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The Gauss-Green integration-by-parts formula with Sobolev traces

Statement

Assume the Axiom of Choice. Let $Ω \subset R^{n}$ , $n \geq 2$ , be a bounded $C^{1}$ domain, $1 < p < \infty$ , $p^{'} = p / (p - 1)$ , $u \in W^{1, p} (Ω; K)$ , $v \in W^{1, p^{'}} (Ω; K)$ , and let $T$ be the trace of The $L^{p}$ trace operator on a bounded $C^{1}$ domain with $ν$ the outward normal of Bounded C1 domains and their outward normals. Then for every $i = 1, \dots, n$ , $\int_{Ω} u \partial_{i} v d x = - \int_{Ω} (\partial_{i} u) v d x + \int_{\partial Ω} (T u) (T v) ν_{i} d S,$ and all three integrals are finite. If $u, v \in C^{1} (\overline{Ω})$ , the boundary term is the classical $\int_{\partial Ω} u v ν_{i} d S$ of the divergence theorem Divergence on a bounded C1 Euclidean domain applied to the field $u v e_{i}$ ; if in addition $T u = \partial_{ν} w$ on $\partial Ω$ for a specified $w \in C^{1} (\overline{Ω})$ , the boundary term is $\int_{\partial Ω} (\partial_{ν} w) (T v) ν_{i} d S$ . The normal derivative of Classical normal derivative is defined on the boundary, and this extra identity is an assumption, not an interior substitution.

Facts & Assumptions

Given: The Axiom of Choice; a bounded $C^{1}$ domain $Ω$ ; $1 < p < \infty$ with conjugate $p^{'}$ ; $u \in W^{1, p} (Ω; K)$ and $v \in W^{1, p^{'}} (Ω; K)$ ; and the trace operator $T$ of The $L^{p}$ trace operator on a bounded $C^{1}$ domain.

[F1]

Divergence theorem: for $F \in C^{1} (\overline{Ω}; R^{n})$ , $\int_{Ω} div F d x = \int_{\partial Ω} F \cdot ν d S$ , with both integrals finite and $ν$ the outward normal. (Divergence on a bounded C1 Euclidean domain)

[F2]

$T : W^{1, r} (Ω) \to L^{r} (\partial Ω)$ is bounded and linear for $1 \leq r < \infty$ , and $T u = u ∣_{\partial Ω}$ when $u$ has a continuous representative on $\overline{Ω}$ . (The $L^{p}$ trace operator on a bounded $C^{1}$ domain, The trace agrees with classical restriction for continuous Sobolev functions)

[F3]

The restrictions to $Ω$ of $C_{c}^{\infty} (R^{n})$ functions are dense in $W^{1, r} (Ω)$ for $1 \leq r < \infty$ . (Ambient smooth restrictions are dense on bounded C^k domains)

[F4]

Holder's inequality holds on $Ω$ and, since the surface measure is finite, on $\partial Ω$ with the same exponents. (Holder's inequality for integrals, including the endpoint cases, Surface integration on compact C1 hypersurfaces)

[F5]

The outward normal $ν$ is continuous on $\partial Ω$ with $∣ ν_{i} ∣ \leq 1$ , and the classical normal derivative of a $C^{1} (\overline{Ω})$ function $w$ is $\partial_{ν} w = D w \cdot ν$ . (Bounded C1 domains and their outward normals, Classical normal derivative)

[F6]

$W^{1, r} (Ω)$ consists of $L^{r}$ classes whose first weak derivatives lie in $L^{r}$ . (Integer-order Sobolev spaces and their norms)

Proof

technique · direct

1.1F1F2F5algebragiven

The smooth case. Let $u, v \in C^{1} (\overline{Ω})$ and $F : = u v e_{i}$ , which is a $C^{1}$ vector field on $\overline{Ω}$ for real scalars; for complex scalars apply the real case to the real and imaginary parts and add, the identity being bilinear. Since $div (u v e_{i}) = \partial_{i} (u v) = (\partial_{i} u) v + u \partial_{i} v$ and $F \cdot ν = u v ν_{i}$ , the divergence theorem [F1] gives $\int_{Ω} (\partial_{i} u) v + u \partial_{i} v = \int_{\partial Ω} u v ν_{i} d S$ , that is $\int_{Ω} u \partial_{i} v = - \int_{Ω} (\partial_{i} u) v + \int_{\partial Ω} u v ν_{i} d S$ . By [F2] the classical restrictions are $T u$ and $T v$ , so the boundary term is $\int_{\partial Ω} (T u) (T v) ν_{i} d S$ ; all integrals are finite because $u, v, \partial_{i} u, \partial_{i} v$ and $ν$ are bounded on $\overline{Ω}$ . If the boundary restriction of $u$ equals $\partial_{ν} w$ for a specified $w \in C^{1} (\overline{Ω})$ , substitute that equality only into the boundary integrand, using [F5].

2.1F2F3F4F5F6step 1.1algebragiven∎

The general case by density and limits. Let $u_{m}, v_{m}$ be restrictions of $C_{c}^{\infty} (R^{n})$ functions with $u_{m} \to u$ in $W^{1, p} (Ω)$ and $v_{m} \to v$ in $W^{1, p^{'}} (Ω)$ , which exist by [F3]. Step 1.1 gives $\int_{Ω} u_{m} \partial_{i} v_{m} = - \int_{Ω} (\partial_{i} u_{m}) v_{m} + \int_{\partial Ω} (T u_{m}) (T v_{m}) ν_{i} d S$ for every $m$ . The volume terms converge: by Holder [F4], $∣ \int_{Ω} u_{m} \partial_{i} v_{m} - \int_{Ω} u \partial_{i} v ∣ \leq ∥ u_{m} - u ∥_{L^{p}} ∥ \partial_{i} v_{m} ∥_{L^{p^{'}}} + ∥ u ∥_{L^{p}} ∥ \partial_{i} v_{m} - \partial_{i} v ∥_{L^{p^{'}}} \to 0$ and $∣ \int_{Ω} (\partial_{i} u_{m}) v_{m} - \int_{Ω} (\partial_{i} u) v ∣ \leq ∥ \partial_{i} u_{m} - \partial_{i} u ∥_{L^{p}} ∥ v_{m} ∥_{L^{p^{'}}} + ∥ \partial_{i} u ∥_{L^{p}} ∥ v_{m} - v ∥_{L^{p^{'}}} \to 0$ . The boundary terms converge: by Holder on $\partial Ω$ and the boundedness of $T$ [F2], $∣ \int_{\partial Ω} [(T u_{m}) (T v_{m}) - (T u) (T v)] ν_{i} d S ∣ \leq ∥ T u_{m} - T u ∥_{L^{p} (\partial Ω)} ∥ T v_{m} ∥_{L^{p^{'}} (\partial Ω)} + ∥ T u ∥_{L^{p} (\partial Ω)} ∥ T v_{m} - T v ∥_{L^{p^{'}} (\partial Ω)} \to 0$ . Hence the identity passes to the limit. Finally each of the three integrals is finite: $u \partial_{i} v$ and $(\partial_{i} u) v$ lie in $L^{1} (Ω)$ by Holder, and $(T u) (T v) ν_{i}$ lies in $L^{1} (\partial Ω)$ by Holder with $∣ ν_{i} ∣ \leq 1$ on the finite-measure boundary.

Source notes

Teschl's Lemma 9.20 (printed p. 210) is the integration-by-parts identity for $W^{1, p}$ functions with boundary traces; Schikorra's proof of Theorem III.3.21 (printed p. 77) obtains the boundary term by the same integration by parts, and Laugesen's Step 1 of Theorem 3.14 (printed p. 63) carries out the boundary calculation behind it. The proof above separates the divergence theorem on smooth fields from the density extension, and it records the finiteness of all three pairings.

Depends on

Used by

The kernel of the trace is the closure of the test functions Theorem

Dependency tree · two levels

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (archived 2025 author manuscript) (standard reference, not scraped)
Armin Schikorra, Partial Differential Equations (University of Pittsburgh, version 4 December 2019) (standard reference, not scraped)
Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, complete 158-page graduate notes) (standard reference, not scraped)