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Inhomogeneous Dirichlet data reduce to zero trace

Statement

Assume the Axiom of Choice. Let $Ω \subset R^{n}$ , $n \geq 2$ , be a bounded $C^{1}$ domain, $1 < p < \infty$ , $θ = 1 - 1 / p$ , and let $R$ be the bounded right inverse of A bounded right inverse of the trace, supported in a prescribed collar. Then for every $g \in W^{θ, p} (\partial Ω)$ and every $u \in W^{1, p} (Ω)$ with $T u = g$ one has $u = R g + v, v \in W_{0}^{1, p} (Ω),$ and conversely every $u = R g + v$ with $v \in W_{0}^{1, p} (Ω)$ has trace $g$ . If $g_{0} \in L^{p} (\partial Ω) ∖ W^{θ, p} (\partial Ω)$ , a set nonempty for $p > 1$ , then no $u \in W^{1, p} (Ω)$ satisfies $T u = g_{0}$ : the inhomogeneous problem is solvable exactly for data in the trace range, not for arbitrary boundary $L^{p}$ data.

Facts & Assumptions

Given: The Axiom of Choice; a bounded $C^{1}$ domain $Ω$ ; $1 < p < \infty$ ; $θ = 1 - 1 / p$ ; a bounded right inverse $R : W^{θ, p} (\partial Ω) \to W^{1, p} (Ω)$ of the trace with $T \circ R = i d$ ; and the identification ${T u = 0} = W_{0}^{1, p} (Ω)$ .

[F1]

$T \circ R = i d$ on $W^{θ, p} (\partial Ω)$ : $T (R g) = g$ for every boundary datum $g$ , and $R$ is linear and bounded. (A bounded right inverse of the trace, supported in a prescribed collar)

[F2]

The kernel of the trace is exactly $W_{0}^{1, p} (Ω)$ , the $W^{1, p}$ -closure of $C_{c}^{\infty} (Ω)$ . (The kernel of the trace is the closure of the test functions, Zero-boundary Sobolev space as a norm closure)

[F3]

$T$ is linear, $T (W^{1, p} (Ω)) = W^{θ, p} (\partial Ω)$ , and the range is a strict subset of $L^{p} (\partial Ω)$ for $p > 1$ . (The sharp trace theorem: boundedness and range in the fractional space)

Proof

technique · direct

1.1F1F2algebra

The decomposition and its converse. Let $g \in W^{θ, p} (\partial Ω)$ and $u \in W^{1, p} (Ω)$ with $T u = g$ . By linearity of $T$ and [F1], $T (u - R g) = T u - T (R g) = g - g = 0$ , so $v : = u - R g$ lies in the kernel of $T$ , which equals $W_{0}^{1, p} (Ω)$ by [F2]; this gives $u = R g + v$ with $v \in W_{0}^{1, p} (Ω)$ . Conversely, if $u = R g + v$ with $v \in W_{0}^{1, p} (Ω)$ , then $T u = T (R g) + T v = g + 0 = g$ by [F1], [F2] and linearity.

1.2F3algebra

Data outside the range are not attained. By [F3] the range of $T$ is exactly $W^{θ, p} (\partial Ω)$ and is a strict subset of $L^{p} (\partial Ω)$ for $p > 1$ , so the set $L^{p} (\partial Ω) ∖ W^{θ, p} (\partial Ω)$ is nonempty and no $u \in W^{1, p} (Ω)$ has trace equal to an element of it.

2.1step 1.1step 1.2algebragiven∎

Conclusion. Step 1.1 proves that the inhomogeneous problem with datum $g$ reduces to the zero-trace problem with remainder $v = u - R g$ , and that conversely every $g$ in the range is attained by $R g + W_{0}^{1, p} (Ω)$ ; step 1.2 shows that data outside the range are not attained at all. This is exactly the asserted statement.

Source notes

Gagliardo's Teorema [1.I] (printed p. 289) identifies the range exactly, so data outside it are not attained; Teschl's Lemmas 9.20-9.21 (printed p. 210) record the reduction of a prescribed trace to a zero-trace remainder, and Kampanou's Theorems 3.3 and 3.5 (printed pp. 23-31) supply the extension used in the reduction. The corollary keeps the two directions separate: existence for data in the range, and non-attainment outside it.

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Sources

Emilio Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in $n$ variabili, Rend. Sem. Mat. Univ. Padova 27 (1957), 284-305 (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (archived 2025 author manuscript) (standard reference, not scraped)
Maria Kampanou, Trace Theorems for Sobolev Spaces (master's thesis, National and Kapodistrian University of Athens, July 2018) (standard reference, not scraped)