ lies in $H_{0}^{1} (Ω)$ . (Zero-boundary Sobolev space as a norm closure, The cutoff difference-quotient commutator estimate).

Proof

technique · direct

1.1F3F5

The localised classes: $ζ u \in H^{1} (Ω)$ with $D_{j} (ζ u) = ζ D_{j} u + u D_{j} ζ$ a.e., and for $v \in H_{0}^{1} (Ω)$ one has $ζ v \in H_{0}^{1} (Ω)$ with $D_{i} (ζ v) = ζ D_{i} v + v D_{i} ζ$ a.e., since $ζ v$ has compact support in $Ω$ .

1.2F1F3given

The weak equation may be tested with $ζ v$ : for $v \in C_{c}^{\infty} (Ω)$ the product $ζ v$ lies in $C_{c}^{\infty} (Ω)$ , so [F1] gives $a (u, ζ v) = \int_{Ω} f \overline{ζ v} d x = \int_{Ω} f \overline{ζ} \overline{v} d x = \int_{Ω} ζ f \overline{v} d x$ because $ζ$ is real-valued.

1.3F2algebra

The commutator terms are bounded: by [F2] and Hölder, $∣ \int_{Ω} a^{i j} u (D_{j} ζ) \overline{D_{i} v} ∣ \leq n M_{a} ∥ D ζ ∥_{\infty} ∥ u ∥_{L^{2}} ∥ D v ∥_{L^{2}}, ∣ \int_{Ω} b^{i} u (D_{i} ζ) \overline{v} ∣ \leq \sqrt{n} M_{b} ∥ D ζ ∥_{\infty} ∥ u ∥_{L^{2}} ∥ v ∥_{L^{2}},$ and $∣ \int_{Ω} a^{i j} D_{j} u (D_{i} ζ) \overline{v} ∣ \leq n M_{a} ∥ D ζ ∥_{\infty} ∥ D u ∥_{L^{2}} ∥ v ∥_{L^{2}},$ so their sum is at most $(2 n M_{a} + \sqrt{n} M_{b}) ∥ D ζ ∥_{\infty} ∥ u ∥_{H^{1}} ∥ v ∥_{H^{1}}$ . Here $\sum_{i} ∣ b^{i} D_{i} ζ ∣ \leq ∣ b ∣ ∣ D ζ ∣ \leq \sqrt{n} M_{b} ∣ D ζ ∣$ follows from the component bounds in [F2]. Moreover $ζ f \in L^{2} (Ω)$ with $∥ ζ f ∥_{L^{2}} \leq ∥ ζ ∥_{\infty} ∥ f ∥_{L^{2} (supp ζ)}$ by Hölder.

2.1F2step 1.1algebra

Expansion of the localised form. For $v \in C_{c}^{\infty} (Ω)$ , inserting the product rule of step 1.1 into the three terms of $a (ζ u, v)$ gives $a (ζ u, v) = \int_{Ω} ζ (a^{i j} D_{j} u \overline{D_{i} v} + b^{i} D_{i} u \overline{v} + c u \overline{v}) d x + \int_{Ω} a^{i j} u (D_{j} ζ) \overline{D_{i} v} d x + \int_{Ω} b^{i} u (D_{i} ζ) \overline{v} d x .$

3.1F2F3step 1.1algebra

The first integral is $a (u, ζ v)$ corrected by one Leibniz term: expanding $a (u, ζ v) = \int_{Ω} (a^{i j} D_{j} u \overline{D_{i} (ζ v)} + b^{i} D_{i} u \overline{ζ v} + c u \overline{ζ v}) d x$ with $D_{i} (ζ v) = ζ D_{i} v + v D_{i} ζ$ shows $a (u, ζ v) = \int_{Ω} ζ (a^{i j} D_{j} u \overline{D_{i} v} + b^{i} D_{i} u \overline{v} + c u \overline{v}) d x + \int_{Ω} a^{i j} D_{j} u (D_{i} ζ) \overline{v} d x,$ the coefficient $a^{i j}$ and the factor $D_{j} u$ being untouched by the conjugation because the Leibniz term sits in the second slot. Hence the first integral of step 2.1 equals $a (u, ζ v) - \int_{Ω} a^{i j} D_{j} u (D_{i} ζ) \overline{v} d x$ .

4.1step 1.2step 2.1step 3.1

Substituting step 3.1 into step 2.1 and inserting the weak equation of step 1.2 yields the displayed identity, first for $v \in C_{c}^{\infty} (Ω)$ : $a (ζ u, v) = \int_{Ω} ζ f \overline{v} d x + \int_{Ω} a^{i j} u (D_{j} ζ) \overline{D_{i} v} d x + \int_{Ω} b^{i} u (D_{i} ζ) \overline{v} d x - \int_{Ω} a^{i j} D_{j} u (D_{i} ζ) \overline{v} d x .$

5.1F4F5step 1.3step 4.1

Both sides of the identity of step 4.1 are continuous in $v \in H_{0}^{1} (Ω)$ : the left side by [F4] and the right side by step 1.3. Since $C_{c}^{\infty} (Ω)$ is dense in $H_{0}^{1} (Ω)$ by [F5], the identity extends from the test functions of step 4.1 to every $v \in H_{0}^{1} (Ω)$ .

6.1F2step 1.3step 5.1algebra∎

The exact identity and the commutator-form bound follow from steps 1.3 and 5.1. With only bounded coefficients the flux term pairs an $L^{2}$ vector field with $D v$ , hence is an $H^{- 1}$ functional. Under $a \in W^{1, \infty}$ , the multiplier rule of The cutoff difference-quotient commutator estimate gives $D_{i} (a^{i j} u D_{j} ζ) = (D_{i} a^{i j}) u D_{j} ζ + a^{i j} D_{i} u D_{j} ζ + a^{i j} u D_{i} D_{j} ζ$ . All terms are $L^{2}$ , yielding the displayed $g_{ζ}$ and its norm bound. No estimate for $L^{2}$ forcing is inferred from an $H^{- 1}$ datum alone.

Source notes

Simon's Lecture 6 (printed pp. 60-62) localises the equation by replacing $u$ with a cutoff multiple, and Teschl's proof of Lemma 10.18 (printed p. 242) reduces to the localised classes $u_{j} = ζ_{j} u$ ; both produce the commutator terms displayed here. The scaffold's display carried only the first two commutator terms and identified the $ζ$ -part of the expansion with $a (u, v)$ ; the correct test class is the localised test $ζ v$ , and the difference contributes the additional term $- \int_{Ω} a^{i j} (D_{i} ζ) D_{j} u \overline{v}$ shown in step 3.1. This term is exactly the first-order commutator that the difference-quotient and Caccioppoli arguments of this page absorb.

Depends on

Used by

A finite partition glues the local interior and boundary H² estimates Lemma

Dependency tree · two levels

54 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

Leon Simon, Lectures on Partial Differential Equations (Stanford, complete 223-page author scan) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, complete 392 pages) (standard reference, not scraped)