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Absorption of lower-order Sobolev terms in the elliptic estimate

Statement

Assume Countable Choice. Let $Ω \subseteq R^{n}$ be open, $n \geq 1$ , $K \in {R, C}$ , and let $m \geq 1$ . For every $ε > 0$ there is $C = C (n, m, ε)$ such that every $u \in H_{0}^{m + 1} (Ω; K)$ satisfies $∥ D^{m} u ∥_{L^{2} (Ω)} \leq ε ∥ D^{m + 1} u ∥_{L^{2} (Ω)} + C ∥ D^{m - 1} u ∥_{L^{2} (Ω)},$ and the same estimate holds for every $u \in H^{m + 1} (Ω; K)$ whose class vanishes almost everywhere outside a compact subset of $Ω$ . In both displays $∥ D^{j} u ∥_{L^{2} (Ω)}^{2} : = \sum_{α \in N_{0}^{n}, ∣ α ∣ = j} ∥ D^{α} u ∥_{L^{2} (Ω)}^{2} .$ In particular, for $u \in H_{0}^{2} (Ω; K)$ , $∥ D u ∥_{L^{2} (Ω)} \leq ε ∥ D^{2} u ∥_{L^{2} (Ω)} + C_{ε} ∥ u ∥_{L^{2} (Ω)} .$ The constants are not asserted sharp, and the estimate is the tool that absorbs commutator terms linear in the highest derivatives.

Facts & Assumptions

Given: Countable Choice; an open set $Ω \subseteq R^{n}$ with $n \geq 1$ ; a scalar field $K \in {R, C}$ ; an integer $m \geq 1$ ; a tolerance $ε > 0$ ; and a class $u$ lying in $H_{0}^{m + 1} (Ω; K)$ or in $H^{m + 1} (Ω; K)$ with compact support in $Ω$ ; write $N_{m} : = # {α \in N_{0}^{n} : ∣ α ∣ = m}$ .

[F1]

Classical derivatives of smooth functions are weak derivatives: for $w \in C^{k} (Ω)$ and $∣ α ∣ \leq k$ and every $φ \in C_{c}^{\infty} (Ω)$ , $\int_{Ω} w D^{α} φ = (- 1)^{∣ α ∣} \int_{Ω} (\partial^{α} w) φ$ , so the componentwise classical derivative represents the weak derivative. (Classical derivatives agree with weak derivatives)

[F2]

The Sobolev norm of Integer-order Sobolev spaces and their norms is $∥ u ∥_{W^{k, p} (Ω)}^{p} = \sum_{∣ α ∣ \leq k} ∥ D^{α} u ∥_{L^{p} (Ω)}^{p}$ for $p < \infty$ , so for every $∣ α ∣ \leq k$ one has $∥ D^{α} u ∥_{L^{p} (Ω)} \leq ∥ u ∥_{W^{k, p} (Ω)}$ , and convergence in $W^{k, p}$ implies convergence of every derivative class of order at most $k$ in $L^{p}$ .

[F3]

$H^{k} = W^{k, 2}$ and $H_{0}^{k}$ is the closure of $C_{c}^{\infty} (Ω)$ in the $W^{k, 2}$ norm; a class lies in $H_{0}^{k} (Ω)$ exactly when it is a limit in $W^{k, 2} (Ω)$ of test functions. (The notation $H^{k}$ and the reserved zero-boundary symbol, Zero-boundary Sobolev space as a norm closure)

[F4]

Young's inequality: for $p, q > 1$ with $1 / p + 1 / q = 1$ and real $A, B \geq 0$ one has $A B \leq A^{p} / p + B^{q} / q$ . (Young's inequality for conjugate real exponents)

[F5]

Zero extension: if $u \in W^{k, p} (Ω)$ vanishes almost everywhere outside a compact subset of $Ω$ , then its extension by zero $E_{0} u$ lies in $W^{k, p} (R^{n})$ , with $D^{α} (E_{0} u) = E_{0} (D^{α} u)$ almost everywhere for $∣ α ∣ \leq k$ and $∥ E_{0} (D^{α} u) ∥_{L^{p} (R^{n})} = ∥ D^{α} u ∥_{L^{p} (Ω)}$ ; in particular the $W^{k, p}$ norms agree. (Compactly supported Sobolev functions extend by zero in every integer order)

[F6]

$C_{c}^{\infty} (R^{n})$ is dense in $W^{k, p} (R^{n})$ for $k \in N_{0}$ and $1 \leq p < \infty$ . (Compactly supported smooth functions are dense in W^{k,p}(R^n))

Proof

technique · direct

1.1F1algebra

Let $u \in C_{c}^{\infty} (Ω)$ and let $α \in N_{0}^{n}$ satisfy $∣ α ∣ = m \geq 1$ ; choose a coordinate $k$ with $α_{k} \geq 1$ . The class $\overline{D^{α} u}$ is of class $C^{\infty}$ on $Ω$ , and the multi-index $e_{k}$ has $∣ e_{k} ∣ = 1$ , so [F1] applied to $w : = \overline{D^{α} u}$ with test function $φ : = D^{α - e_{k}} u \in C_{c}^{\infty} (Ω)$ gives $\int_{Ω} ∣ D^{α} u ∣^{2} d x = \int_{Ω} \overline{D^{α} u} D^{α} u d x = - \int_{Ω} (\partial_{k} \overline{D^{α} u}) D^{α - e_{k}} u d x = - \int_{Ω} D^{α - e_{k}} u \overline{D^{α + e_{k}} u} d x,$ because $\partial_{k} \overline{D^{α} u} = \overline{\partial_{k} D^{α} u} = \overline{D^{α + e_{k}} u}$ .

2.1F2step 1.1algebra

For the same $u \in C_{c}^{\infty} (Ω)$ and each such $α$ , Cauchy-Schwarz and the component bounds of [F2] give $∣ \int_{Ω} D^{α - e_{k}} u \overline{D^{α + e_{k}} u} d x ∣ \leq ∥ D^{α - e_{k}} u ∥_{L^{2} (Ω)} ∥ D^{α + e_{k}} u ∥_{L^{2} (Ω)} \leq ∥ D^{m - 1} u ∥_{L^{2} (Ω)} ∥ D^{m + 1} u ∥_{L^{2} (Ω)} .$ Summing the resulting estimates $∣ D^{α} u ∣_{L^{2}}^{2} \leq ∥ D^{m - 1} u ∥ ∥ D^{m + 1} u ∥$ over the finitely many $α$ with $∣ α ∣ = m$ gives $∥ D^{m} u ∥_{L^{2} (Ω)}^{2} \leq N_{m} ∥ D^{m - 1} u ∥_{L^{2} (Ω)} ∥ D^{m + 1} u ∥_{L^{2} (Ω)} .$

3.1F4step 2.1algebra

Write $A : = ∥ D^{m + 1} u ∥_{L^{2} (Ω)}$ and $B : = ∥ D^{m - 1} u ∥_{L^{2} (Ω)}$ , and set $x : = ε A$ and $y : = N_{m} B / (4 ε)$ . Then $(x + y)^{2} - 4 x y = (x - y)^{2} \geq 0$ , so $(ε A + N_{m} B / (4 ε))^{2} \geq 4 x y = N_{m} A B$ . Taking square roots and using step 2.1 gives $∥ D^{m} u ∥_{L^{2} (Ω)} \leq ε ∥ D^{m + 1} u ∥_{L^{2} (Ω)} + \frac{N_{m}}{4 ε} ∥ D^{m - 1} u ∥_{L^{2} (Ω)};$ equivalently, [F4] with $p = q = 2$ absorbs the geometric mean at the cost of the constant $C = C (n, m, ε)$ .

4.1F2F3step 3.1

Let $u \in H_{0}^{m + 1} (Ω)$ and choose $φ_{j} \in C_{c}^{\infty} (Ω)$ with $∥ u - φ_{j} ∥_{W^{m + 1, 2} (Ω)} \to 0$ , as [F3] permits. Step 3.1 applied to $φ_{j}$ gives $∥ D^{m} φ_{j} ∥ \leq ε ∥ D^{m + 1} φ_{j} ∥ + C ∥ D^{m - 1} φ_{j} ∥$ , and [F2] gives $∥ D^{α} φ_{j} - D^{α} u ∥_{L^{2} (Ω)} \to 0$ for every $∣ α ∣ \leq m + 1$ ; passing to the limit $j \to \infty$ in the estimate (norms are continuous) yields the displayed inequality for $u$ .

5.1F5F6step 3.1step 4.1∎

Let $u \in H^{m + 1} (Ω)$ vanish almost everywhere outside a compact subset of $Ω$ . By [F5], $E_{0} u \in W^{m + 1, 2} (R^{n}) = H^{m + 1} (R^{n})$ with norm-preserving zero extensions of all derivatives of order at most $m + 1$ , and by [F6] the test functions are dense in $W^{m + 1, 2} (R^{n})$ , so $E_{0} u \in H_{0}^{m + 1} (R^{n})$ ; step 4.1 on $R^{n}$ therefore gives $∥ D^{m} E_{0} u ∥ \leq ε ∥ D^{m + 1} E_{0} u ∥ + C ∥ D^{m - 1} E_{0} u ∥$ , and [F5] rewrites every term as the corresponding norm of $u$ on $Ω$ . Combining this with step 4.1 proves the two displays; for $m = 1$ the lower-order term is $∥ D^{0} u ∥_{L^{2} (Ω)} = ∥ u ∥_{L^{2} (Ω)}$ , which is the stated $H_{0}^{2}$ instance.

Source notes

The interpolation estimate is Simon's Lemma 6 (printed pp. 52-53) in the form $∥ u ∥_{m - 1} \leq ε ∥ u ∥_{m} + C_{ε} ∥ u ∥_{0}$ ; the sharp constant $N_{m} / (4 ε)$ above is not asserted to be optimal and the proof only needs finitely many multi-indices. Hunter uses the same absorption as the Cauchy inequality with $ϵ$ inside the final step of Theorem 4.27 (printed p. 113).

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Sources

Leon Simon, Lectures on Partial Differential Equations (Stanford, complete 223-page author scan) (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)