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A jump boundary datum is outside the trace range for $p \geq 2$

Statement refuted

Assume the Axiom of Choice. The claim that for $p \geq 2$ every boundary datum $g \in L^{p} (\partial Ω)$ on a bounded $C^{1}$ domain $Ω \subset R^{2}$ is the trace of some $u \in W^{1, p} (Ω)$ is false. Let a boundary chart containing the closed straight segment $[0, 1]$ strictly inside its patch be given and let $g = 1_{(0, 1)}$ be the jump function on that segment, extended by zero. Then $g \in L^{p} (\partial Ω)$ for every $p$ , but for every $2 \leq p < \infty$ one has $g \notin W^{1 - 1 / p, p} (\partial Ω)$ : the chart computation gives a divergent Slobodeckij seminorm, and by The sharp trace theorem: boundedness and range in the fractional space no $u \in W^{1, p} (Ω)$ has $T u = g$ . In particular the Dirichlet datum is not arbitrary in $L^{p} (\partial Ω)$ , and the trace range depends on $p$ : for $1 < p < 2$ the same jump function does belong to $W^{1 - 1 / p, p} (\partial Ω)$ because $1 - 1 / p < 1 / p$ .

Facts & Assumptions

Given: The Axiom of Choice; a bounded $C^{1}$ domain $Ω \subset R^{2}$ with a boundary chart containing $[0, 1]$ strictly inside a straight patch; the jump function $g = 1_{(0, 1)}$ on that segment, extended by zero; and $1 < p < \infty$ with $θ = 1 - 1 / p$ .

[F1]

On a straight chart the boundary norm is that of the Euclidean Slobodeckij space on $R$ : $[g]_{θ, p}^{p} = \int_{R} \int_{R} ∣ g (x) - g (y) ∣^{p} ∣ x - y ∣^{- 1 - p θ} d x d y$ , an extended nonnegative integral, and the boundary space is the set of $L^{p}$ classes with finite norm. (The Gagliardo--Slobodeckij space on Euclidean space, The fractional Sobolev space on a compact $C^{1}$ boundary)

[F2]

Assume Countable Choice. For nonnegative measurable functions on a product of sigma-finite spaces the double integral equals the iterated integrals. (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F3]

The trace range of $T : W^{1, p} (Ω) \to L^{p} (\partial Ω)$ is exactly $W^{θ, p} (\partial Ω)$ for $1 < p < \infty$ . (The sharp trace theorem: boundedness and range in the fractional space)

[F4]

$L^{p} (\partial Ω)$ is the quotient of the boundary-measurable functions by the almost-everywhere zero functions, so a bounded function supported in a finite-measure boundary is an $L^{p}$ class. (The space $L^{p} (μ)$ as the quotient by null functions)

Counterexample

1.1F1F2algebragiven

The full seminorm of the line jump. Let $g = 1_{(0, 1)}$ on $R$ and put $q = p θ > 0$ . Symmetry and Tonelli give $[g]_{θ, p}^{p} = 2 \int_{0}^{1} (\int_{- \infty}^{0} (x - y)^{- 1 - q} d y + \int_{1}^{\infty} (y - x)^{- 1 - q} d y) d x = (2 / q) \int_{0}^{1} (x^{- q} + (1 - x)^{- q}) d x = (4 / q) \int_{0}^{1} x^{- q} d x$ . Thus it is finite exactly when $q < 1$ , and infinite when $q \geq 1$ ; at $q = 1$ the logarithmic divergence is explicit. Since $q = p - 1$ , the threshold is exactly $p = 2$ .

2.1F1F3F4step 1.1algebra∎

Transfer to the boundary and conclusion. Because $\partial Ω$ has finite surface measure and the straight chart has bounded density, $g \in L^{p} (\partial Ω)$ is a bounded function on a finite-measure boundary, hence an $L^{p}$ class by [F4]; choose a subordinate cutoff equal to one near the closed segment. In that atlas its representation is exactly the line jump from step 1.1, while all other localised representations are bounded Lipschitz multiples and coordinate transforms of it. The multiplier and atlas estimates of Chart independence of the fractional boundary norm therefore make the norm finite when $p < 2$ and infinite when $p \geq 2$ , independently of the atlas. By [F3] the trace range equals the boundary space, so for $p \geq 2$ there is no $u \in W^{1, p} (Ω)$ with $T u = g$ . For $1 < p < 2$ , step 1.1 gives $p θ < 1$ and $[g]_{θ, p} < + \infty$ , so $g$ is in the trace range in that exponent range: the range genuinely depends on $p$ .

Source notes

Hunter's Section 3.9 (printed p. 73) identifies the trace range for $1 < p < \infty$ with the Besov space $B^{1 - 1 / p, p}$ , which is not all of $L^{p}$ ; Gagliardo's Teorema [1.I] (printed p. 289) states the two-sided condition whose boundary class is a strict subspace of $L^{p}$ , and Schikorra's Section V.2 (printed pp. 97-98) records the derivative loss. The computation above is elementary and separates the cases $p < 2$ and $p \geq 2$ explicitly.

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Sources

John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis, complete 242-page two-quarter notes) (standard reference, not scraped)
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)
Armin Schikorra, Partial Differential Equations (University of Pittsburgh, version 4 December 2019) (standard reference, not scraped)

A jump boundary datum is outside the trace range for p≥2p≥2